Patent Number: 
Section: description

In the various drawings, like parts are indicated by like references. FIG. 1 schematically depicts a lithographic projection apparatus according to a particular embodiment of the invention. The apparatus comprises: a radiation system Ex, IL, for supplying a projection beam PB of radiation (e.g. EUV radiation), which in this particular case also comprises a radiation source LA; a first object table (mask table) MT provided with a mask holder for holding a mask MA (e.g. a reticle), and connected to first positioning means PM for accurately positioning the mask with respect to item PL; a second object table (substrate table) WT provided with a substrate holder for holding a substrate W (e.g. a resist-coated silicon wafer), and connected to second positioning means PW for accurately positioning the substrate with respect to item PL; a projection system (xe2x80x9clensxe2x80x9d) PL (e.g. a mirror group) for imaging an irradiated portion of the mask MA onto a target portion C (e.g. comprising one or more dies) of the substrate W. As here depicted, the apparatus is of a reflective type (i.e. has a reflective mask). However, in general, it may also be of a transmissive type, for example (with a transmissive mask). Alternatively, the apparatus may employ another kind of patterning structure, such as a programmable mirror array of a type as referred to above. The source LA (e.g. a discharge or laser-produced plasma source) produces a beam of radiation. This beam is fed into an illumination system (illuminator) IL, either directly or after having traversed conditioning means, such as a beam expander Ex, for example. The illuminator IL may comprise adjusting means AM for setting the outer and/or inner radial extent (commonly referred to as "sgr"-outer and "sgr"-inner, respectively) of the intensity distribution in the beam. In addition, it will generally comprise various other components, such as an integrator IN and a condenser CO. In this way, the beam PB impinging on the mask MA has a desired uniformity and intensity distribution in its cross-section. It should be noted with regard to FIG. 1 that the source LA may be within the housing of the lithographic projection apparatus (as is often the case when the source LA is a mercury lamp, for example), but that it may also be remote from the lithographic projection apparatus, the radiation beam which it produces being led into the apparatus (e.g. with the aid of suitable directing mirrors); this latter scenario is often the case when the source LA is an excimer laser. The current invention and Claims encompass both of these scenarios. The beam PB subsequently intercepts the mask MA, which is held on a mask table MT. Having been selectively reflected by the mask MA, the beam PB passes through the lens PL, which focuses the beam PB onto a target portion C of the substrate W. With the aid of the second positioning means PW (and interferometric measuring means IF), the substrate table WT can be moved accurately, e.g. so as to position different target portions C in the path of the beam PB. Similarly, the first positioning means PM can be used to accurately position the mask MA with respect to the path of the beam PB, e.g. after mechanical retrieval of the mask MA from a mask library, or during a scan. In general, movement of the object tables MT, will be realized with the aid of a long-stroke module (course positioning) and a short-stroke module (fine positioning), which are not explicitly depicted in FIG. 1. However, in the case of a wafer stepper (as opposed to a step-and-scan apparatus) the mask table MT may just be connected to a short stroke actuator, or may be fixed. The depicted apparatus can be used in two different modes: 1. In step mode, the mask table MT is kept essentially stationary, and an entire mask image is projected at once (i.e. a single xe2x80x9cflashxe2x80x9d) onto a target portion C. The substrate table WT is then shifted in the x and/or y directions so that a different target portion C can be irradiated by the beam PB; 2. In scan mode, essentially the same scenario applies, except that a given target portion C is not exposed in a single xe2x80x9cflashxe2x80x9d. Instead, the mask table MT is movable in a given direction (the so-called xe2x80x9cscan directionxe2x80x9d, e.g. the y direction) with a speed v, so that the projection beam PB is caused to scan over a mask image; concurrently, the substrate table WT is simultaneously moved in the same or opposite direction at a speed V=Mv, in which M is the magnification of the lens PL (typically, M={fraction (1/4)} or {fraction (1/5)}). In this manner, a relatively large target portion C can be exposed, without having to compromise on resolution. Mirror System Classification According to the present invention, a mirror system of n mirrors is classified by reference to the direction of the reflected beam compared to the incident beam at each mirror surface. Having defined the object height to be positive along a Y-axis and a suitable reference plane, e.g. the plane containing the optical axis Z of the projection system and the Y-axis (as shown in FIG. 1), the chief ray CR at a mirror is defined as having a positive angle of incidence xcex1, if the angle between the normal of the surface and the incident ray is counterclockwise (as shown in FIG. 2) and a negative angle of incidence if the angle between the normal and the incident ray is clockwise. Further, one should view this reference plane along a positive X direction, the X, Y, Z directions making up a right-handed orthogonal coordinate system, as shown in FIG. 1. The chief ray is defined as that ray emerging from the object point, which goes through the center of the stop and therefore also through the center of the entrance and exit pupils, i.e. at a height equal to zero from the optical axis. (NB this assignment is arbitrary, the scheme can be implemented with either relative direction of reflection regarded as positive, provided that the assignment is consistent.) By assigning the binary digit xe2x80x9c1xe2x80x9d to a negative angle of incidence and xe2x80x9c0xe2x80x9d to a positive angle of incidence of the chief ray, a mirror system is assigned a binary number defined by the sequence of binary digits assigned to each mirror in the system in sequence along the optical path of the beam from object to source. For convenience, this binary number is expressed in decimal notation. The various classes of the incidence angle classification system are further characterized by indicating the sign of the magnification of the system. Herein, this is indicated by the appropriate sign in parentheses after the class number, e.g. 6(xe2x88x92). The sign is obtained by dividing the magnification M by its absolute value |M|. A system has a positive magnification if the object and image are the same side of the optical axis and a negative magnification if they are opposite sides. The decimal incidence angle classification, C, can thus be expressed as:                     C        =                              ∑                          i              =              1                        n                    ⁢                                                    a                i                            ·                              2                                  (                                      n                    -                    i                                    )                                                      ⁢                          (                              M                                  "LeftBracketingBar"                  M                  "RightBracketingBar"                                            )                                                          (        1        )             where: ai=1 if the angle of incidence of the chief ray at mirror i is negative, ai=0 if the angle of incidence of the chief ray at mirror i is positive, M is the magnification of the projection system, and index i numbers the mirrors of the system in series from object to source. FIG. 2 shows the four possible arrangements of incident chief rays CR and mirrors M. In A the incident chief ray is travelling left to right and has an angle of incidence xcex1 greater than 0, so ai=0. In B the incident chief ray is travelling right to left and has an angle of incidence xcex1 less than 0, so ai=1. In C the incident chief ray is travelling right to left and has an angle of incidence xcex1 greater than 0, so ai=0. In D the incident chief ray is travelling left to right with an angle of incidence xcex1 less than 0, so ai=1. Note that although convex mirrors are shown, the same assignments apply with concave or plane mirrors. While the incidence angle classification C does not wholly define a mirror system, the basic layout of a system is inherent in its incidence angle classification. By reference to whether the reflection at a given mirror is positive or negative, the orientation of that mirror and the position of the succeeding mirror, above or below the beam, can be determined. Thus a given classification number can be used by the designer of a mirror system to set out the system prior to optimization of that system, e.g. using commercially available ray tracing software such as CODE V(TM) by Optical Research Associates, Pasadena, Calif., USA. It should be noted that previous classifications of mirror systems based on whether the curvature, and hence power, of each mirror in the system is positive or negative, do not give any information as to the layout of a mirror system. It will also be noted that the incidence angle classification of a given mirror system can readily be determined from simple inspection of the beam path. Using the above classification system and numerical simulations, the present inventors have determined that only certain classes contain mirror systems usable as the projection system in a lithographic projection system. For four-mirror systems, feasible projection systems exist in classes 2(xe2x88x92), 6(xe2x88x92), 9(+), 9(xe2x88x92) and 10(xe2x88x92). For six-mirror systems, feasible projection systems exist in classes 5(+), 6(xe2x88x92), 9(+), 13(+), 18(xe2x88x92), 21(+), 22(xe2x88x92), 25(+), 26(xe2x88x92), 29(+), 34(xe2x88x92), 37(+), 38(xe2x88x92), 41(+), 42(xe2x88x92), 45(+) and 54(xe2x88x92). For eight-mirror systems, feasible projection systems exist in class 2(+), 5(+), 9(+), 12(+), 13(+), 18(+), 18(xe2x88x92), 19(+), 20(+), 21(+), 22(+), 23(+), 25(+), 26(+), 34(xe2x88x92), 36(+), 37(+), 38(xe2x88x92), 45(+), 46(+), 49(+), 52(+), 53(+), 54(+), 54(xe2x88x92), 55(xe2x88x92), 58(xe2x88x92), 68(+), 69(+), 73(+), 74(+), 77(+), 82(+), 82(xe2x88x92), 85(+), 88(+), 89(+), 90(xe2x88x92), 92(+), 93(+), 97(+), 100(xe2x88x92), 101(+), 102(xe2x88x92), 104(+), 105(+), 106(+), 106(xe2x88x92), 107(+), 108(+), 109(+), 109(xe2x88x92), 110(+), 110(xe2x88x92), 111(+), 113(+), 116(+), 117(+), 118(+), 118(xe2x88x92), 120(+), 121(+), 122(xe2x88x92), 123(xe2x88x92), 132(+), 133(+), 134(xe2x88x92), 137(+), 138(+), 141(+), 145(+), 145(xe2x88x92), 146(+), 146(xe2x88x92), 147(+), 148(+), 148(xe2x88x92), 149(+), 150(+), 150(xe2x88x92), 151(+), 151(xe2x88x92), 152(xe2x88x92), 153(+), 154(+), 154(xe2x88x92), 155(+), 155(xe2x88x92), 156(+), 157(+), 159(+), 161(+), 162(xe2x88x92), 163(xe2x88x92), 164(+), 165(+), 166(+), 166(xe2x88x92), 167(+), 168(+), 169(+), 170(+), 170(xe2x88x92), 171(+), 172(+), 173(+), 174(+), 175(+), 176(+), 177(+), 178(xe2x88x92), 179(+), 180(+), 180(xe2x88x92), 181(+), 181(xe2x88x92), 182(+), 182(xe2x88x92), 183(+), 183(xe2x88x92), 184(+), 185(+), 185(xe2x88x92), 186(xe2x88x92), 187(+), 187(xe2x88x92), 188(xe2x88x92), 189(+), 196(+), 197(+), 201(+), 203(+), 205(+), 209(+), 214(xe2x88x92), 216(+), 217(+), 218(+), 218(xe2x88x92), 225(+), 228(+), 229(+), 230(+), 232(+), 233(+), 235(+), 236(+), 237(+), 238(xe2x88x92), 243(+), 246(+), 247(+), 248(+), 250(xe2x88x92). Design Methodology As mentioned above, a given class defines an outline layout for a mirror system for which a functional projection system can be designed. A methodology according to the present invention for such a design process is described below. In the design process according to the invention, the mirrors in a system are defined by xe2x80x9cthicknessesxe2x80x9d and curvatures, defined as shown in FIG. 3. (NB the term xe2x80x9cthicknessxe2x80x9d is used by analogy to refractive systems which are conventionally defined in terms of surface curvatures, thicknesses between surfaces and the refractive indices of the media between surfaces.) Thus, thickness d1 is the distance between the object, the mask MA in the present case of a projection system in a lithographic apparatus, and the intersection of the (imaging extended) first mirror M1 with the optical axis OA. The distance between the intersections of the (imaging extended) first mirror M1 and the (imaging extended) second mirror M2 with the optical axis OA is d1. Note that since the second mirror is situated between the first mirror M1 and the object (mask MA), thickness d1 is negative. In general, thickness di is the distance between the intersections of mirror Mi and mirror Mi+1 with the optical axis OA. For an n-mirror system, the thickness dn is the distance between the last mirror Mn and the image plane, where the substrate W is positioned in a lithographic projection apparatus. In specific embodiments described below, an additional thickness dn+1 is given, this represents the distance between the position of the image calculated using a first order approximation and using a real ray tracing algorithm. In a first step the design method identifies conceivable systems under a number of constraints by using a paraxial approach described below. Those systems should not present obscuration, as is also described further below. The paraxial approach and the constraints yield a limited number of variables that are sampled to identify solutions. In a further step those solutions are checked using a real ray tracing method, referred to above, in which the paraxial assumptions are not present and in which also multilayer coatings of the reflectors may be modelled. Paraxial Approach The present inventors have developed an approach to designing mirror systems which starts with a paraxial approximation of a mirror system using matrix formalism. In a paraxial approximation the sine of an angle is approximated as the angle, i.e. sin xcex1=xcex1, and are the mirrors considered as being flat, while the actual curvature of a mirror is considered only to affect the angle of an incident ray, not its point of intersection with the supposedly xe2x80x98flatxe2x80x99 surface. In a matrix formalism, such as described in xe2x80x9cIntroduction to Opticsxe2x80x9d by Frank and Leno Pedrotti, Prentice Hall 1993; ISBN: 0135015456, incorporated herein by reference, the description of an optical system consist of an accumulation of translation and reflection (and/or refraction in a catadioptric or refractive system) matrices Mtrans,Mrefl which are defined as follows:                               M          trans                =                  [                                                    1                                                              d                  i                                                                                    0                                            1                                              ]                                    (        2        )                                          M          refl                =                  [                                                    1                                            0                                                                                                          -                    2                                    ·                                      c                    i                                                                                                -                  1                                                              ]                                    (        3        )             where di is the distance to the next surface and ci the curvature of the surface, which is positive if the center of the sphere is on the right side of the surface. The path of a ray is given by a vector made of a height (distance from the optical axis) and an angle: [height, angle]. The multiplication of the vector with one or more matrices gives the ray after the corresponding translations or reflections. The system matrix is the multiplication of all the matrices in the system. The first matrix is the reflection matrix of the first surface, the penultimate matrix is the translation matrix of the thickness preceding the last reflective surface and the last matrix is the reflection matrix of the last reflective surface. The effective focal length, the back focal length, the front focal length and the paraxial image distance can be derived from the system matrix as follows. If the system matrix is defined as:                               M          system                =                  [                                                    a                                            b                                                                    c                                            d                                              ]                                    (        4        )             then the effective focal length is given by:                     efl        =                  -                      1            c                                              (        5        )             the back focal length is given by:                     bfl        =                  -                      a            c                                              (        6        )             the front focal length is given by:                     ffl        =                  d          c                                    (        7        )             and the paraxial image distance, i.e. the distance between the last reflective surface and the image plane, is given by:                               d          n                =                                            a              ·                              d                0                                      +            b                                              c              ·                              d                0                                      +            d                                              (        8        )             where       d    0    =            ad      -      cb      -              magn        ·        d                    magn      ·      c       and magn is the magnification of the system. The system matrix for the first part of the system, from the object plane to the stop (pupil) can be represented as:                               M                      1            ⁢            st                          =                  [                                                                      a                                      1                    ⁢                    st                                                                                                b                                      1                    ⁢                    st                                                                                                                        c                                      1                    ⁢                    st                                                                                                d                                      1                    ⁢                    st                                                                                ]                                    (        9        )             so that the distance, Lenpup, to the entrance pupil is given by:                                           [                                                                                a                                          1                      ⁢                      st                                                                                                            b                                          1                      ⁢                      st                                                                                                                                        c                                          1                      ⁢                      st                                                                                                            d                                          1                      ⁢                      st                                                                                            ]                    ·                      [                                                            1                                                                      L                    enpup                                                                                                0                                                  1                                                      ]                    ·                      [                                                            0                                                                                                  A                    enpup                                                                        ]                          =                  "AutoLeftMatch"                      xe2x80x83                    ⁢                      [                                          "AutoLeftMatch"                                                                                                                              (                                                                                                                    a                                                                  1                                  ⁢                                  st                                                                                            ·                                                              L                                enpup                                                                                      +                                                          b                                                              1                                ⁢                                st                                                                                                              )                                                ·                                                  A                          enpup                                                                                                                                                                                                  (                                                                                                                    c                                                                  1                                  ⁢                                  st                                                                                            ·                                                              L                                enpup                                                                                      +                                                          d                                                              1                                ⁢                                st                                                                                                              )                                                ·                                                  A                          enpup                                                                                                                    ]                            =                              [                                                                            0                                                                                                                          A                        stop                                                                                            ]                                                                        (        10        )             The second part of the system, from stop to image surface can be represented as:                               M                      2            ⁢            nd                          =                  [                                                                      a                                      2                    ⁢                    nd                                                                                                b                                      2                    ⁢                    nd                                                                                                                        c                                      2                    ⁢                    nd                                                                                                d                                      2                    ⁢                    nd                                                                                ]                                    (        11        )             so that the distance, Lexpup, to the exit pupil is given by:                                           [                                                            1                                                                      L                                          exp                      ⁢                                              xe2x80x83                                            ⁢                      up                                                                                                                    0                                                  1                                                      ]                    ·                      [                                                                                a                                          2                      ⁢                      nd                                                                                                            b                                          2                      ⁢                      nd                                                                                                                                        c                                          2                      ⁢                      nd                                                                                                            d                                          2                      ⁢                      nd                                                                                            ]                    ·                      [                                                            0                                                                                                  A                    stop                                                                        ]                          =                  "AutoLeftMatch"                                    [                                                                                                                  (                                                                              b                                                          2                              ⁢                              nd                                                                                +                                                                                    L                                                              exp                                ⁢                                                                  xe2x80x83                                                                ⁢                                up                                                                                      ·                                                          d                                                              2                                ⁢                                nd                                                                                                                                    )                                            ·                                              A                        stop                                                                                                                                                                                D                                                  2                          ⁢                          nd                                                                    ·                                              A                        stop                                                                                                        ]                        =                          [                                                                    0                                                                                                              A                                              exp                        ⁢                                                  xe2x80x83                                                ⁢                        up                                                                                                        ]                                                          (        12        )             The distances to the entrance and exit pupils, if xcex9enpupxe2x89xa00, are then given by:                               L          enpup                =                                            -                                                b                                      1                    ⁢                    st                                                                    a                                      1                    ⁢                    st                                                                        ⁢                          xe2x80x83                        ⁢            and            ⁢                                          xe2x80x83                            ⁢                              xe2x80x83                                      ⁢                          L                              exp                ⁢                                  xe2x80x83                                ⁢                up                                              =                      -                                          b                                  2                  ⁢                  nd                                                            d                                  2                  ⁢                  nd                                                                                        (        13        )             Constraints Given the above, various constraints that must be applied to the system can be used to determine equations for the curvatures and thicknesses of certain surfaces of the system as functions of the constraints and other curvatures and thicknesses. Some example constraints G1 to G4 are shown in FIG. 5. A first constraint G1 is minimum deviation from telecentricity on the object side that still enables obscuration-free illumination of the object, which may determine the curvature of the first surface or the thickness between mirrors 1 and 2. Another constraint G3, is perfect telecentricity on the image side, which may determine the curvature of the last surface or the thickness between the final and penultimate mirrors. This telecentricity requirement is equivalent to the requirement that the exit pupil is at infinity. The requirement that the object and the image are conjugated and have a prescribed value of the transverse magnification fixes the values of the object (G2) and image (G4) distances. The object distance G2, the first thickness, can be solved as a function of the desired magnification of the system: the paraxial image distance is inserted in the thickness of the surface immediately preceding the image plane and the object distance is modified to satisfy:                     M        =                              Image            ⁢                          xe2x80x83                        ⁢            Height                                Object            ⁢                          xe2x80x83                        ⁢            Height                                              (        14        )             In current lithography apparatus, M is usually set as xc2x10.20 or xc2x10.25, i.e. reduction by a factor of 5 or 4 respectively. A minimum deviation from telecentricity at the object side is an important requirement in lithography. The reflective object (mask MA) is illuminated with a beam coming from the radiation system. The chief ray angle at the object must be such that the incident illuminating beam does not interfere with the beam reflected from the object and going into the projection system. The angle of the chief ray together with the numerical aperture on the object side should be almost zero and the angles of all rays must be smaller or larger than zero, to fulfill these two requirements. For telecentricity on the image side, the angle of the chief ray relative to the optical axis has to be zero. The size of the last mirror increases quickly as a function of the distance between the image and the last mirror, due to the relatively large numerical aperture. A system with zero or an even number of intermediate images has a negative magnification. To have an overall system with a positive magnification the number of intermediate images has to be odd. The working distance at the object side is the minimum distance between the object plane and the surface closest to the object, most of the time the second mirror. On the image side the working distance is the minimum distance between the image plane and the plane closest to the image, most often the penultimate mirror. The working distances provide room for mirror supports and for mechanical movements of the object and the image and must not be too small. An example of applying the above constraints in a six-mirror system is described below. This may be carried out in practice using software such as Maple(TM) produced by Waterloo Maple Inc. 57 Erb Street W. Waterloo, Ontario Canada N2L 6C2. First is a derivation of the formulas used for a six-mirror system, but the formulas are also valid for other numbers of mirrors, using the paraxial approach. In the matrix notation a ray is defined by the vector: [height, angle in radians]. After a distance di the ray [y,a] will be:                     [                                                            y                +                                                      d                    i                                    ⁢                  a                                                                                        a                                      ]                            (        15        )             using the matrix given in equation (2). After a mirror with curvature ci the ray [y,a] will have the same height but a different angle:                     [                                            v                                                                                                                -                                          2                                              c                        i                                                                              ⁢                  v                                -                a                                                    ]                            (        16        )             using the matrix given in equation (3). To derive formulas used later on, firstly the distance between mirror 5 and 6 is solved by requiring telecentricity in the image of the ray going through the optical axis in the stop surface. The following matrix A is from the stop surface to after the 5th mirror, as we don""t now where we will locate the stop surface we take an unknown 2 by 2 matrix:                     A        :=                  [                                                    a                                            b                                                                    c                                            d                                              ]                                    (        17        )             From the 5th mirror we travel a distance la to the 6th mirror, la is the variable to solve now.                     L6        :=                  [                                                    1                                            la                                                                    0                                            1                                              ]                                    (        18        )             Matrix MC is of the 6th mirror surface:                     MC        :=                  [                                                    1                                            0                                                                                                          -                    2                                    ⁢                  c6                                                                              -                  1                                                              ]                                    (        19        )             The ray going through the center of the stop with an arbitrary angle ap is:                     Y        :=                  [                                                    0                                                                    ap                                              ]                                    (        20        )             That ray after the 6th mirror will be:                     Y_image        :=                  [                                                                                          (                                          b                      +                                              la                        ⁢                                                  xe2x80x83                                                ⁢                        d                                                              )                                    ⁢                  ap                                                                                                                          (                                                                                            -                          2                                                ⁢                        c6                                            +                                                                        (                                                                                                                    -                                2                                                            ⁢                              c6                              ⁢                                                              xe2x80x83                                                            ⁢                              la                                                        -                            1                                                    )                                                ⁢                        d                                                              )                                    ⁢                  ap                                                              ]                                    (        21        )             in which the angle is equal to zero since telecentricity is required and the solution opl for the distance la between mirror five and six is now                     opl        :=                              -                          1              2                                ⁢                                                    2                ⁢                c6                ⁢                                  xe2x80x83                                ⁢                b                            +              d                                      c6              ⁢                              xe2x80x83                            ⁢              d                                                          (        22        )             The matrix from the stop surface to after the 6th mirror is now:                     B        :=                  [                                                                                          1                    2                                    ⁢                                                                                    2                        ⁢                        c6                        ⁢                                                  xe2x80x83                                                ⁢                        ad                                            -                                              2                        ⁢                        c6                        ⁢                                                  xe2x80x83                                                ⁢                        bc                                            -                      cd                                                              c6                      ⁢                                              xe2x80x83                                            ⁢                      d                                                                                                                                        -                                          1                      2                                                        ⁢                                      d                    c6                                                                                                                                            -                    2                                    ⁢                                      xe2x80x83                                    ⁢                                                            c6                      ⁡                                              (                                                  da                          -                          bc                                                )                                                              d                                                                              0                                              ]                                    (        23        )             The next solve is the distance d between the object and the first mirror and the solve ya for the angle of the chief ray (going through the center of the stop) between the object and the first mirror. The ray Ya in the object point yob with the desired angle ya is given by the vector:                     Ya        :=                  [                                                    yob                                                                    ya                                              ]                                    (        24        )             and the distance l between the object and the first mirror surface by the matrix:                     L        :=                  [                                                    1                                            l                                                                    0                                            1                                              ]                                    (        25        )             The first mirror surface is given by:                     MC        :=                  [                                                    1                                            0                                                                                                          -                    2                                    ⁢                  cl                                                                              -                  1                                                              ]                                    (        26        )             The distance m between the first mirror and the second mirror is given by:                     M        :=                  [                                                    1                                            m                                                                    0                                            1                                              ]                                    (        27        )             The unknown matrix from the second mirror surface to the stop position is defined as                     A        :=                  [                                                    e                                            f                                                                    g                                            h                                              ]                                    (        28        )             The chief ray in the stop surface is now:                     Y_stop        :=                  xe2x80x83                ⁢                  
                ⁢                  xe2x80x83                ⁢                  [                      xe2x80x83                    ⁢                                                                                                                (                                              e                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          e                                ⁢                                                                  xe2x80x83                                                                ⁢                                m                                                            +                              f                                                        )                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c1                                                                    )                                        ⁢                    yob                                    +                                                            (                                                                                                    (                                                          e                              -                                                              2                                ⁢                                                                  (                                                                                                            e                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      m                                                                        +                                    f                                                                    )                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c1                                                                                      )                                                    ⁢                          l                                                -                        em                        -                        f                                            )                                        ⁢                    ya                                                                                                                                                                  (                                              g                        -                                                  2                          ⁢                                                      (                                                                                          g                                ⁢                                                                  xe2x80x83                                                                ⁢                                m                                                            +                              h                                                        )                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c1                                                                    )                                        ⁢                    yob                                    +                                                            (                                                                                                    (                                                          g                              -                                                              2                                ⁢                                                                  (                                                                                                            g                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      m                                                                        +                                    h                                                                    )                                                                ⁢                                c1                                                                                      )                                                    ⁢                          l                                                -                                                  g                          ⁢                                                      xe2x80x83                                                    ⁢                          m                                                -                        h                                            )                                        ⁢                    ya                                                                                ⁢                      xe2x80x83                    ]                                    (        29        )             and in the image the chief ray is:                                                         Y_image              :=                              xe2x80x83                            ⁢                              [                                  (                                                                                    1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                                                    (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                a                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                            -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                b                                ⁢                                                                  xe2x80x83                                                                ⁢                                c                                                            -                                                              c                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                                                      )                                                    ⁢                          e                                                                          c6                          ⁢                                                      xe2x80x83                                                    ⁢                          d                                                                                      -                                                                  1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                        d                          ⁢                                                      xe2x80x83                                                    ⁢                          g                                                c6                                                              -                                                                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  (                                                                                    (                                                                                                            1                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                                                                                            (                                                                                                            2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c6                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      a                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d                                                                        -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c6                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      b                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c                                                                        -                                                                          c                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d                                                                                                        )                                                                ⁢                                e                                                                                            c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                                                                              -                                                                                    1                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                                                          d                                ⁢                                                                  xe2x80x83                                                                ⁢                                g                                                            c6                                                                                                      )                                            ⁢                      m                                        +                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                                                    (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                a                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                            -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                b                                ⁢                                                                  xe2x80x83                                                                ⁢                                c                                                            -                                                              c                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                                                      )                                                    ⁢                          f                                                                          c6                          ⁢                                                      xe2x80x83                                                    ⁢                          d                                                                                      -                                                                  1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                        d                          ⁢                                                      xe2x80x83                                                    ⁢                          h                                                c6                                            )                      c1                                                        )                                ⁢                yob                            +                                                                                          xe2x80x83                            ⁢                              (                                  (                                                                                    1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                                                    (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                a                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                            -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                b                                ⁢                                                                  xe2x80x83                                                                ⁢                                c                                                            -                                                              c                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                                                      )                                                    ⁢                          e                                                                          c6                          ⁢                                                      xe2x80x83                                                    ⁢                          d                                                                                      -                                                                  1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                        d                          ⁢                                                      xe2x80x83                                                    ⁢                          g                                                c6                                                              -                                                                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  (                                                                                    (                                                                                                            1                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                                                                                            (                                                                                                            2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c6                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      a                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d                                                                        -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c6                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      b                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c                                                                        -                                                                          c                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d                                                                                                        )                                                                ⁢                                e                                                                                            c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                                                                              -                                                                                    1                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                                                          d                                ⁢                                                                  xe2x80x83                                                                ⁢                                g                                                            c6                                                                                                      )                                            ⁢                      m                                        +                                                                                                                                          xe2x80x83                            ⁢                                                                    1                    2                                    ⁢                                      xe2x80x83                                    ⁢                                                                                    (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c6                            ⁢                                                          xe2x80x83                                                        ⁢                            a                            ⁢                                                          xe2x80x83                                                        ⁢                            d                                                    -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c6                            ⁢                                                          xe2x80x83                                                        ⁢                            b                            ⁢                                                          xe2x80x83                                                        ⁢                            c                                                    -                                                      c                            ⁢                                                          xe2x80x83                                                        ⁢                            d                                                                          )                                            ⁢                      f                                                              c6                      ⁢                                              xe2x80x83                                            ⁢                      d                                                                      -                                                      1                    2                                    ⁢                                      xe2x80x83                                    ⁢                                                            d                      ⁢                                              xe2x80x83                                            ⁢                      h                                        c6                                    )                  c1                  )                  l                                -                                                                                                        xe2x80x83                            ⁢                                                                    (                                                                                            1                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                                                                            (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c6                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  a                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d                                                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c6                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  b                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c                                                                -                                                                  c                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d                                                                                            )                                                        ⁢                            e                                                                                c6                            ⁢                                                          xe2x80x83                                                        ⁢                            d                                                                                              -                                                                        1                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                                              d                            ⁢                                                          xe2x80x83                                                        ⁢                            g                                                    c6                                                                                      )                                    ⁢                  m                                -                                                                                                                                              xe2x80x83                                    ⁢                                                                                    1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                                                    (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                a                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                            -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                b                                ⁢                                                                  xe2x80x83                                                                ⁢                                c                                                            -                                                              c                                ⁢                                                                  xe2x80x83                                                                ⁢                                d                                                                                      )                                                    ⁢                          f                                                                          c6                          ⁢                                                      xe2x80x83                                                    ⁢                          d                                                                                      +                                                                  1                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                        d                          ⁢                                                      xe2x80x83                                                    ⁢                          h                                                c6                                                                              )                                ⁢                yz                            ]                                                                                          xe2x80x83                            ⁢                              [                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                                                                        c6                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          d                                ⁢                                                                  xe2x80x83                                                                ⁢                                a                                                            -                                                              b                                ⁢                                                                  xe2x80x83                                                                ⁢                                c                                                                                      )                                                    ⁢                                                      xe2x80x83                                                    ⁢                          e                                                d                                                              -                                          2                      ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                                                                                            c6                                  ⁡                                                                      (                                                                                                                  d                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        a                                                                            -                                                                              b                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c                                                                                                              )                                                                                                  ⁢                                e                                ⁢                                                                  xe2x80x83                                                                ⁢                                m                                                            d                                                                                -                                                                                                                                                                                                            xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                                                            c6                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              d                            ⁢                                                          xe2x80x83                                                        ⁢                            a                                                    -                                                      b                            ⁢                                                          xe2x80x83                                                        ⁢                            c                                                                          )                                            ⁢                      f                                        d                                    )                  c1                  )                  yob                                +                                  (                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                                              c6                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                d                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  a                                                                -                                                                  b                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c                                                                                            )                                                        ⁢                                                          xe2x80x83                                                        ⁢                            e                                                    d                                                                    -                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                            2                      ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                                                          c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                            d                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      a                                                                        -                                                                          b                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c                                                                                                        )                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                e                                ⁢                                                                  xe2x80x83                                                                ⁢                                m                                                            d                                                                                -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                                                                                          c6                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                            d                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      a                                                                        -                                                                          b                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c                                                                                                        )                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                f                                                            d                                                                                                      )                                            ⁢                      c1                                        )                                                  ⁢                l                            +                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                                                                        c6                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          d                                ⁢                                                                  xe2x80x83                                                                ⁢                                a                                                            -                                                              b                                ⁢                                                                  xe2x80x83                                                                ⁢                                c                                                                                      )                                                    ⁢                                                      xe2x80x83                                                    ⁢                          e                          ⁢                                                      xe2x80x83                                                    ⁢                          m                                                d                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                                                                        c6                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          d                                ⁢                                                                  xe2x80x83                                                                ⁢                                a                                                            -                                                              b                                ⁢                                                                  xe2x80x83                                                                ⁢                                c                                                                                      )                                                    ⁢                                                      xe2x80x83                                                    ⁢                          f                                                d                                                                              )                                ⁢                yz                            ]                                                          (        30        )             The height of the chief ray in the image is by definition magn*height in the object surface (yob), we solve l from equation (30) to impose this reduction to the system giving:                                                                         Y_image                ⁢                _l                            :=                              xe2x80x83                            ⁢                              -                                  (                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      a                      ⁢                                              xe2x80x83                                            ⁢                      d                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      b                      ⁢                                              xe2x80x83                                            ⁢                      c                                        -                                          xe2x80x83                                        ⁢                                          yob                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c                      ⁢                                              xe2x80x83                                            ⁢                      d                                        -                                                                                                                                          xe2x80x83                            ⁢                                                yob                  ⁢                                      xe2x80x83                                    ⁢                                      d                    2                                    ⁢                                      xe2x80x83                                    ⁢                  g                                -                                  xe2x80x83                                ⁢                                  4                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                                +                                                                                                        xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                                      d                    2                                    ⁢                                      xe2x80x83                                    ⁢                  g                                -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                                      d                    2                                    ⁢                                      xe2x80x83                                    ⁢                  h                                -                                                                                                        xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                                +                                  ya                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                                                                                        xe2x80x83                            ⁢                                                ya                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                                      d                    2                                    ⁢                                      xe2x80x83                                    ⁢                  g                                -                                  2                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                                +                                  ya                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                                                                                                          xe2x80x83                                ⁢                                                                            ya                      ⁢                                              xe2x80x83                                            ⁢                                              d                        2                                            ⁢                                              xe2x80x83                                            ⁢                      h                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      d                                                        )                                            /                                                                                          xe2x80x83                            ⁢                              (                                  (                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      a                      ⁢                                              xe2x80x83                                            ⁢                      d                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      b                      ⁢                                              xe2x80x83                                            ⁢                      c                                        -                                          e                      ⁢                                              xe2x80x83                                            ⁢                      c                      ⁢                                              xe2x80x83                                            ⁢                      d                                        -                                                                  d                        2                                            ⁢                      g                                        -                                          4                      ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      m                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      a                      ⁢                                              xe2x80x83                                            ⁢                      d                                        +                                                                                                                                          xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  m                  ⁢                                      xe2x80x83                                    ⁢                                      d                    2                                    ⁢                                      xe2x80x83                                    ⁢                  g                                -                                                                                                                                              xe2x80x83                                    ⁢                                                                                    4                        ⁢                                                  xe2x80x83                                                ⁢                        c1                        ⁢                                                  xe2x80x83                                                ⁢                        f                        ⁢                                                  xe2x80x83                                                ⁢                        c6                        ⁢                                                  xe2x80x83                                                ⁢                        a                        ⁢                                                  xe2x80x83                                                ⁢                        d                                            +                                              4                        ⁢                                                  xe2x80x83                                                ⁢                        c1                        ⁢                                                  xe2x80x83                                                ⁢                        f                        ⁢                                                  xe2x80x83                                                ⁢                        c6                        ⁢                                                  xe2x80x83                                                ⁢                        b                        ⁢                                                  xe2x80x83                                                ⁢                        c                                            +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c1                        ⁢                                                  xe2x80x83                                                ⁢                        f                        ⁢                                                  xe2x80x83                                                ⁢                        c                        ⁢                                                  xe2x80x83                                                ⁢                        d                                            +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c1                        ⁢                                                  xe2x80x83                                                ⁢                                                  d                          2                                                ⁢                        h                                                              )                                                  ⁢                ya                            )                                                          (        31        )             and we force the height of the chief ray to be zero in the stop surface in equation (29), as it should be by definition to solve the distance m                               Y_stop          ⁢          _m                :=                  -                      xe2x80x83                    ⁢                                                                      -                  yob                                ⁢                                  xe2x80x83                                ⁢                e                            +                              2                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                f                ⁢                                  xe2x80x83                                ⁢                c1                            -                              ya                ⁢                                  xe2x80x83                                ⁢                l                ⁢                                  xe2x80x83                                ⁢                e                            +                              2                ⁢                ya                ⁢                                  xe2x80x83                                ⁢                l                ⁢                                  xe2x80x83                                ⁢                f                ⁢                                  xe2x80x83                                ⁢                c1                            +                              ya                ⁢                                  xe2x80x83                                ⁢                f                                                    e              ⁢                              xe2x80x83                            ⁢                              (                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    yob                    ⁢                                          xe2x80x83                                        ⁢                    c1                                    +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    ya                    ⁢                                          xe2x80x83                                        ⁢                    c1                    ⁢                                          xe2x80x83                                        ⁢                    l                                    +                  ya                                )                                                                        (        32        )             The solution for the distance m between the first and the second mirror now becomes:                               Y_stop          ⁢          _m                :=                              1                          4              ⁢                              xe2x80x83                                              ⁢                      xe2x80x83                    ⁢                                                                                                                                        -                        4                                            ⁢                                              xe2x80x83                                            ⁢                      f                      ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                                        -                                                                                                                                          e                      ⁢                                              xe2x80x83                                            ⁢                      ya                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      h                                        +                                          ya                      ⁢                                              xe2x80x83                                            ⁢                      f                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      g                                                                                                                          e                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            ⁢                              xe2x80x83                                                                        (        33        )             The solution for the distance m between the object and the first mirror now is:                               Y_image          ⁢          _l                :=                              -                          1              2                                ⁢                                                                                                                ya                      ⁢                                              xe2x80x83                                            ⁢                      f                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      g                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      f                      ⁢                                              xe2x80x83                                            ⁢                      g                                        -                                          e                      ⁢                                              xe2x80x83                                            ⁢                      ya                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      h                                        -                                                                                                                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      h                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                                                                                                          ya              ⁢                              xe2x80x83                            ⁢                              (                                                      f                    ⁢                                          xe2x80x83                                        ⁢                    g                                    -                                      e                    ⁢                                          xe2x80x83                                        ⁢                    h                                                  )                            ⁢                              xe2x80x83                            ⁢              c1              ⁢                              xe2x80x83                            ⁢              d                                                          (        34        )             We substitute the just derived expressions in the matrices L and M of equations (25) and (27).                     M        :=                  [                                                    1                                                                                  1                                          4                      ⁢                                              xe2x80x83                                                                              ⁢                                      xe2x80x83                                    ⁢                                                                                                                                                                                                        -                                4                                                            ⁢                                                              xe2x80x83                                                            ⁢                              f                              ⁢                                                              xe2x80x83                                                            ⁢                              c1                              ⁢                                                              xe2x80x83                                                            ⁢                              magn                              ⁢                                                              xe2x80x83                                                            ⁢                              yob                              ⁢                                                              xe2x80x83                                                            ⁢                              c6                                                        +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              e                              ⁢                                                              xe2x80x83                                                            ⁢                              magn                              ⁢                                                              xe2x80x83                                                            ⁢                              yob                              ⁢                                                              xe2x80x83                                                            ⁢                              c6                                                        -                                                                                                                                                                                                          e                              ⁢                                                              xe2x80x83                                                            ⁢                              ya                              ⁢                                                              xe2x80x83                                                            ⁢                              d                              ⁢                                                              xe2x80x83                                                            ⁢                              h                                                        +                                                          ya                              ⁢                                                              xe2x80x83                                                            ⁢                              f                              ⁢                                                              xe2x80x83                                                            ⁢                              d                              ⁢                                                              xe2x80x83                                                            ⁢                              g                                                                                                                                                                                          e                        ⁢                                                  xe2x80x83                                                ⁢                        c1                        ⁢                                                  xe2x80x83                                                ⁢                        magn                        ⁢                                                  xe2x80x83                                                ⁢                        yob                        ⁢                                                  xe2x80x83                                                ⁢                        c6                                            ⁢                                              xe2x80x83                                                                                                                                                0                                            1                                              ]                                    (        35        )                                L        :=                  [                                                    1                                                                                  -                                          1                      2                                                        ⁢                                                                                                                                                                        ya                              ⁢                                                              xe2x80x83                                                            ⁢                              f                              ⁢                                                              xe2x80x83                                                            ⁢                              d                              ⁢                                                              xe2x80x83                                                            ⁢                              g                                                        +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d                              ⁢                                                              xe2x80x83                                                            ⁢                              yob                              ⁢                                                              xe2x80x83                                                            ⁢                              c1                              ⁢                                                              xe2x80x83                                                            ⁢                              f                              ⁢                                                              xe2x80x83                                                            ⁢                              g                                                        -                                                          e                              ⁢                                                              xe2x80x83                                                            ⁢                              ya                              ⁢                                                              xe2x80x83                                                            ⁢                              d                              ⁢                                                              xe2x80x83                                                            ⁢                              h                                                        -                                                                                                                                                                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d                              ⁢                                                              xe2x80x83                                                            ⁢                              e                              ⁢                                                              xe2x80x83                                                            ⁢                              yob                              ⁢                                                              xe2x80x83                                                            ⁢                              c1                              ⁢                                                              xe2x80x83                                                            ⁢                              h                                                        +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              e                              ⁢                                                              xe2x80x83                                                            ⁢                              magn                              ⁢                                                              xe2x80x83                                                            ⁢                              yob                              ⁢                                                              xe2x80x83                                                            ⁢                              c6                                                                                                                                                                  ya                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              f                            ⁢                                                          xe2x80x83                                                        ⁢                            g                                                    -                                                      e                            ⁢                                                          xe2x80x83                                                        ⁢                            h                                                                          )                                            ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      d                                                                                                                          0                                            1                                              ]                                    (        36        )             And as a check we calculate the chief ray after the 6th surface with the new expressions. We see that the angle is always zero and that the height is the object height multiplied with magnification.                     Y_image        :=                  [                                                                      magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                                                                                    0                                              ]                                    (        37        )             The final solve is the distance n between the last mirror surface and the image surface. In the image surface all rays from the same object point come together in a point with a height=magnification*object height. First we define a ray Yb from the object point yob and an arbitrary angle yb.                     Yb        :=                  [                                                    yob                                                                    yb                                              ]                                    (        38        )                                N        :=                  [                                                    1                                            n                                                                    0                                            1                                              ]                                    (        39        )             In the image this ray Yb will become:                                                         Y_image              :=                              xe2x80x83                            ⁢                              [                                                      -                                          1                      4                                                        ⁢                                      (                                                                                            -                          f                                                ⁢                                                  xe2x80x83                                                ⁢                                                  g                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                  d                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                  ya                          2                                                                    +                                              g                        ⁢                                                  xe2x80x83                                                ⁢                                                  d                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        e                        ⁢                                                  xe2x80x83                                                ⁢                                                  ya                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        h                                            -                                              g                        ⁢                                                  xe2x80x83                                                ⁢                        yb                        ⁢                                                  xe2x80x83                                                ⁢                                                  d                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        e                        ⁢                                                  xe2x80x83                                                ⁢                        ya                        ⁢                                                  xe2x80x83                                                ⁢                        h                                            +                                                                                                                                                              xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  h                                -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                                                            magn                      ⁢                                              xe2x80x83                                                              2                                    ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  e                                -                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  h                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  h                                +                                                                                                        xe2x80x83                            ⁢                                                                    e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                    ⁢                                      xe2x80x83                                    ⁢                  h                                -                                  4                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                    ⁢                                      xe2x80x83                                    ⁢                  h                                +                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                    ⁢                                      xe2x80x83                                    ⁢                  h                                +                                  xe2x80x83                                ⁢                                  2                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                    ⁢                                      xe2x80x83                                    ⁢                  h                                -                                                                                                        xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                    ⁢                                      xe2x80x83                                    ⁢                  h                                -                                  2                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  h                                -                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  h                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  h                                ⁢                                  xe2x80x83                                -                                                                                                        xe2x80x83                            ⁢                                                yb                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  h                                +                                  f                  ⁢                                      xe2x80x83                                    ⁢                                      g                    2                                    ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                                      d                    2                                    ⁢                                      xe2x80x83                                    ⁢                  ya                                +                                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  ya                                -                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                                  +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  ya                                -                                                                                                        xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                                  -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                                +                                                                                                        xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                                  +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                                +                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                                  -                                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                                      ya                    2                                                  +                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  a                  ⁢                                      xe2x80x83                                    ⁢                  ya                                -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  yb                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  n                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  ya                                -                                                                                                                                              xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      f                      ⁢                                              xe2x80x83                                            ⁢                      g                      ⁢                                              xe2x80x83                                            ⁢                      yb                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      a                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      ya                                        )                                                  /                                  (                                      ya                    ⁢                                          xe2x80x83                                        ⁢                    magn                    ⁢                                          xe2x80x83                                        ⁢                    e                    ⁢                                          xe2x80x83                                        ⁢                                          c6                      2                                                        )                                            ]                                                                                          xe2x80x83                            ⁢                              [                                  -                                      xe2x80x83                                    ⁢                                                                                    (                                                                              -                            ya                                                    +                          yb                                                )                                            ⁢                                              (                                                                              f                            ⁢                                                          xe2x80x83                                                        ⁢                            g                                                    -                                                      e                            ⁢                                                          xe2x80x83                                                        ⁢                            h                                                                          )                                            ⁢                                              (                                                                              d                            ⁢                                                          xe2x80x83                                                        ⁢                            a                                                    -                                                      b                            ⁢                                                          xe2x80x83                                                        ⁢                            c                                                                          )                                                              magn                                                  ]                                                                        (        40        )             The expression for the image distance n is, given that the image height is equal to magn. yob:                                                                         Y_image                ⁢                _n                            :=                              xe2x80x83                            ⁢                                                -                                      1                    4                                                  ⁢                                  (                                                                                    -                                                  d                          2                                                                    ⁢                                              xe2x80x83                                            ⁢                      g                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      ya                      ⁢                                              xe2x80x83                                            ⁢                      h                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                                              e                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      a                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      ya                      ⁢                                              xe2x80x83                                            ⁢                      h                                        -                                                                                                                                          xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                                                            magn                      ⁢                                              xe2x80x83                                                              2                                    ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  e                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                                -                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  g                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                                -                                  2                  ⁢                                      xe2x80x83                                    ⁢                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  b                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  h                                +                                                                                                        xe2x80x83                            ⁢                                                e                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                  g                                ⁢                                  xe2x80x83                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  h                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                                -                                                      e                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  ya                  ⁢                                      xe2x80x83                                    ⁢                  h                                +                                                                                                                          xe2x80x83                                ⁢                                                                                                    d                        2                                            ⁢                                              xe2x80x83                                            ⁢                                              g                        2                                            ⁢                                              xe2x80x83                                            ⁢                      ya                      ⁢                                              xe2x80x83                                            ⁢                      f                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      e                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      a                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      ya                      ⁢                                              xe2x80x83                                            ⁢                      f                      ⁢                                              xe2x80x83                                            ⁢                      g                                                        )                                            /                                                                                          xe2x80x83                            ⁢                              (                                                      (                                                                                            -                          e                                                ⁢                                                  xe2x80x83                                                ⁢                        d                        ⁢                                                  xe2x80x83                                                ⁢                        a                        ⁢                                                  xe2x80x83                                                ⁢                        h                                            +                                              e                        ⁢                                                  xe2x80x83                                                ⁢                        b                        ⁢                                                  xe2x80x83                                                ⁢                        c                        ⁢                                                  xe2x80x83                                                ⁢                        h                                            +                                              f                        ⁢                                                  xe2x80x83                                                ⁢                        g                        ⁢                                                  xe2x80x83                                                ⁢                        d                        ⁢                                                  xe2x80x83                                                ⁢                        a                                            -                                              f                        ⁢                                                  xe2x80x83                                                ⁢                        g                        ⁢                                                  xe2x80x83                                                ⁢                        b                        ⁢                                                  xe2x80x83                                                ⁢                        c                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  e                  ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  ya                                )                                                                        (        41        )             6-mirror system, stop on mirror 2 Now we use these derivations in the first part: solving the variables of a six-mirror system with the stop position on mirror two, defining thicknesses as d:=[d0,d1,d2,d3,d4,d5,d6] and the curvatures as c=[c1,c2,c3,c4,c5,c6]. The stop (pupil) position is on the second surface. A limitation on the Petzval sum (i.e. the sum of curvatures in the system, curvatures of odd surfaces being subtracted from curvatures of even surfaces, or vice versa), e.g. to be zero, can be introduced and used to solve the curvature of the stop surface. However, a zero Petzval sum is not essential and an non-zero value can be accommodated. Now we define all the matrices in the system, the reflectance (even subscripts) and translation (odd subscripts) matrices, from the object to the image.                                                                         M                1                            :=                              [                                                                            1                                                              d0                                                                                                  0                                                              1                                                                      ]                                                                                        M                2                            :=                              [                                                                            1                                                              0                                                                                                                                                    -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c1                                                                                                            -                        1                                                                                            ]                                                                                                        M                3                            :=                              [                                                                            1                                                              d1                                                                                                  0                                                              1                                                                      ]                                                                                        M                4                            :=                              [                                                                            1                                                              0                                                                                                                                                    -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c2                                                                                                            -                        1                                                                                            ]                                                                                                        M                5                            :=                              [                                                                            1                                                              d2                                                                                                  0                                                              1                                                                      ]                                                                                        M                6                            :=                              [                                                                            1                                                              0                                                                                                                                                    -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                                                                                                            -                        1                                                                                            ]                                                                                                        M                7                            :=                              [                                                                            1                                                              d3                                                                                                  0                                                              1                                                                      ]                                                                                        M                8                            :=                              [                                                                            1                                                              0                                                                                                                                                    -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                                                                                                            -                        1                                                                                            ]                                                                                                        M                9                            :=                              [                                                                            1                                                              d4                                                                                                  0                                                              1                                                                      ]                                                                                        M                10                            :=                              [                                                                            1                                                              0                                                                                                                                                    -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                                                                                                            -                        1                                                                                            ]                                                                                                        M                11                            :=                              [                                                                            1                                                              d5                                                                                                  0                                                              1                                                                      ]                                                                                        M                12                            :=                              [                                                                            1                                                              0                                                                                                                                                    -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c6                                                                                                            -                        1                                                                                            ]                                                                                                        M                13                            :=                              [                                                                            1                                                              d6                                                                                                  0                                                              1                                                                      ]                                                                        xe2x80x83                                                          (        42        )             The first solve is the exit pupil in infinity or telecentricity in the image. The angle of the ray going through the optical axis in the stop surface should be zero in the image. The matrix from the stop surface to the fifth mirror generated by multiplication of the appropriate M matrices derived just above, is given by:                               |                      1            -                          2              ⁢                              xe2x80x83                            ⁢              d3              ⁢                              xe2x80x83                            ⁢              c3                        +                          d4              ⁢                              xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                                          c4                      ⁡                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d3                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                                                                          )                                                                              +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                                                  )                                                    ,                              d2            +                          d3              ⁡                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    -                  1                                )                                      +                          d4              ⁡                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                                          c4                      ⁡                                              (                                                  d2                          +                                                      d3                            ⁡                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                -                                1                                                            )                                                                                                      )                                                                              +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                                              |                      
                    ⁢                      [                                                            -                  2                                ⁢                                  xe2x80x83                                ⁢                c5                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d3                      ⁢                                              xe2x80x83                                            ⁢                      c3                                        +                                          d4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c4                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                                                                          )                                                                              )                                            +                              2                ⁢                                  xe2x80x83                                ⁢                                  c4                  (                                                            1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d3                        ⁢                                                  xe2x80x83                                                ⁢                        c3                                            -                                              2                        ⁢                                                  c3                          .                                                      -                            2                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                  c5                          ⁡                                                      (                                                          d2                              +                                                              d3                                ⁡                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        -                                    1                                                                    )                                                                                            +                                                              d4                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                                                              xe2x80x83                                                                            ⁢                                                                              c4                                        ⁡                                                                                  (                                                                                      d2                                            +                                                                                          d3                                              ⁡                                                                                              (                                                                                                                                                                                                            -                                                      2                                                                                                        ⁢                                                                                                          xe2x80x83                                                                                                        ⁢                                                    c3                                                    ⁢                                                                                                          xe2x80x83                                                                                                        ⁢                                                    d2                                                                                                    -                                                  1                                                                                                )                                                                                                                                                                              )                                                                                                                                                      +                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        +                                    1                                                                    )                                                                                                                      )                                                                                              +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                                                  c4                          ⁡                                                      (                                                          d2                              +                                                              d3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        -                                    1                                                                    )                                                                                                                      )                                                                                              -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        |                                                                                                          (        43        )             The matrix from the second mirror surface to the stop surface is given by:                     [                                            1                                      0                                                                                            -                  2                                ⁢                                  xe2x80x83                                ⁢                c2                                                                    -                1                                                    ]                            (        44        )             So as we derived the distance between mirror 5 and 6, we solve this new found value in the appropriate matrix and the vector of distances.                                                         d5              :=                              xe2x80x83                            ⁢                                                -                                      1                    2                                                  ⁢                                  (                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                                          4                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      d3                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      d3                                        -                                          4                      ⁢                                              xe2x80x83                                            ⁢                      c6                      ⁢                                              xe2x80x83                                            ⁢                      d4                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        +                                                                                                                                          xe2x80x83                            ⁢                                                8                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d3                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d3                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c6                  ⁢                                      xe2x80x83                                    ⁢                  d4                                -                                                                                                        xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d3                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d3                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d2                                -                                                                                                        xe2x80x83                            ⁢                                                8                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d3                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d3                                -                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                -                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d3                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                -                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d3                                -                                                                                                                          xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                            /                              (                                  c6                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                                              4                        ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d3                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                                                                                                                                                              xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d3                                -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d2                                +                                  8                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c                  ⁢                                      xe2x80x83                                    ⁢                  d3                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                +                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                  d3                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                                -                                                                                                                          xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        +                                          4                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                      d3                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                      d3                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        +                    1                                    )                                            )                                                          (        45        )             The distance between mirror one and two is derived as:                     d1        :=                              -                          1              4                                ⁢                                                                                                                                        -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                                        -                                          angle                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c5                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d2                              +                                                              d3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        -                                    1                                                                    )                                                                                            +                                                                                                                                                                                                                                            d4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c4                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d2                              +                                                              d3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        -                                    1                                                                    )                                                                                                                      )                                                                          +                                                                                                                                                                                                            2                        ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      d2                            +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    -                                  1                                                                )                                                                                                              )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                                                                      c1              ⁢                              xe2x80x83                            ⁢              magn              ⁢                              xe2x80x83                            ⁢              yob              ⁢                              xe2x80x83                            ⁢              c6                                                          (        46        )             and the distance between the object and the first mirror is:                                                         d0              :=                              xe2x80x83                            ⁢                                                1                  2                                ⁢                                  (                                                            -                      angle                                        ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      d2                            +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    -                                  1                                                                )                                                                                      +                                                                                                                                                                                                                                                        xe2x80x83                                ⁢                                                      d4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c4                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d2                              +                                                              d3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        -                                    1                                                                    )                                                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                +                        1                                            )                                                        )                                            +                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  d2                          +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                -                                1                                                            )                                                                                                      )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c5                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              d2                        +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            -                              1                                                        )                                                                          +                                                                                                                                                                                    xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  d2                          +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                -                                1                                                            )                                                                                                      )                                                              +                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      d2                    +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    -                          1                                                )                                                                              )                                            -                                                                                                                                                      xe2x80x83                                        ⁢                                                                                            2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                -                        1                                            )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    magn                    ⁢                                          xe2x80x83                                        ⁢                    yob                    ⁢                                          xe2x80x83                                        ⁢                    c6                                                  )                            /                              (                                  angle                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c5                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  d2                          +                                                                                                                                                                                                            xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  d2                          +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                -                                                                                                                                                                                                                                                                                                                                                                                                              xe2x80x83                                                ⁢                                                  1                          )                                                                    )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        +                    1                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      d2                    +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    -                          1                                                )                                                                              )                                            -                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                            )                                                          (        47        )             The distance between mirror six and the image surface is:                                                         d6              :=                              xe2x80x83                            ⁢                                                -                                      1                    4                                                  ⁢                                  (                                      2                    ⁢                                          xe2x80x83                                        ⁢                    angle                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                          d3c3                                                +                                                  d4                          (                                                      -                                                          "AutoLeftMatch"                                                                                                2                                  ⁢                                                                      c4                                    ⁡                                                                          (                                                                              1                                        -                                                                                  "AutoLeftMatch"                                                                                      xe2x80x83                                                                                    ⁢                                                                                      2                                            ⁢                                            d3c3                                                                                                                                                              )                                                                                                                                      +                                                                                                                                                                                                                                                                                                                                      xe2x80x83                                    ⁢                                                            2                      ⁢                      c3                                        )                                                  )                            ⁢                              xe2x80x83                            ⁢                              c6                (                                  xe2x80x83                                ⁢                                                      -                    2                                    ⁢                  c5                  (                                      d2                    +                                          "AutoLeftMatch"                                              xe2x80x83                                            ⁢                                              "AutoLeftMatch"                                                                              d3                            ⁡                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                  c3d2                                                                -                                1                                                            )                                                                                +                                                      d4                            (                                                                                          -                                2                                                            ⁢                              c4                              (                                                              d2                                +                                                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                            ⁢                                                                        d3                          ⁡                                                      (                                                                                                                            -                                  2                                                                ⁢                                c3d2                                                            -                              1                                                        )                                                                          )                                                              +                                          2                      ⁢                      c3d2                                        +                    1                                    )                                )                            +                              2                ⁢                c4                (                                  d2                  +                                                                                                                                                                                    xe2x80x83                                        ⁢                                                                  d3                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c3d2                                                        -                            1                                                    )                                                                    )                                                        -                                      2                    ⁢                    c3d2                                    -                  1                                )                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d3c3                                                        +                                                          d4                              ⁡                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                                                          c4                                      ⁡                                                                              (                                                                                  1                                          -                                                                                      2                                            ⁢                                            d3c3                                                                                                                          )                                                                                                                                              +                                                                      2                                    ⁢                                    c3                                                                                                  )                                                                                                              )                                                                                      +                                                                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              c4                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d3c3                                                                                )                                                                                      -                                          2                      ⁢                      c3                                                        )                                            ⁢              c6              (                              d2                +                                  d3                  ⁡                                      (                                                                                            -                          2                                                ⁢                        c3d2                                            -                      1                                        )                                                  +                                                                                                                          xe2x80x83                                ⁢                                                      d4                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                                                      c4                            ⁡                                                          (                                                              d2                                +                                                                  d3                                  ⁡                                                                      (                                                                                                                                                            -                                          2                                                                                ⁢                                        c3d2                                                                            -                                      1                                                                        )                                                                                                                              )                                                                                                      +                                                  2                          ⁢                          c3d2                                                +                        1                                            )                                                        )                                            -                                                                                          xe2x80x83                            ⁢                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d3c3                                                        +                                                          d4                              ⁡                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                                                          c4                                      ⁡                                                                              (                                                                                  1                                          -                                                                                      2                                            ⁢                                            d3c3                                                                                                                          )                                                                                                                                              +                                                                      2                                    ⁢                                    c3                                                                                                  )                                                                                                              )                                                                                      +                                                                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              c4                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d3c3                                                                                )                                                                                      -                                          2                      ⁢                      c3                                                        )                                            ⁢                              (                                                      -                    2                                    ⁢                  c5                  (                                      d2                    +                                          d3                      ⁡                                              (                                                                                                            -                              2                                                        ⁢                            c3d2                                                    -                          1                                                )                                                              +                                                                                                                                                            xe2x80x83                                ⁢                                                      d4                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                                                      c4                            ⁡                                                          (                                                              d2                                +                                                                  d3                                  ⁡                                                                      (                                                                                                                                                            -                                          2                                                                                ⁢                                        c3d2                                                                            -                                      1                                                                        )                                                                                                                              )                                                                                                      +                                                  2                          ⁢                          c3d2                                                +                        1                                            )                                                        )                                            +                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              c4                        ⁡                                                  (                                                      d2                            +                                                          d3                              ⁡                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                    c3d2                                                                    -                                  1                                                                )                                                                                                              )                                                                                      -                                          2                      ⁢                      c3d2                                        -                    1                                    )                                            +                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                    c5                    (                                          d2                      +                                              d3                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c3d2                                                        -                            1                                                    )                                                                    +                                                                                                                                                                                xe2x80x83                                ⁢                                                      d4                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                                                      c4                            ⁡                                                          (                                                              d2                                +                                                                  d3                                  ⁡                                                                      (                                                                                                                                                            -                                          2                                                                                ⁢                                        c3d2                                                                            -                                      1                                                                        )                                                                                                                              )                                                                                                      +                                                  2                          ⁢                          c3d2                                                +                        1                                            )                                                        )                                            +                                                                                                                                                      xe2x80x83                                        ⁢                                                                  "AutoLeftMatch"                                                                              2                            ⁢                                                          c4                              ⁡                                                              (                                                                  d2                                  +                                                                      d3                                    ⁡                                                                          (                                                                                                                                                                    -                                            2                                                                                    ⁢                                          c3d2                                                                                -                                        1                                                                            )                                                                                                                                      )                                                                                                              -                                                      2                            ⁢                            c3d2                                                    -                          1                                                                    )                                        "AutoRightMatch"                                    2                                ⁢                c2                            +                                                                                          xe2x80x83                            ⁢                              4                ⁢                                  c1                  (                                                            -                      2                                        ⁢                    c5                    (                                          d2                      +                                              d3                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c3d2                                                        -                            1                                                    )                                                                    +                                                                                                                                                                                xe2x80x83                                ⁢                                                      d4                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                                                      c4                            ⁡                                                          (                                                              d2                                +                                                                  d3                                  ⁡                                                                      (                                                                                                                                                            -                                          2                                                                                ⁢                                        c3d2                                                                            -                                      1                                                                        )                                                                                                                              )                                                                                                      +                                                  2                          ⁢                          c3d2                                                +                        1                                            )                                                        )                                            +                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  c4                          ⁡                                                      (                                                          d2                              +                                                              d3                                ⁡                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                      c3d2                                                                        -                                    1                                                                    )                                                                                                                      )                                                                                              -                                              2                        ⁢                        c3d2                                            -                      1                                        )                                                  ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            +                                                                                                            xe2x80x83                                ⁢                                                      4                    ⁢                                          xe2x80x83                                        ⁢                                          magn                      2                                        ⁢                                          xe2x80x83                                        ⁢                    yob                    ⁢                                          xe2x80x83                                        ⁢                                          c6                      2                                                        )                                            /                              (                                  (                                      (                                                                  -                        2                                            ⁢                      c5                      (                                              1                        -                                                  2                          ⁢                          d3c3                                                +                                                  d4                          (                                                                                    -                              2                                                        ⁢                            c4                            (                                                          1                              -                                                                                                                                                                                                                                                                                                                                                                                                            xe2x80x83                                                    ⁢                                                                                    2                              ⁢                              d3c3                                                        )                                                                          +                                                  2                          ⁢                          c3                                                                    )                                        )                                    +                                      2                    ⁢                                          c4                      ⁡                                              (                                                  1                          -                                                      2                            ⁢                            d3c3                                                                          )                                                                              -                                      2                    ⁢                    c3                                                  )                            ⁢                              (                                  d2                  +                                      d3                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                          c3d2                                                -                        1                                            )                                                        +                                                                                                                                          xe2x80x83                                ⁢                                                      d4                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                                                      c4                            ⁡                                                          (                                                              d2                                +                                                                  d3                                  ⁡                                                                      (                                                                                                                                                            -                                          2                                                                                ⁢                                        c3d2                                                                            -                                      1                                                                        )                                                                                                                              )                                                                                                      +                                                  2                          ⁢                          c3d2                                                +                        1                                            )                                                        )                                            -                                                                                          xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                  c5                  (                                      d2                    +                                          d3                      ⁡                                              (                                                                                                            -                              2                                                        ⁢                            c3d2                                                    -                          1                                                )                                                              +                                          d4                      (                                                                        -                          2                                                ⁢                        c4                        (                                                  d2                          +                                                                                                                                                                                                                                                                                                xe2x80x83                                            ⁢                                                                        d3                          ⁡                                                      (                                                                                                                            -                                  2                                                                ⁢                                c3d2                                                            -                              1                                                        )                                                                          )                                                              +                                          2                      ⁢                      c3d2                                        +                    1                                    )                                )                            +                              2                ⁢                                  c4                  ⁡                                      (                                          d2                      +                                              d3                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c3d2                                                        -                            1                                                    )                                                                                      )                                                              -                                                                                                                                                                              xe2x80x83                                            ⁢                                                                                                    2                            ⁢                            c3d2                                                    -                          1                                                )                                                              ⁢                                          (                                              1                        -                                                  2                          ⁢                          d3c3                                                +                                                  d4                          ⁡                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  c4                                  ⁡                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                        d3c3                                                                                                              )                                                                                                                              +                                                              2                                ⁢                                c3                                                                                      )                                                                                              )                                                        )                                ⁢                                  c6                  2                                ⁢                                  xe2x80x83                                ⁢                angle                            )                                                          (        48        )             The variable angle is identical to ya introduced in equation (24) above. 6-Mirror System, Stop on Mirror 3 The original derivations can similarly be used to solve the variables of a six-mirror system with the stop position on mirror three, as will now be shown. The first solve is pupil in infinity or telecentricity in the image. The angle of the ray going through the optical axis in the stop surface should be zero in the image. The matrix from the stop surface to the fifth mirror generated by multiplication of the appropriate M matrices derived just above, is given by:                     [                                                            1                -                                  2                  ⁢                  d4c4                                                                                    d3                +                                  d4                  ⁡                                      (                                                                                            -                          2                                                ⁢                        c4d3                                            -                      1                                        )                                                                                                                                                                -                    2                                    ⁢                                      c5                    ⁡                                          (                                              1                        -                                                  2                          ⁢                          d4c4                                                                    )                                                                      +                                  2                  ⁢                  c4                                                                                                                          -                    2                                    ⁢                                      c5                    ⁡                                          (                                              d3                        +                                                  d4                          ⁡                                                      (                                                                                                                            -                                  2                                                                ⁢                                c4d3                                                            -                              1                                                        )                                                                                              )                                                                      +                                  2                  ⁢                  c4d3                                +                1                                                    ]                            (        50        )             The matrix from the second mirror surface to the stop surface is given by:                     [                                                            1                -                                  2                  ⁢                  d2c2                                                                                    -                d2                                                                                                                              -                    2                                    ⁢                                      c3                    ⁡                                          (                                              1                        -                                                  2                          ⁢                          d2c2                                                                    )                                                                      +                                  2                  ⁢                  c2                                                                                                      2                  ⁢                  c3d2                                +                1                                                    ]                            (        51        )             So, as derived above, the distance between mirror 5 and 6 is, solved this new found value in the appropriate matrix and the vector of distances:                     d5        :=                              -                          1              2                                ⁢                      xe2x80x83                    ⁢                                                                                                                2                      ⁢                      c6d3                                        -                                          4                      ⁢                      c6d4c4d3                                        -                                          2                      ⁢                      c6d4                                        -                                                                                                                                          2                      ⁢                      c5d3                                        +                                          4                      ⁢                      c5d4c4d3                                        +                                          2                      ⁢                      c5d4                                        +                                          2                      ⁢                      c4d3                                        +                    1                                                                                      c6              ⁡                              (                                                                            -                      2                                        ⁢                    c5d3                                    +                                      4                    ⁢                    c5d4c4d3                                    +                                      2                    ⁢                    c5d4                                    +                                      2                    ⁢                    c4d3                                    +                  1                                )                                                                        (        51        )             The distance between mirror one and two was:                                                         d1              :=                              xe2x80x83                            ⁢                                                -                                      1                    4                                                  ⁢                                  (                                                                                    -                        4                                            ⁢                      d2c1                      ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                                        -                                          2                      ⁢                                              (                                                  1                          -                                                      2                            ⁢                            d2c2                                                                          )                                            ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                                        +                                                                                                                                          xe2x80x83                            ⁢                                                (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      d3                            +                                                          d4                              ⁡                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                    c4d3                                                                    -                                  1                                                                )                                                                                                              )                                                                                      +                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                        c4d3                                            +                      1                                        )                                                  ⁢                                  (                                                            2                      ⁢                      c3d2                                        +                    1                                    )                                            +                              angle                ⁢                                  xe2x80x83                                ⁢                                  d2                  (                                                            -                      2                                        ⁢                                          c5                      (                                              d3                        +                                                  d4                          (                                                                                                                    -                                2                                                            ⁢                              c4d3                                                        -                                                                                                                                                                                                                                                                                                                                                                                xe2x80x83                                                    ⁢                                                      1                            )                                                                          )                                            +                                              2                        ⁢                        c4d3                                            +                      1                                        )                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  c3                          ⁡                                                      (                                                          1                              -                                                              2                                ⁢                                d2c2                                                                                      )                                                                                              +                                              2                        ⁢                        c2                                                              )                                                  )                            /                              (                                  (                                      1                    -                                                                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                      d2c2                                        )                                                  ⁢                c1                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            )                                                          (        52        )             And the distance between the object and the first mirror:                                                         d0              :=                              xe2x80x83                            ⁢                                                -                                      1                    2                                                  ⁢                                  (                                                            (                                              1                        -                                                  2                          ⁢                          d2c2                                                                    )                                        ⁢                                          xe2x80x83                                        ⁢                    angle                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      c5                            ⁡                                                          (                                                              d3                                +                                                                  d4                                  ⁡                                                                      (                                                                                                                                                            -                                          2                                                                                ⁢                                        c4d3                                                                            -                                      1                                                                        )                                                                                                                              )                                                                                                      +                                                                                                                                                                                                                          xe2x80x83                                    ⁢                                                                                    2                        ⁢                        c4d3                                            +                      1                                        )                                                  ⁢                                  (                                                            2                      ⁢                      c3d2                                        +                    1                                    )                                            +                              angle                ⁢                                  xe2x80x83                                ⁢                                  d2                  (                                                            -                      2                                        ⁢                    c5                    (                                          d3                      +                                              d4                        (                                                                                                            -                              2                                                        ⁢                            c4d3                                                    -                                                                                                                                                                                                                                                                                                                          xe2x80x83                                                ⁢                                                  1                          )                                                                    )                                        +                                          2                      ⁢                      c4d3                                        +                    1                                    )                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c3                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d2c2                                                                                )                                                                                      +                                          2                      ⁢                      c2                                                        )                                            +                                                                                                                                                      xe2x80x83                                        ⁢                                                                  2                        ⁢                        c1                        ⁢                                                  xe2x80x83                                                ⁢                        yob                        ⁢                                                  xe2x80x83                                                ⁢                                                  d2                          (                                                                                    -                              2                                                        ⁢                            c5                            (                                                          d3                              +                                                              d4                                ⁡                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                      c4d3                                                                        -                                    1                                                                    )                                                                                                                                                                                        )                                                        +                                      2                    ⁢                    c4d3                                    +                  1                                )                            ⁢                              (                                                      -                    2                                    ⁢                                      c3                    (                                          1                      -                                                                                                                                                                                                                          xe2x80x83                                        ⁢                                                                  2                        ⁢                        d2c2                                            )                                                        +                                      2                    ⁢                    c2                                                  )                            +                              2                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  (                                                            -                      2                                        ⁢                                          c5                      (                                              d3                        +                                                                                                                                                                                                                                                                        xe2x80x83                                            ⁢                                                                        d4                          ⁡                                                      (                                                                                                                            -                                  2                                                                ⁢                                c4d3                                                            -                              1                                                        )                                                                          )                                                              +                                          2                      ⁢                      c4d3                                        +                    1                                    )                                ⁢                                  (                                                            2                      ⁢                      c3d2                                        +                    1                                    )                                            -                                                                                                            xe2x80x83                                ⁢                                                      2                    ⁢                                          (                                              1                        -                                                  2                          ⁢                          d2c2                                                                    )                                        ⁢                                          xe2x80x83                                        ⁢                    magn                    ⁢                                          xe2x80x83                                        ⁢                    yob                    ⁢                                          xe2x80x83                                        ⁢                    c6                                    )                                            /                              (                                  angle                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          (                                                                                                    -                            2                                                    ⁢                                                      c3                            ⁡                                                          (                                                              1                                -                                                                  2                                  ⁢                                  d2c2                                                                                            )                                                                                                      +                                                                                                                                                                                                                                                                        xe2x80x83                                            ⁢                                                                        2                          ⁢                          c2                                                )                                                              ⁢                    d2                                    +                                                            (                                                                        2                          ⁢                          c3d2                                                +                        1                                            )                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                          d2c2                                                                    )                                                                      )                            ⁢              c1              (                                                -                  2                                ⁢                                  c5                  (                                      d3                    +                                                                                                                                                                                                      xe2x80x83                                        ⁢                                                                  d4                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c4d3                                                        -                            1                                                    )                                                                    )                                                        +                                      2                    ⁢                    c4d3                                    +                  1                                )                            )                                                          (        53        )             And the distance between mirror six and the image surface:                                                         d6              :=                              xe2x80x83                            ⁢                                                -                                      xe2x80x83                                    ⁢                                      1                    4                                                  ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    angle                    ⁢                                          xe2x80x83                                        ⁢                                                                  (                                                  1                          -                                                      2                            ⁢                            d2c2                                                                          )                                            2                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                          d4c4                                                                    )                                        ⁢                    c6                    (                                                                  -                        2                                            ⁢                      c5                      (                                              d3                        +                                                                                                                                                                                                                                                xe2x80x83                                        ⁢                                                                  d4                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c4d3                                                        -                            1                                                    )                                                                    )                                                        +                                      2                    ⁢                    c4d3                                    +                  1                                )                            ⁢                              (                                                      2                    ⁢                    c3d2                                    +                                                                                                                                          xe2x80x83                                ⁢                                  1                  )                                            -                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d4c4                                                        )                                ⁢                c6                (                                                      -                    2                                    ⁢                  c5                  (                                      d3                    +                                                                                                                                                                                                      xe2x80x83                                        ⁢                                                                  d4                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c4d3                                                        -                            1                                                    )                                                                    )                                                        +                                      2                    ⁢                    c4d3                                    +                  1                                )                            ⁢              d2              (                                                                    -                    2                                    ⁢                                      c3                    ⁡                                          (                                              1                        -                                                  2                          ⁢                          d2c2                                                                    )                                                                      +                                                                                                                          xe2x80x83                                ⁢                                                      2                    ⁢                    c2                                    )                                            +                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                                      (                                          1                      -                                              2                        ⁢                        d2c2                                                              )                                    2                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d4c4                                                                                )                                                                                      +                                                                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                      c4                                        )                                                  ⁢                                  c6                  ⁡                                      (                                          d3                      +                                              d4                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c4d3                                                        -                            1                                                    )                                                                                      )                                                  ⁢                                  (                                                            2                      ⁢                      c3d2                                        +                    1                                    )                                            +                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d4c4                                                                                )                                                                                      +                                          2                      ⁢                      c4                                                        )                                ⁢                                  c6                  (                                      d3                    +                                                                                                                                                                                xe2x80x83                                    ⁢                                                            d4                      ⁡                                              (                                                                                                            -                              2                                                        ⁢                            c4d3                                                    -                          1                                                )                                                              )                                                  ⁢                                  d2                  ⁡                                      (                                                                                            -                          2                                                ⁢                                                  c3                          ⁡                                                      (                                                          1                              -                                                              2                                ⁢                                d2c2                                                                                      )                                                                                              +                                              2                        ⁢                        c2                                                              )                                                              +                                                                                          xe2x80x83                            ⁢                              angle                ⁢                                  xe2x80x83                                ⁢                                                      (                                          1                      -                                              2                        ⁢                        d2c2                                                              )                                    2                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d4c4                                                                                )                                                                                      +                                                                                                                                                            xe2x80x83                                ⁢                                                      2                    ⁢                    c4                                    )                                            ⁢                              (                                                                            -                      2                                        ⁢                                          c5                      ⁡                                              (                                                  d3                          +                                                      d4                            ⁡                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                  c4d3                                                                -                                1                                                            )                                                                                                      )                                                                              +                                      2                    ⁢                    c4d3                                    +                  1                                )                            ⁢                              (                                                      2                    ⁢                    c3d2                                    +                  1                  +                                                                                                                        xe2x80x83                            ⁢                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d4c4                                                                                )                                                                                      +                                          2                      ⁢                      c4                                                        )                                ⁢                                  (                                                            -                      2                                        ⁢                    c5                    (                                          d3                      +                                                                                                                                                                                                                          xe2x80x83                                        ⁢                                                                  d4                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                              c4d3                                                        -                            1                                                    )                                                                    )                                                        +                                      2                    ⁢                    c4d3                                    +                  1                                )                            ⁢              d2              (                                                                    -                    2                                    ⁢                                      c3                    ⁡                                          (                                              1                        -                                                  2                          ⁢                          d2c2                                                                    )                                                                      +                                                                                                                          xe2x80x83                                ⁢                                                      2                    ⁢                    c2                                    )                                            +                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      d3                            +                                                          d4                              ⁡                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                    c4d3                                                                    -                                  1                                                                )                                                                                                              )                                                                                      +                                          2                      ⁢                      c4d3                                        +                                                                                                                                                                                                      xe2x80x83                                        ⁢                                          1                      )                                                        2                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c3                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d2c2                                                                                )                                                                                      +                                          2                      ⁢                      c2                                                        )                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  (                                                            2                      ⁢                      c3d2                                        +                    1                                    )                                            +                                                                                          xe2x80x83                            ⁢                              angle                ⁢                                  xe2x80x83                                ⁢                                                      (                                                                                            -                          2                                                ⁢                                                  c5                          ⁡                                                      (                                                          d3                              +                                                              d4                                ⁡                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                      c4d3                                                                        -                                    1                                                                    )                                                                                                                      )                                                                                              +                                              2                        ⁢                        c4d3                                            +                      1                                        )                                    2                                ⁢                                  (                                                            -                      2                                        ⁢                    c3                    (                                          1                      -                                                                                                                                                                                                                                                                                                        "AutoLeftMatch"                                                          xe2x80x83                                                        ⁢                                                          2                              ⁢                              d2c2                                                                                )                                                +                                                  2                          ⁢                          c2                                                                    )                                        "AutoRightMatch"                                    2                                ⁢                d2                            -                              4                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      d3                            +                                                          d4                              ⁡                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                    c4d3                                                                    -                                  1                                                                )                                                                                                              )                                                                                      +                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                        c4d3                                            +                      1                                        )                                                  ⁢                                  (                                                                                    -                        2                                            ⁢                                              c3                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d2c2                                                                                )                                                                                      +                                          2                      ⁢                      c2                                                        )                                ⁢                d2c1                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            -                                                                                          xe2x80x83                            ⁢                              4                ⁢                                  c1                  ⁡                                      (                                                                                            -                          2                                                ⁢                                                  c5                          ⁡                                                      (                                                          d3                              +                                                              d4                                ⁡                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                      c4d3                                                                        -                                    1                                                                    )                                                                                                                      )                                                                                              +                                              2                        ⁢                        c4d3                                            +                      1                                        )                                                  ⁢                                  (                                                            2                      ⁢                      c3d2                                        +                                                                                                                                                                                xe2x80x83                                    ⁢                                      1                    )                                                  ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            +                              4                ⁢                                  xe2x80x83                                ⁢                                  magn                  2                                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                                      c6                    2                                    (                                      1                    -                                                                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                      d2c2                                        )                                                  )                            /                              (                                  (                                      (                                                                                            -                          2                                                ⁢                                                  c5                          ⁡                                                      (                                                          d3                              +                                                              d4                                ⁡                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                      c4d3                                                                        -                                    1                                                                    )                                                                                                                      )                                                                                              +                                              2                        ⁢                        c4d3                                            +                                                                                                                                                                                                    xe2x80x83                                    ⁢                                      1                    )                                                  ⁢                                  (                                      1                    -                                          2                      ⁢                      d4c4                                                        )                                ⁢                                  d2                  ⁡                                      (                                                                                            -                          2                                                ⁢                                                  c3                          ⁡                                                      (                                                          1                              -                                                              2                                ⁢                                d2c2                                                                                      )                                                                                              +                                              2                        ⁢                        c2                                                              )                                                              -                              (                                                      -                    2                                    ⁢                  c5                  (                                      1                    -                                                                                                                                                                                                      xe2x80x83                                        ⁢                                                                  2                        ⁢                        d4c4                                            )                                                        +                                      2                    ⁢                    c4                                                  )                            ⁢                              (                                  d3                  +                                      d4                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                          c4d3                                                -                        1                                            )                                                                      )                            ⁢              d2              (                                                -                  2                                ⁢                c3                (                                  1                  -                                                                                                                                                                                    xe2x80x83                                        ⁢                                                                  2                        ⁢                        d2c2                                            )                                                        +                                      2                    ⁢                    c2                                                  )                            -                                                (                                                            2                      ⁢                      c3d2                                        +                    1                                    )                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              c5                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d4c4                                                                                )                                                                                      +                                                                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                      c4                                        )                                                  ⁢                                  (                                      d3                    +                                          d4                      ⁡                                              (                                                                                                            -                              2                                                        ⁢                            c4d3                                                    -                          1                                                )                                                                              )                                            +                                                (                                                            2                      ⁢                      c3d2                                        +                    1                                    )                                ⁢                                  (                                      1                    -                                                                                                                                                            xe2x80x83                                ⁢                                                      2                    ⁢                    d2c2                                    )                                            ⁢                              (                                                                            -                      2                                        ⁢                                          c5                      ⁡                                              (                                                  d3                          +                                                      d4                            ⁡                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                  c4d3                                                                -                                1                                                            )                                                                                                      )                                                                              +                                      2                    ⁢                    c4d3                                    +                                                                                                                                                                                                            xe2x80x83                                            ⁢                                              1                        )                                                              ⁢                                          (                                              1                        -                                                  2                          ⁢                          d4c4                                                                    )                                                        )                                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                                        )                                ⁢                                  c6                  2                                ⁢                                  xe2x80x83                                ⁢                angle                            )                                                          (        54        )             6-Mirror System, Stop on Mirror 4 Similarly, the original derivations can be used to solve the variables of a six-mirror system with the stop position on mirror four. Again, the first solve is pupil in infinity or telecentricity in the image. The angle of the ray going through the optical axis in the stop surface should be zero in the image. The matrix from the stop surface to the fifth mirror generated by multiplication of the appropriate M matrices derived above is given by:                     [                                            1                                      d4                                                                                            -                  2                                ⁢                c5                                                                                                          -                    2                                    ⁢                  c5d4                                -                1                                                    ]                            (        55        )             The matrix from the second mirror surface to the stop surface is given by:                     [                                                                                                                        1                      -                                              2                        ⁢                        d2c2                                            +                                              d3                        (                                                                              -                            2                                                    ⁢                          c3                          (                                                      1                            -                                                                                                                                                                                                                                                                                        2                            ⁢                            d2c2                                                    )                                                +                                                  2                          ⁢                          c2                                                                    )                                                                                                                                            -                  d2                                +                                  d3                  ⁡                                      (                                                                  2                        ⁢                        c3d2                                            +                      1                                        )                                                                                                                                                                                                              -                        2                                            ⁢                                              c4                        (                                                  1                          -                                                      2                            ⁢                            d2c2                                                    +                                                      d3                            (                                                                                          -                                2                                                            ⁢                                                              c3                                (                                                                  1                                  -                                                                                                                                                                                                                                                                                                                                                                                                                                                2                                ⁢                                d2c2                                                            )                                                        +                                                          2                              ⁢                              c2                                                                                )                                                )                                            +                                              2                        ⁢                                                  c3                          ⁡                                                      (                                                          1                              -                                                              2                                ⁢                                d2c2                                                                                      )                                                                                              -                                              2                        ⁢                        c2                                                                                                                                                                                                                                    -                        2                                            ⁢                                              c4                        (                                                                              -                            d2                                                    +                                                      d3                            (                                                                                          2                                ⁢                                c3d2                                                            +                                                                                                                                                                                                                                                                                        1                          )                                                )                                            -                                              2                        ⁢                        c3d2                                            -                      1                                                                                                          ]                            (        56        )             So, as derived above, the distance between mirror 5 and 6, solved in the appropriate matrix and the vector of distances, is:                     d5        :=                              -                          xe2x80x83                        ⁢                          1              2                                ⁢                      xe2x80x83                    ⁢                                                                      -                  2                                ⁢                c6d4                            +                              2                ⁢                c5d4                            +              1                                      c6              ⁡                              (                                                      2                    ⁢                    c5d4                                    +                  1                                )                                                                        (        57        )             The distance between mirror one and two is:                                                         d1              :=                              xe2x80x83                            ⁢                                                -                                      xe2x80x83                                    ⁢                                      1                    4                                                  ⁢                                  (                                                            4                      ⁢                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁡                                                          (                                                                                                2                                  ⁢                                  c3d2                                                                +                                1                                                            )                                                                                                      )                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c6                                        -                                          2                      ⁢                                              (                                                  1                          -                                                                                                                                                                                                                                                  xe2x80x83                                    ⁢                                                                                    2                        ⁢                        d2c2                                            +                                              d3                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                                                              c3                                ⁡                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                      d2c2                                                                                                        )                                                                                                                      +                                                          2                              ⁢                              c2                                                                                )                                                                                      )                                                  ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            +                                                                                          xe2x80x83                            ⁢                              (                                  1                  -                                      2                    ⁢                    d2c2                                    +                                      d3                    (                                                                                            -                          2                                                ⁢                                                  c3                          ⁡                                                      (                                                          1                              -                                                              2                                ⁢                                d2c2                                                                                      )                                                                                              +                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                            2                      ⁢                      c2                                        )                                                  )                            ⁢                              xe2x80x83                            ⁢              angle              ⁢                              xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                    c5d4                                    -                  1                                )                            ⁢                              (                                                      -                    2                                    ⁢                  c4                  (                                                            -                      d2                                        +                                          d3                      (                                                                        2                          ⁢                          c3d2                                                +                                                                                                                                                                                                                                                                        xe2x80x83                                            ⁢                                              1                        )                                                              )                                    -                                      2                    ⁢                    c3d2                                    -                  1                                )                            -                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁡                                              (                                                                              2                            ⁢                            c3d2                                                    +                          1                                                )                                                                              )                                ⁢                                  (                                                                                    -                        2                                            ⁢                      c5d4                                        -                                                                                                                                                                                                      xe2x80x83                                        ⁢                                          1                      )                                                        ⁢                                      (                                                                  -                        2                                            ⁢                                              c4                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d2c2                                                        +                                                          d3                              (                                                                                                                                    -                                    2                                                                    ⁢                                                                      c3                                    ⁡                                                                          (                                                                              1                                        -                                                                                  2                                          ⁢                                          d2c2                                                                                                                    )                                                                                                                                      +                                                                  2                                  ⁢                                  c2                                                                                                                                              )                                                                                                                    )                            +                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  c3                          ⁡                                                      (                                                          1                              -                                                              2                                ⁢                                d2c2                                                                                      )                                                                                              -                                              2                        ⁢                        c2                                                              )                                                  )                            /                              (                                  (                                      1                    -                                          2                      ⁢                      d2c2                                        +                                          d3                      (                                                                        -                          2                                                ⁢                        c3                        (                                                  1                          -                                                                                                                                                                                                                                                                                                                          xe2x80x83                                                ⁢                                                                              2                            ⁢                            d2c2                                                    )                                                                    +                                              2                        ⁢                        c2                                                              )                                    )                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            )                                                          (        58        )             And the distance between the object and the first mirror:                                                         d0              :=                              xe2x80x83                            ⁢                                                -                                      1                    2                                                  ⁢                                  (                                      (                                          1                      -                                              2                        ⁢                        d2c2                                            +                                              d3                        (                                                                                                            -                              2                                                        ⁢                                                          c3                              ⁡                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                    d2c2                                                                                                  )                                                                                                              +                                                                                                                                                                                                                                                  xe2x80x83                                    ⁢                                                            2                      ⁢                      c2                                        )                                                  )                            ⁢                              xe2x80x83                            ⁢              angle              ⁢                              xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                    c5d4                                    -                  1                                )                            ⁢                              (                                                      -                    2                                    ⁢                  c4                  (                                                            -                      d2                                        +                                                                                                                                                                                                      xe2x80x83                                        ⁢                                                                  d3                        ⁡                                                  (                                                                                    2                              ⁢                              c3d2                                                        +                            1                                                    )                                                                    )                                                        -                                      2                    ⁢                    c3d2                                    -                  1                                )                            -                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                                                                                                                                            xe2x80x83                                ⁢                                                      d3                    ⁡                                          (                                                                        2                          ⁢                          c3d2                                                +                        1                                            )                                                        )                                            ⁢                              (                                                                            -                      2                                        ⁢                    c5d4                                    -                  1                                )                            ⁢                              (                                                      -                    2                                    ⁢                  c4                  (                                      1                    -                                          2                      ⁢                      d2c2                                        +                                                                                                                                                            xe2x80x83                                ⁢                                                      d3                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                                                      c3                            ⁡                                                          (                                                              1                                -                                                                  2                                  ⁢                                  d2c2                                                                                            )                                                                                                      +                                                  2                          ⁢                          c2                                                                    )                                                        )                                            +                              2                ⁢                                  c3                  ⁡                                      (                                          1                      -                                              2                        ⁢                        d2c2                                                              )                                                              -                                                                                                            xe2x80x83                                ⁢                                                      2                    ⁢                    c2                                    )                                            -                              2                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁡                                              (                                                                              2                            ⁢                            c3d2                                                    +                          1                                                )                                                                              )                                ⁢                                  (                                                                                    -                        2                                            ⁢                      c5d4                                        -                                                                                                                                                            xe2x80x83                                ⁢                                  1                  )                                            ⁢                              (                                                      -                    2                                    ⁢                  c4                  (                                      1                    -                                          2                      ⁢                      d2c2                                        +                                          d3                      (                                                                                                    -                            2                                                    ⁢                                                      c3                            ⁡                                                          (                                                              1                                -                                                                  2                                  ⁢                                  d2c2                                                                                            )                                                                                                      +                                                                                                                                                                                                                                                                        xe2x80x83                                            ⁢                                                                        2                          ⁢                          c2                                                )                                                              )                                    +                                      2                    ⁢                                          c3                      ⁡                                              (                                                  1                          -                                                      2                            ⁢                            d2c2                                                                          )                                                                              -                                      2                    ⁢                    c2                                                  )                            +                              2                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                                                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                      d2c2                                        +                                          d3                      ⁡                                              (                                                                                                            -                              2                                                        ⁢                                                          c3                              ⁡                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                    d2c2                                                                                                  )                                                                                                              +                                                      2                            ⁢                            c2                                                                          )                                                                              )                                            ⁢                              (                                                                            -                      2                                        ⁢                    c5d4                                    -                                                                                                                                                              xe2x80x83                                    ⁢                                      1                    )                                                  ⁢                                  (                                                                                    -                        2                                            ⁢                                              c4                        ⁡                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁡                                                              (                                                                                                      2                                    ⁢                                    c3d2                                                                    +                                  1                                                                )                                                                                                              )                                                                                      -                                          2                      ⁢                      c3d2                                        -                    1                                    )                                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  (                                      1                    -                                          2                      ⁢                      d2c2                                        +                                          d3                      (                                                                                                    -                            2                                                    ⁢                                                      c3                            ⁡                                                          (                                                              1                                -                                                                  2                                  ⁢                                  d2c2                                                                                            )                                                                                                      +                                                                                                                                                                                                                                                                        xe2x80x83                                            ⁢                                                                        2                          ⁢                          c2                                                )                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                                )                            /                              (                                  angle                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          -                                              (                                                                              -                            2                                                    ⁢                          c4                          (                                                      1                            -                                                                                                                                                                                                                                                        xe2x80x83                                ⁢                                                                            2                      ⁢                      d2c2                                        +                                          d3                      ⁡                                              (                                                                                                            -                              2                                                        ⁢                                                          c3                              ⁡                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                    d2c2                                                                                                  )                                                                                                              +                                                      2                            ⁢                            c2                                                                          )                                                                              )                                            +                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              c3                        ⁡                                                  (                                                      1                            -                                                          2                              ⁢                              d2c2                                                                                )                                                                                      -                                          2                      ⁢                      c2                                                        )                                            ⁢                              (                                                      -                    d2                                    +                                      d3                    (                                                                  2                        ⁢                        c3d2                                            +                                                                                                                                                                                                    xe2x80x83                                    ⁢                                      1                    )                                                  )                            +                                                (                                                                                    -                        2                                            ⁢                                              c4                        ⁡                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁡                                                              (                                                                                                      2                                    ⁢                                    c3d2                                                                    +                                  1                                                                )                                                                                                              )                                                                                      -                                          2                      ⁢                      c3d2                                        -                    1                                    )                                ⁢                                  (                                      1                    -                                                                                                                                                                                                      xe2x80x83                                        ⁢                                                                                            2                          ⁢                          d2c2                                                +                                                  d3                          ⁡                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  c3                                  ⁡                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                        d2c2                                                                                                              )                                                                                                                              +                                                              2                                ⁢                                c2                                                                                      )                                                                                              )                                                        )                                ⁢                                  c1                  ⁡                                      (                                                                                            -                          2                                                ⁢                        c5d4                                            -                      1                                        )                                                              )                                                          (        59        )             And the distance between mirror six and the image surface:                                                         d6              :=                              xe2x80x83                            ⁢                                                -                                      xe2x80x83                                    ⁢                                      1                    4                                                  ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    angle                    ⁢                                          xe2x80x83                                        ⁢                                                                  (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                  +                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                                      )                                            2                                                                                                                                                              xe2x80x83                            ⁢                                                c6                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    +                                  1                                                                )                                                                                                              )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                                  +                                                                                                        xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                                              )                                                                                                                        xe2x80x83                            ⁢                              c6                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c5                      ⁢                                              xe2x80x83                                            ⁢                      d4                                        -                    1                                    )                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                                                                                                        xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                                              )                                                        +                                                                                                                                                              xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  angle                                            ⁢                              xe2x80x83                                                                                                        xe2x80x83                            ⁢                                                                    (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                                      )                                    2                                ⁢                                  xe2x80x83                                ⁢                c5                ⁢                                  xe2x80x83                                ⁢                c6                ⁢                                  xe2x80x83                                ⁢                d4                                                                                                        xe2x80x83                            ⁢                                                (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    ⁢                                                      xe2x80x83                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                  angle                                                                                                                        xe2x80x83                            ⁢                                                (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                          c3                              ⁡                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                                              +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                                              )                                ⁢                                  xe2x80x83                                ⁢                c5                ⁢                                  xe2x80x83                                ⁢                c6                ⁢                                  xe2x80x83                                ⁢                d4                            ⁢                              xe2x80x83                                                                                                        xe2x80x83                            ⁢                                                (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                                                                  c4                        (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                                    ⁢                                              xe2x80x83                                                                                                                                                                                                                                                xe2x80x83                                        ⁢                                                                  d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                                      )                                    2                                                                                                                        xe2x80x83                            ⁢                                                c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                  c4                          ⁡                                                      (                                                                                          -                                d2                                                            +                                                              d3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                            2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        +                                    1                                                                    )                                                                                                                      )                                                                                              -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                                  +                                                                                                        xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                                              )                                                                                                                        xe2x80x83                            ⁢                                                c5                  ⁡                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                                  ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                                                                                                        xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  c3                                  ⁡                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                                              +                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                                              )                                                        +                                                                                                                                          xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            +                              angle                ⁢                                  xe2x80x83                                ⁢                                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                    2                                                                                                                        xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                                          c4                      ⁡                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                  +                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                                      )                                                                              +                                                                                                                                          xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                                                                                                                                                                                                                xe2x80x83                                        ⁢                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            )                                                        )                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        ⁢                                          xe2x80x83                                        -                    1                                    )                                            -                                                                                          xe2x80x83                            ⁢                              angle                ⁢                                  xe2x80x83                                ⁢                                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                    2                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                                                                                                                                                    (                                                                                                    xe2x80x83                                            ⁢                                                                        d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          )                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            2                                                                                          xe2x80x83                            ⁢                                                (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                                                                                                                          xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                                              )                                                        +                                                                                                                                                              xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    ⁢                                                      xe2x80x83                                                    +                          1                                                )                                                                              )                                            ⁢                              xe2x80x83                                                                                                        xe2x80x83                            ⁢                                                c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                                -                                  4                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                                                                                                                          xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                                          c4                      ⁡                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁡                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                                              -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    -                  1                                )                                                                                                        xe2x80x83                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                                      )                                                                                                        xe2x80x83                            ⁢                                                magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                                +                                  4                  ⁢                                      xe2x80x83                                    ⁢                                      magn                    2                                    ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                                                            c6                      2                                        (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                            d3                      ⁡                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                          c3                              ⁡                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                                              +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              )                                                  )                            /                              (                                  (                                      -                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c5                          ⁢                                                      xe2x80x83                                                    ⁢                          d4                                                -                        1                                            )                                                                                                                                                              xe2x80x83                            ⁢                                                (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                                                                                                                                                                                                                xe2x80x83                                        ⁢                                                                  d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                c5                ⁢                                  xe2x80x83                                ⁢                d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                                                                                                        xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                ⁢                                                  xe2x80x83                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                                              )                                                        +                                                                                                                                          xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            +                              2                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                              -                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                                  ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                                              )                                            ⁢                              xe2x80x83                                                                                                        xe2x80x83                            ⁢                                                c5                  ⁢                                      xe2x80x83                                    ⁢                  d4                                +                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                                                                                                        xe2x80x83                            ⁢                                                                    (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                                      )                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                        d4                                            -                      1                                        )                                                  )                                                                                                        xe2x80x83                            ⁢                                                                    (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                                      )                                    ⁢                                      xe2x80x83                                    ⁢                                      c6                    2                                    ⁢                                      xe2x80x83                                    ⁢                  angle                                )                                                                        (        60        )             6-Mirror System, Stop on Mirror 5 Again, we use the original derivations to solve the variables of a six-mirror system with the stop position on mirror five. As before, the first solve is pupil in infinity or telecentricity in the image. The angle of the ray going through the optical axis in the stop surface should be zero in the image. The matrix from the stop surface to the fifth mirror generated by multiplication of the appropriate M matrices derived above is given by:                     [                                            1                                      0                                                          0                                      1                                      ]                            (        61        )             The matrix from the second mirror surface to the stop surface is given by:                                                         [                              1                -                                  xe2x80x83                                ⁢                                  2                  ⁢                                      xe2x80x83                                    ⁢                  d2                  ⁢                                      xe2x80x83                                    ⁢                  c2                                +                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                                                                                        xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                  +                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                                      )                                                              +                                                                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            ,                                                -                  d2                                +                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                                  +                                                                                                        xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        -                          d2                                                +                                                  "AutoLeftMatch"                                                                                    d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                                                                                                    (                                                                                                                  2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c3                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                                                            +                                                                              "AutoLeftMatch"                                        1                                                                                                                                              )                                                                )                                                                                      -                                                                                                                                                                                                                                                        xe2x80x83                                ⁢                                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                                              xe2x80x83                                            ⁢                                              "AutoLeftMatch"                        1                                                              )                                    ]                                            [                              xe2x80x83                            ⁢                              -                                  "AutoLeftMatch"                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c5                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                    xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    )                                            +                              "AutoLeftMatch"                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                      "AutoLeftMatch"                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              xe2x80x83                                                        ⁢                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            +                                                                                          xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                                      )                                                  -                                                                                                        xe2x80x83                            ⁢                                                                    2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  ,                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    +                                                                                                                                                                                xe2x80x83                                ⁢                                                      d4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c4                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          -                                d2                                                            +                                                              d3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                            2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                                                        +                                    1                                                                    )                                                                                                                      )                                                                          -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                -                        1                                            )                                                        )                                            +                                                                                          xe2x80x83                            ⁢                                                                    "AutoLeftMatch"                    "AutoRightMatch"                                    ⁢                                      "AutoLeftMatch"                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    +                                  1                                                                )                                                                                                              )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                                                      ]                                                                        (        62        )             So, as derived above, the distance between mirror 5 and 6, solved in the appropriate matrix and the vector of distances: is:                     d5        :=                              -                          xe2x80x83                        ⁢                          1              2                                ⁢                      1            c6                                              (        63        )             The distance between mirror one and two was:                                                         d1              :=                              xe2x80x83                            ⁢                                                -                                      1                    4                                                  ⁢                                  (                                      4                    ⁢                                          (                                                                        -                          d2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          +                                                  d4                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c4                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      -                                    d2                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                              xe2x80x83                                                ⁢                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          )                                            -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                    )                                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                                                                                                                                                                            xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                                                                            xe2x80x83                                                    ⁢                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    )                                                )                                            +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    )                                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            +                                                                                          xe2x80x83                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                                                                                                        xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                                                xe2x80x83                                                ⁢                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                )                                            )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            ⁢                              xe2x80x83                            ⁢              angle              ⁢                              xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                                                                                                                                              xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    +                                  1                                                                )                                                                                                              )                                                                    -                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                            +                                                                                                            xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                            -                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                              +                                                                                                                                          xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                              -                                                                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                                                                                                                                                                                                                                                                                                            xe2x80x83                                        ⁢                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        )                                    +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            +                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                                                                                                                                                                      xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            -                                                                                                                                xe2x80x83                                    ⁢                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                                                                                                                                                                                                                                                                                          xe2x80x83                                                ⁢                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                )                                            )                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            /                              (                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                                                                                                                          xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                                                                                                                                                                                              xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                                      )                            +                                                                                                                                                      xe2x80x83                                        ⁢                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    )                                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            )                                                          (        64        )             And the distance between the object and the first mirror:                                                         d0              :=                              xe2x80x83                            ⁢                                                -                                      1                    2                                                  ⁢                                  (                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                                                                                                                                              xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                  +                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                                      )                                                              +                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            ⁢                              xe2x80x83                            ⁢              angle                                                                                          xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    +                                              d4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c4                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                -                                  d2                                                                +                                                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                            ⁢                                                                        d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          xe2x80x83                                                        ⁢                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        +                    1                                    )                                            -                                                                                          xe2x80x83                            ⁢                              angle                (                                                      -                    d2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                +                        1                                            )                                                        +                                      d4                    (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    +                                                                                                                                                                                                                                                                        xe2x80x83                                        ⁢                                                                  d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    -                  1                                )                            )                                                                                          xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                                                                                                                                              xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                  +                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                                      )                                                              +                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                                                                                                                                                                                      xe2x80x83                                        ⁢                                                                  d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                              +                                          d4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c4                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          -                                d2                                                            +                                                                                                                                                                                                                                                                                                                              xe2x80x83                                        ⁢                                                                  d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                            )                                                                                          xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            ⁢                                              xe2x80x83                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                                                                                                                                              xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    ⁢                                                      xe2x80x83                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                  +                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                                      )                                                              +                                                                                                                                                                                xe2x80x83                                    ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                                                                                                                                                                                      xe2x80x83                                        ⁢                                                                  d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            +                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              +                                                                                                                                          xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                      xe2x80x83                                            ⁢                                                                        2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                )                                                              )                                    +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            )                                                                                          xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    +                                              d4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c4                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                -                                  d2                                                                +                                                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                            ⁢                                                                        d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                                                                                                            xe2x80x83                                ⁢                                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            d2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                                                +                                1                                                            )                                                                                                      )                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        ⁢                                          xe2x80x83                                        +                    1                                    )                                            -                                                                                          xe2x80x83                            ⁢                              2                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              +                                                                                                                                          xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                                                                                                          xe2x80x83                                                        ⁢                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                            )                                                                                )                                                +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                        )                                    ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c6                                )                            /                                                                                          xe2x80x83                            ⁢                              (                                  angle                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          -                                              (                                                                              -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c5                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                            +                                                              d3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                                                                    -                                        2                                                                            ⁢                                                                              xe2x80x83                                                                            ⁢                                      c3                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                                                              (                                                                                  1                                          -                                                                                      2                                            ⁢                                                                                          xe2x80x83                                                                                        ⁢                                            d2                                            ⁢                                                                                          xe2x80x83                                                                                        ⁢                                            c2                                                                                                                          )                                                                                                              +                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                  xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                                                xe2x80x83                                                ⁢                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    )                                                                    )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            +                                                                                          xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              c3                                ⁡                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                                      )                                                  -                                                                                                        xe2x80x83                            ⁢                                                                    2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                                                                                                        xe2x80x83                            ⁢                              (                                                      -                    d2                                    ⁢                                      xe2x80x83                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                +                        1                                            )                                                        +                                      d4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                                                                                                                                                                                                                                                                                          xe2x80x83                                            ⁢                                                                        d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                                                                                          xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    +                                              d4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c4                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                -                                  d2                                                                +                                                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                            ⁢                                                                        d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                                                                                          xe2x80x83                            ⁢                                                                    2                    ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        -                          d2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                                              )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                                                                                                        xe2x80x83                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                                                                                                        xe2x80x83                            ⁢                              d4                (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                                              )                                                        +                                                                                                                                                                                                            xe2x80x83                                            ⁢                                                                                                    2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              )                                    )                                ⁢                c1                            )                                                          (        65        )             And the distance between mirror six and the image surface:                                                         d6              :=                              xe2x80x83                            ⁢                                                -                                      1                    4                                                  ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    angle                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                    xe2x80x83                                ⁢                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c2                                )                            +                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                                                xe2x80x83                                                ⁢                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                )                                            )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                2                            ⁢                              xe2x80x83                            ⁢              c6              ⁢                              xe2x80x83                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                                                                                                                                              xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    +                                  1                                                                )                                                                                                              )                                                                    -                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                            +                                                                                                            xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                angle                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                                                                                                                                                                                                                                                                                              xe2x80x83                                        ⁢                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        )                                    +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            +                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        xe2x80x83                                                        ⁢                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                        )                                                    +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            ⁢                              xe2x80x83                            ⁢              c6              ⁢                              xe2x80x83                            ⁢                              (                                                      -                    d2                                    +                                                                                                                        xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    +                                  1                                                                )                                                                                                              )                                                                    -                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                                                                                                                                                                                                                                                                                                            xe2x80x83                                        ⁢                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        )                                    +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            +                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        xe2x80x83                                                        ⁢                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                        )                                                    +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            +                                                                                          xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  1                                  -                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c2                                                                                                  )                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                                      )                                                  -                                                                                                                          xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            +                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c5                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                                                                                                                                                    xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                                                                                                                                                                                              xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            -                                                                                                                                xe2x80x83                                    ⁢                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                                                                                                                                                                                                                                                                xe2x80x83                                            ⁢                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                            )                                        )                                    -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                                                                                                        xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                                                                                                                                                                                              xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            -                                                                                                                                xe2x80x83                                    ⁢                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    )                                )                            ⁢                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    +                                              d4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c4                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                -                                  d2                                                                +                                                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                            ⁢                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    )                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                                                                                                                                                                                                                                                                                                xe2x80x83                                            ⁢                      1                      )                                        )                                    +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                            -                              angle                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c5                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                                                                                                                                                                      xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            -                                                                                                                                xe2x80x83                                    ⁢                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                                              )                                            -                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                2                            ⁢                              xe2x80x83                            ⁢                              (                                                      -                    d2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                +                        1                                            )                                                        +                                                                                                                                          xe2x80x83                                ⁢                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    +                                  1                                                                )                                                                                                              )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                                  )                            +                                                                                          xe2x80x83                            ⁢                              4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c5                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                                                                                                                                                    xe2x80x83                            ⁢                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                      d3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                    2                                                                    ⁢                                                                      xe2x80x83                                                                    ⁢                                  c3                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                          1                                      -                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        d2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        c2                                                                                                              )                                                                                                  +                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                                      )                                                              +                                                                                                                                                                                xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                                                                                                                                            xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            -                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            +                                                                                                            xe2x80x83                                ⁢                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c2                                )                            ⁢                              xe2x80x83                            ⁢                              (                                                      -                    d2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                +                        1                                            )                                                        +                                      d4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                                ⁢                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          )                                            -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                    )                                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            -                                                                                          xe2x80x83                            ⁢                              4                ⁢                                  xe2x80x83                                ⁢                c1                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c5                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        -                          d2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                                            +                              1                                                        )                                                                          +                                                  d4                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c4                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      -                                    d2                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                            ⁢                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    )                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                                                                                                                                                                                      xe2x80x83                                        ⁢                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                              )                                    +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                                                                                                        xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                                                                                                                                                                                              xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            -                                                                                                                                                      xe2x80x83                                        ⁢                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        )                                    )                                ⁢                                  xe2x80x83                                ⁢                magn                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                c6                            +                              4                ⁢                                  xe2x80x83                                ⁢                                  magn                  2                                ⁢                                  xe2x80x83                                ⁢                yob                ⁢                                  xe2x80x83                                ⁢                                  c6                  2                                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                                                                                                                          xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                                                                                                                                                                                              xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            -                                                                                                                                                      xe2x80x83                                        ⁢                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        )                                    )                                )                            /                              (                                  (                                      -                                          (                                                                        -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c5                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                                                            -                                      2                                                                        ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          (                                                                              1                                        -                                                                                  2                                          ⁢                                                                                      xe2x80x83                                                                                    ⁢                                          d2                                          ⁢                                                                                      xe2x80x83                                                                                    ⁢                                          c2                                                                                                                    )                                                                                                        +                                                                                                                                                                                                                                                                                                                                                  xe2x80x83                                ⁢                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c2                                )                            +                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                                                xe2x80x83                                                ⁢                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                )                                            )                                        +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        +                                                                                                                                                            xe2x80x83                                ⁢                                  d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  )                            -                              2                ⁢                                  xe2x80x83                                ⁢                c3                ⁢                                  xe2x80x83                                ⁢                                  (                                      1                    -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                            +                                                                                                            xe2x80x83                                ⁢                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c2                                )                            ⁢                              (                                                      -                    d2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                +                        1                                            )                                                        +                                      d4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                                                                                                                                                                                                                                                                                          xe2x80x83                                            ⁢                                              d3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        +                            1                                                    )                                                                    )                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                              (                                                      -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c5                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        d2                                            +                                                                                                                                                              xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c4                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    -                              d2                                                        +                                                          d3                              ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                      2                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    c3                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    d2                                                                    +                                  1                                                                )                                                                                                              )                                                                    -                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        -                    1                                    )                                )                            +                              2                ⁢                                  xe2x80x83                                ⁢                c4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      d2                                        +                                          d3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                                              )                                            +                                                                                                            xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                      d3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                                                                                                                                                                      xe2x80x83                                ⁢                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c2                                )                            +                              d4                ⁢                                  xe2x80x83                                ⁢                                  (                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c4                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                +                                                  d3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                                                            -                                  2                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                c3                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                            +                                                                                                                                                                                                                                                                                                                                                                                                            xe2x80x83                                                    ⁢                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    )                                                )                                            +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    )                                )                            ⁢                              (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                    +                                                                                                                        xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                  d4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                    +                                                                                                                                                                                                            xe2x80x83                            ⁢                                                d3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              )                                                  +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  -                                                                                                                                                                    xe2x80x83                                        ⁢                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                        )                                    )                                ⁢                                  xe2x80x83                                ⁢                                  c6                  2                                ⁢                                  xe2x80x83                                ⁢                angle                            )                                                          (        66        )             4-Mirror System, Stop on mirror 2 Yet again, we can use these derivations to solve the variables of a four-mirror system with the stop position on mirror two. As usual, the first solve is pupil in infinity or telecentricity in the image. The angle of the ray going through the optical axis in the stop surface should be zero in the image. The matrix from the stop surface to the third mirror generated by multiplication of the appropriate M matrices derived above, is given by:                     [                                            1                                      d2                                                                                            -                  2                                ⁢                                  xe2x80x83                                ⁢                c3                                                                                                          -                    2                                    ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                  d2                                -                1                                                    ]                            (        67        )             The matrix from the second mirror surface to the stop surface is given by:                     [                                            1                                      0                                                                                            -                  2                                ⁢                                  xe2x80x83                                ⁢                c2                                                                    -                1                                                    ]                            (        68        )             So as derived above, the distance between mirror 3 and 4, solved in the appropriate matrix and the vector of distances is:                     d3        :=                              -                          1              2                                -                      2            ⁢                          xe2x80x83                        ⁢            c4            ⁢                          xe2x80x83                        ⁢            d2                    +                      2            ⁢                          xe2x80x83                        ⁢            c3            ⁢                          xe2x80x83                        ⁢            d2                    +                      1                          c4              ⁢                              xe2x80x83                            ⁢                              (                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                )                                                                        (        69        )             The distance between mirror one and two was:                     d1        :=                              -                          1              4                                -                      2            ⁢                          xe2x80x83                        ⁢            magn            ⁢                          xe2x80x83                        ⁢            yob            ⁢                          xe2x80x83                        ⁢            c4                    -                                    angle              ⁢                              xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    -                  1                                )                                                    c1              ⁢                              xe2x80x83                            ⁢              magn              ⁢                              xe2x80x83                            ⁢              yob              ⁢                              xe2x80x83                            ⁢              c4                                                          (        70        )             And the distance between the object and the first mirror:                     d0        :=                              1            2                    ⁢                                                                                                                -                                              angle                        ⁡                                                  (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                              c3                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                                                        ⁢                                                          xe2x80x83                                                        -                            1                                                    )                                                                                      -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    ⁢                                                      xe2x80x83                                                    -                          1                                                )                                                              -                                                                                                                    2                    ⁢                                          xe2x80x83                                        ⁢                    magn                    ⁢                                          xe2x80x83                                        ⁢                    yob                    ⁢                                          xe2x80x83                                        ⁢                    c4                                                                                      angle              ⁢                              xe2x80x83                            ⁢              c1              ⁢                              xe2x80x83                            ⁢                              (                                                                            -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    ⁢                                      xe2x80x83                                    -                  1                                )                                                                        (        71        )             And the distance between mirror four and the image surface:                                                         d4              :=                              xe2x80x83                            ⁢                                                -                                      1                    4                                                  ⁢                                  xe2x80x83                                ⁢                                  (                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      angle                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    -                          1                                                )                                                              +                                          4                      ⁢                                              xe2x80x83                                            ⁢                      angle                      ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                      d2                                        +                                                                                                                                          xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  angle                  ⁢                                      xe2x80x83                                    ⁢                  c3                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            -                      1                                        )                                                  +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  angle                  ⁢                                      xe2x80x83                                    ⁢                                                            (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                -                        1                                            )                                        2                                    ⁢                                      xe2x80x83                                    ⁢                  c2                                +                                                                                                                          xe2x80x83                                ⁢                                                                            4                      ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    -                          1                                                )                                            ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c4                                        +                                          4                      ⁢                                              xe2x80x83                                            ⁢                                              magn                        2                                            ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                                              c4                        2                                                                              )                                            /                                                                                          xe2x80x83                            ⁢                              (                                                      c4                    2                                    ⁢                                      xe2x80x83                                    ⁢                  angle                                )                                                                        (        72        )             4-Mirror System, Stop on Mirror 3 Again, we use the original derivations to solve the variables of a four-mirror system with the stop position on mirror three. The first solve is pupil in infinity or telecentricity in the image. The angle of the ray going through the optical axis in the stop surface should be zero in the image. The matrix from the stop surface to the third mirror generated by multiplication of the appropriate M matrices derived above is given by:                     "AutoLeftMatch"                  [                                                    1                                            0                                                                    0                                            1                                              ]                                    (        73        )             The matrix from the second mirror surface to the stop surface is given by:                     "AutoLeftMatch"                  [                                                                      1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                                                                -                  d2                                                                                                                                                -                      2                                        ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        +                                      2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                                                                                    2                    ⁢                                          xe2x80x83                                        ⁢                    c3                    ⁢                                          xe2x80x83                                        ⁢                    d2                                    +                  1                                                              ]                                    (        74        )             So as we derived the distance between mirror 3 and 4 is:                     d3        :=                              -                          1              2                                ⁢                      1            c4                                              (        75        )             And we solve this new found value in the appropriate matrix and the vector of distances. The distance between mirror one and two was:                     d1        :=                              -                          1              4                                ⁢                                                                                                                                        -                        4                                            ⁢                                              xe2x80x83                                            ⁢                      d2                      ⁢                                              xe2x80x83                                            ⁢                      c1                      ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c4                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c4                                        +                                                                                                                                                                  (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            ⁢                                              xe2x80x83                                            ⁢                      angle                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                              +                                                                                                                    angle                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                                                                                          (                                  1                  -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    d2                    ⁢                                          xe2x80x83                                        ⁢                    c2                                                  )                            ⁢                              xe2x80x83                            ⁢              c1              ⁢                              xe2x80x83                            ⁢              magn              ⁢                              xe2x80x83                            ⁢              yob              ⁢                              xe2x80x83                            ⁢              c4                                                          (        76        )             And the distance between the object and the first mirror:                                                         d0              :=                              xe2x80x83                            ⁢                                                -                                      1                    2                                                  ⁢                                  (                                                                                    (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            ⁢                                              xe2x80x83                                            ⁢                      angle                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                              +                                                                                                                                          xe2x80x83                            ⁢                                                angle                  ⁢                                      xe2x80x83                                    ⁢                  d2                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                  +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  d2                                            ⁢                              xe2x80x83                                                                                                        xe2x80x83                            ⁢                                                (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      c3                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      c2                                                        )                                +                                  2                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                                                                                                                                            xe2x80x83                                ⁢                                                                            (                                                                        2                          ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                                                +                        1                                            )                                        -                                          2                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            ⁢                                              xe2x80x83                                            ⁢                      magn                      ⁢                                              xe2x80x83                                            ⁢                      yob                      ⁢                                              xe2x80x83                                            ⁢                      c4                                                        )                                            /                                                                                          xe2x80x83                            ⁢                              (                                  angle                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            (                                                                                                                    -                                2                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              c3                                ⁡                                                                  (                                                                      1                                    -                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      d2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      c2                                                                                                        )                                                                                                                      +                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                                                                                                                                                                                                    xe2x80x83                                    ⁢                                                                                    (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                              )                                                  ⁢                c1                            )                                                          (        77        )             And the distance between mirror four and the image surface:                                                         d4              :=                              xe2x80x83                            ⁢                                                -                                      1                    4                                                  ⁢                                  (                                                                                    -                        2                                            ⁢                                              xe2x80x83                                            ⁢                      angle                      ⁢                                              xe2x80x83                                            ⁢                                                                        (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                2                                            ⁢                      c4                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                                              -                                                                                                                                          xe2x80x83                            ⁢                                                2                  ⁢                                      xe2x80x83                                    ⁢                  angle                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              xe2x80x83                                            ⁢                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  c4                  ⁢                                      xe2x80x83                                    ⁢                                      d2                    ⁡                                          (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                angle                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                                  +                                                                                                        xe2x80x83                            ⁢                                                angle                  ⁢                                      xe2x80x83                                    ⁢                                                            (                                                                                                    -                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                          c3                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          1                              -                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                d2                                ⁢                                                                  xe2x80x83                                                                ⁢                                c2                                                                                      )                                                                          +                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                        2                                    ⁢                  d2                                -                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                                            -                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      1                            -                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              d2                              ⁢                                                              xe2x80x83                                                            ⁢                              c2                                                                                )                                                                    +                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  d2                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c4                                -                                                                                                        xe2x80x83                            ⁢                                                4                  ⁢                                      xe2x80x83                                    ⁢                  c1                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  2                        ⁢                                                  xe2x80x83                                                ⁢                        c3                        ⁢                                                  xe2x80x83                                                ⁢                        d2                                            +                      1                                        )                                    ⁢                                      xe2x80x83                                    ⁢                                      (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  magn                  ⁢                                      xe2x80x83                                    ⁢                  yob                  ⁢                                      xe2x80x83                                    ⁢                  c4                                +                                                                                                                          xe2x80x83                                ⁢                                                      4                    ⁢                                          xe2x80x83                                        ⁢                                          magn                      2                                        ⁢                                          xe2x80x83                                        ⁢                    yob                    ⁢                                          xe2x80x83                                        ⁢                                          c4                      2                                        ⁢                                          xe2x80x83                                        ⁢                                          (                                              1                        -                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          d2                          ⁢                                                      xe2x80x83                                                    ⁢                          c2                                                                    )                                                        )                                            /                                                                                          xe2x80x83                            ⁢                              (                                  (                                                                                    (                                                                                                            -                              2                                                        ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              1                                -                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  d2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  c2                                                                                            )                                                                                +                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                            ⁢                      d2                                        +                                                                  (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            c3                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                                                    +                          1                                                )                                            ⁢                                              (                                                  1                          -                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            d2                            ⁢                                                          xe2x80x83                                                        ⁢                            c2                                                                          )                                                                              )                                                                                                                        xe2x80x83                            ⁢                                                                    (                                          1                      -                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        d2                        ⁢                                                  xe2x80x83                                                ⁢                        c2                                                              )                                    ⁢                                      c4                    2                                    ⁢                                      xe2x80x83                                    ⁢                  angle                                )                                                                        (        78        )             Obscuration A particular problem in designing mirror systems, not encountered in refractive lens systems, is ensuring that the beam is not obscured in its zigzag course by other mirrors. Because of the necessary zigzag path, as the projection beam proceeds between successive mirrors I and I+1 on the optical path in many cases it will pass by at least one other mirror J. Thus, for an optical system not to be obscured it is necessary to ensure that the position and extent of the intervening mirror J is such that it does not intersect any part of the beam between mirrors I and I+1. This is shown in FIG. 4 in which it can be seen that mirror lies wholly below the beam between I and I+1 whereas mirror Jxe2x80x2 partially intersects the beam. The arrangement of mirror is not permitted. In a model of a potential projection system, obscuration can be detected by the following procedure: 1. For each pair of successive mirrors I, I+1 on the optical path, check if there exists a 20 mirror J (with J not equal to I, I+1) having a position on the optical axis (Z axis) between I and I+1 2. If J exists, calculate the distance from the optical axis (Y position) of the extreme rays from I to I+1 at the position of mirror j on the optical axis. 3. Check that the top and bottom of mirror J are both above (i.e. have greater Y position) or both below (i.e. have smaller Y position) both the extreme rays from I to I+1. If the check in (3) fails, then mirror at least partially obscures the beam from I to I+1 and the mirror system must be modified or rejected. Preferred Four-mirror Systems FIG. 6 shows a mirror system in class 9(+) which can be used in the lithography apparatus of FIG. 1. In this class, the stop can be positioned on mirror 2 or 3; in the system of FIG. 6, the stop position is on surface 2. The ring field of this system is defined on the object side, between 114 and 118 arbitrary units, with a numerical aperture of 0.25 (0.05 on the object side). The magnification is 0.2 and an intermediate image is formed between mirrors 3 and 4. The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 1 below. The values found for the curvatures and thicknesses can be re-scaled using a scaling factor. If the thicknesses are multiplied by that factor, the curvatures should be divided by it, and vice versa. FIG. 7 also shows a mirror system in class 9(+). In this case the stop is on mirror 3 and the intermediate image is between mirrors 1 and 2. The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 2 below. FIG. 8 shows an example of a class 2(xe2x88x92) system having the stop located on the third mirror. From the object (mask MA, surface 0) all the rays go, with a negative angle (a zero angle is parallel to the optical axis), to the first convex mirror M1. The convex mirror M1 reflects the beam upward to a large concave mirror M2. The position of the second mirror M2 has to be above the beam between the object (mask MA) and mirror M1. The beam then goes under mirror M1 to the stop surface mirror M3. From the stop surface the beam is reflected to concave mirror M4. Mirror M4 takes care of a telecentric illumination of the image (surface 5). The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 3 below. A class 6(xe2x88x92) system, shown in FIG. 9, consists of two mirror pairs, in a symmetrical design. From the object (mask MA, surface 0) all the rays go, with a negative angle (a zero angle is parallel to the optical axis), to the first convex mirror M1. The object is illuminated as telecentric as possible, this being a requirement for lithography. The convex mirror M1 reflects the beam upward to a large concave mirror M2. The position of this mirror has to be above the beam between the object and mirror M1. So far this design resembles the class 2(xe2x88x92) design (shown in FIG. 8). The beam then goes over mirror M1 to the stop surface on mirror M3, limited by the top of mirror M1. From the stop surface M3 the beam is reflected to concave mirror M4. Mirror M4 takes care of a telecentric illumination of the image (surface 5). The ring field of this system is defined on the image side, between xe2x88x9222.8 and xe2x88x9223.8, resulting in a Strehl ratio of at least 0.972 with a numerical aperture of 0.15. The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 4 below. A class 9(xe2x88x92) system with a stop on the second surface is shown in FIG. 10. The ring field of this system is defined on the object side, between 114 and 118 arbitrary units, with a numerical aperture of 0.2 (0.05 on the object side). The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 5 below. FIG. 11 shows a system in class 10(xe2x88x92). The ring field of this system is defined on the object side, between 114 and 118 arbitrary units, with a numerical aperture of 0.2 (0.05 on the object side). The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 6 below. Preferred Six-mirror Systems All the six-mirror systems found to be feasible have, as they have a positive magnification, an intermediate image. FIG. 12 shows a six-mirror system in class 9(+) in which the stop can be positioned on mirror 2, 3, 4 and 5. The system has the intermediate image located between mirror 2 and five. The ring field of this system is defined on the object side, between 114 and 118 arbitrary units, with a numerical aperture of 0.24 (0.06 on the object side). The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 7 below. A class 37(+) six-mirror system can have the stop positioned on mirror 2, 3, 4 or 5 and has the intermediate image located between mirror 2 and five. The ring field of such a system is defined on the image side, between 27 and 30 arbitrary units, with a numerical aperture of 0.24. The system shown in FIG. 13 has the stop on surface 2. This system consists of a mirror pair near the object and four-mirrors grouped near the image. From the object (mask MA, surface 0) all the rays go, with a negative angle, to the first concave mirror M1. The concave mirror M1 reflects the beam downward to mirror M2 which is almost flat. The top of mirror M2 is restricted to be below the beam between the object and mirror M,. The beam between mirror M2 and M3 limits the bottom of the small mirror M4, while the beam between mirror M4 and M5 limits the top of mirror M4. Finally, the beam between the last mirror M6 and the image limits the top of mirror M5. The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 8 below. For comparison, FIG. 14 shows a class 37(+) six-mirror system with the stop on surface 5. The ring field of this system is defined on the image side, between 27 and 30 arbitrary units, with a numerical aperture of 0.24. The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 9 below. Preferred Eight-mirror System An eight-mirror system in class 165(+) with stop on surface 3 is shown in FIG. 15. The ring field of this system is defined on the object side, between 116 and 124 arbitrary units, with a numerical aperture of 0.24 (0.06 on the object side). The first order curvatures and thicknesses of this system, in arbitrary units, are given in Table 10 below. An eight mirror system in class 169(+) is shown in FIG. 16, the curvatures and thicknesses of its elements are shown in Table 11. This system has a ring field in the object side between 114 and 118 arbitrary units, a numerical aperture of 0.4, distortion  less than 2.9 nm and an rms wavefront error  less than 0.3xcex. An eight mirror system in class 181(+) is shown in FIG. 17, the curvatures and thicknesses of its elements are shown in Table 12. Again, the ring field on the object side is between 114 and 118 units and the numerical aperture is 0.4. However, the distortion is  less than 1.9 nm and the rms wavefront error  less than 0.5xcex. An eight mirror system in class 150(xe2x88x92) is shown in FIG. 18, the curvatures and thicknesses of its elements are shown in Table 13. This system provides a distortion  less than 2.6 nm and an rms wavefront error  less than 0.19xcex. An eight mirror system in class 182(xe2x88x92) is shown in FIG. 19, the curvatures and thicknesses of its elements are shown in Table 14. This system likewise has a ring field on the object side between 114 and 118 arbitrary units, a numerical aperture of 0.4, an rms wavefront error of  less than 1xcex and a distortion  less than 2.18 nm. While we have described above specific embodiments of the invention it will be appreciated that the invention may be practiced otherwise than described. The description is not intended to limit the invention.