Abstract:
A progressive power ophthalmic lens having a spherical distance viewing area, a progressive power portion with a decreasing radius of curvature along an umbilic meridian, and a spherical near viewing area. Power distribution functions are optimized for smooth transition along the principal directions of gaze. A statistically smoothed polynomial function of the general form Pm=b 1  +b 2  X+b 3  X 2  +. . . +b 9  X 8  defines the mean power at any point on the lens surface to regulate curvature variation. Thereby providing continuity of the visual field at the boundaries of the principal lens zones and creating an integrated visual field as the eye traverses the lens.

Description:
BACKGROUND OF THE INVENTION 
     This invention relates to eyeglass lenses. More particularly, the present invention relates to progressive lenses for the correction of presbyopia. 
     The original multifocal ophthalmic lenses intended for the correction of presbyopia were bifocals with the distance prescription in the carrier lens and the near prescription in a segment which occupied some or all of the lower portion of the lens. Such lenses have the disadvantage of an abrupt discontinuity in optical power at the boundary between distance and near viewing zones and are limited to providing correction for essentially two distances. Aesthetically, the line at the boundary is often considered unattractive. Since then, many multifocal surfaces with continuously variable surface power have been described all with varying degrees of the inevitable unwanted surface astigmatism. In 1959 Kanolt described a lens composed of four different analytic polynomial surfaces of unspecified origin which are continuous at their common boundaries (U.S. Pat. No. 2,878,721). Continuity extends at least through the first derivatives and preferably the second order derivatives. This is necessary to achieve continuity of the physical surface and power transition. In this early embodiment, there was no umbilical meridian and therefore astigmatism was present in the transitional zones between the distance and near viewing portions of the lens. 
     In 1972 Maitenaz disclosed a progressive power ophthalmic lens design which reduced the levels of surface astigmatism to tolerable levels (U.S. Pat. No. 3,687,528), thus making the concept of progressive addition lenses more practical. This lens was constructed around a central umbilicus whose surface could be defined by an infinite set of spheres of variable radius with their centers located on an involute and their surfaces tangential to the lens surface. Another characteristic of one preferred embodiment was that the surface not directly on the umbilicus meridian was described as a set of orthogonal conic sections whose eccentricity evolved gradually along the path of power transition in a way that was intended to reduce surface astigmatism and provide areas of useful width for viewing at all distances. These goals were substantially achieved, although the intermediate viewing zone was quite narrow. Successive generations of this original lens followed. 
     In 1986, Fueter et al. described a progressive surface (U.S. Pat. No. 4,606,622) which was intended to enhance binocular vision through a concept known as horizontal symmetrics. In this lens, off axis points on each lens corresponding to the simultaneous lines of sight of the left and right eyes are balanced in the sense that both eyes view approximately equal amounts of aberration in terms of mean power and astigmatism errors in lateral gaze. This has the effect of improving orthoscopy by reducing vertical and horizontal prism imbalance. 
     The concept of mathematically optimizing smoothness was introduced by Winthrop in his 1989 invention (U.S. Pat. No. 4,861,153). By variationally minimizing an auxiliary function closely related to the mean square gradient of surface curvature, the lens successfully achieved the lowest maximum astigmatism and the smoothest gradients of astigmatism and mean power prior to that time. The bipolar boundary conditions assured that the aberrations would be distributed over the entire area of the lens in a particularly smooth way. 
     The general framework for a progressive lens has become an ophthalmic lens with distance and near viewing areas which are essentially aberration free and are connected by a transitional corridor which is also essentially free of astigmatism. Adjacent to the transitional corridor are areas with varying amounts of unwanted astigmatism and mean power errors. Numerous progressive lenses have been proposed. Each such lens attempts to optimize the balance of the level of unwanted surface astigmatism, distribution of aberration on the surface, and the useful width of the various viewing zones. The specific distributions of mean power and astigmatism are the features which distinguish each particular progressive lens and make them unique. 
     SUMMARY OF THE INVENTION 
     Briefly stated, the present invention is a progressive power ophthalmic surface such that the distribution of mean power along a path which is parallel to the eye path in both lateral gaze and convergence for near objects corresponds to a statistically smoothed polynomial. The invention in a preferred form is a multifocal ophthalmic lens having a wide spherical distance viewing zone and an essentially aberration free near viewing zone which are connected by an asymmetric corridor which provides a continuously varying increase in focal power as a function of the addition. The corridor parallels an umbilicus which asymmetrically bisects the progressive zone and which is angled to correspond to the path of the eye during convergence and along which the variation in refractive power is determined by the specified power addition. The curvatures in the principal meridians are equal along the meridional umbilic so that the surface along the umbilical curve can be described by an infinite set of osculating spheres of decreasing radius of curvature. Power curves along sections parallel to the corridor are proportional to the refractive power increase along the umbilical curve. 
     The mean power at any point on the surface of a progressive power ophthalmic lens in accordance with the present invention is defined by the general expression Pm=(1/2) (n-1)/(1/r 1  +1/r 2 ) wherein r 1  and r 2  are the radii of surface curvature in the principal meridians and n is the index of refraction. More particularly, the functions of mean power are parallel to selected directions of gaze and correspond to a unique eighth order polynomial of the form P=b 1  +b 2  X+b 3  X 2  +. . . b 9  X 8  wherein b 1  through b 9  are non-zero coefficients. 
     It is an object of the present invention to provide an improved progressive power ophthalmic lens. 
     It is another object of the present invention to provide an improved progressive power ophthalmic lens having an optically smooth transition between the essentially spherical distance and near viewing areas of the lens and the asymmetric corridor which connects the distance and near viewing areas. 
     Other objects and advantages of the invention will become apparent from the drawings and specification. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention may be better understood and its numerous objects and advantages will become apparent to those skilled in the art by reference to the accompanying drawings wherein like reference numerals refer to like elements in the several figures and in which: 
     FIGS. 1A and 1B are views in vertical elevation and coordinate representation respectively, of a progressive power ophthalmic lens in accordance with the present invention; 
     FIGS. 2A through 2D are graphs showing the effect of increasing the order of a polynomial equation defining a progressive power ophthalmic lens having a 2.00 diopter progression; 
     FIGS. 3A through 3D are graphs showing the effect of increasing the order of a polynomial equation defining a progressive power ophthalmic lens having a 2.50 diopter progression; 
     FIGS. 4A, 4B, 4C and 4D depict, respectively, contours of constant surface astigmatism, contours of constant mean surface power, an isometric plot of surface astigmatism, and an isometric plot of surface mean power of a progressive power ophthalmic lens in accordance with the present invention having a 2.00 diopter progression; 
     FIGS. 5A through 5H are graphs showing the horizontal cross-sections of mean power taken every 5 mm along the X-axis of the lens of FIG. 4; 
     FIGS. 6A through 6M are graphs showing the mean power curves parallel to the eye path during convergence taken every 5 mm along the boundary between the distance and near portions of the lens of FIG. 4; 
     FIG. 7 is a graph showing the meridional power curves from 0 to 35 mm along the main corridor for a family of progressive power ophthalmic lenses in accordance with the present invention; 
     FIGS. 8A through 8C are graphs showing the astigmatism profiles of horizontal sections at 10, 20, and 30 mm into the progressive zone of the lens of FIG. 4; 
     FIGS. 9A, 9B, 9C and 9D depict, respectively, contours of constant surface astigmatism, contours of constant mean surface power, an isometric plot of surface astigmatism, and an isometric plot of surface mean power of a progressive power ophthalmic lens in accordance with the present invention having a 2.50 diopter progression; 
     FIGS. 10A through 10H are graphs showing the horizontal cross-sections of mean power taken every 5 mm along the X-axis of the lens of FIG. 9; 
     FIGS. 11A through 11M are graphs showing the mean power curves parallel to the eye path during convergence taken every 5 mm along the boundary between the distance and near portions of the lens of FIG. 9; 
     FIG. 12 is a graph showing the meridional power curves from 0 to 35 mm along the main corridor for a family of progressive power ophthalmic lenses in accordance with the present invention; and 
     FIGS. 13A through 13C are graphs showing the astigmatism profiles of horizontal sections at 10, 20, and 30 mm into the progressive zone of the lens of FIG. 9. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     A progressive addition ophthalmic lens 10 in accordance with this invention has a wide essentially spherical area for distance viewing 12 and a essentially spherical area for near viewing 14 connected by an umbilicus along which the principal curvatures are equal. This creates a corridor 16 between distance viewing and near viewing areas 12, 14 paralleling the umbilicus (FIG. 1A) in which the surface astigmatism is low. There is a gradual increase in surface power along the corridor 16 according to the reading addition. During convergence, the eye path coincides with the line of the corridor 16 which runs asymmetrically in a nasal direction between the distance and reading areas 12, 14. 
     In a conventional progressive lens spline functions are often used to create continuity in both surface height and power transition between the spherical reading and distance areas 14, 12 and the aspheric regions which border them. They have been an effective modeling tool in surface construction. However, the present invention utilizes another stochastic method which can more easily achieve smoothing over a relatively long distance. This method is somewhat similar in concept to an ordinary least squares fit in linear regression theory where ##EQU1## wherein n(x) is the normal probability distribution function, μ is the mean, σ 2  is the variance and x is an independent variable. 
     In the above equation, a change in u translates the distribution curve up and down the axis and the variance controls the width of the curve. In polynomial regression theory, f(x i ) is regarded as if it were the outcome of an experiment whose outcomes are normally distributed and is conducted at x═x i  where x is an independent variable of known value. The mean is now regarded as a function of the independent variable and the variance is held constant. Therefore, μ is defined as a polynomial of the form ##EQU2## 
     The coefficients a i  are found by determining the choice of the function which maximizes the probability of the n+1 observations by least squares methods. The resulting polynomial regression curve is unique and will depend on the order &#34;j&#34; chosen. The order must be sufficiently high so as to correspond accurately to required boundary conditions and other constraints. 
     In a first example, 11 points are specified as power requirements (FIGS. 2A through 2D). In FIG. 2A, a second order solution is found for a horizontal cross-section 20 mm down into the progressive zone of a 2.00 diopter progression. Clearly, the second order solution is inadequate to model the surface power requirements. In FIG. 2B, a fourth order solution more nearly reflects design requirements but still not closely enough. In FIG. 2C, the sixth order solution is almost adequate. Finally, in FIG. 2D, the corresponding eighth order solution conforms well to specified parameters. Going to even higher orders will produce no benefit and increases the possibility of oscillatory behavior. In a second example (FIGS. 3A through 3D) the power specifications and curves 20 mm down into the progressive portion of a 2.50 diopter addition are examined. The results of second example are qualitatively similar to the first. Only the eighth order solution produces a satisfactory model. 
     Given the general parameters and boundary conditions, it is possible to iteratively construct a very general power grid from a fairly small number of points consistent with these constraints. These points can serve as data points to be used in a curve fitting procedure to determine the equation for a power function along a chosen cross-section of the lens. It is most crucial to maintain smoothness of the power curves along the eye path to reduce the swim effects caused by distortion created by the lens surface. It is therefore desirable to select power functions along the principal paths of eye movement which are as smooth as possible and still consistent with the given constraints. This can be achieved by taking advantage of the statistical smoothing properties inherent in curve fitting by polynomial regression. 
     The mean power at any point on the surface 18 is defined by the general expression: 
     
         Pm=(1/2)(n-1)/(1/r.sub.1 +1/r.sub.2) 
    
     wherein r 1  and r 2  are the radii of surface curvature in the principal meridians (FIG. 1B) and n is the index of refraction. Therefore, the mean power is directly related to the curvature functions which can be used to completely describe a lens surface. More particularly, the functions of mean power are parallel to selected directions of gaze and correspond to a unique eighth order polynomial of the form: 
     
         P=b.sub.1 +b.sub.2 X+b.sub.3 X.sup.2 +. . . b.sub.9 X.sup.8 
    
     wherein b 1  through b 9  are non-zero coefficients and X is the ordinate value of the coordinate axis, as shown in FIG. 1B. 
     In a preferred embodiment, each power curve for each cross-section of the lens surface parallel to the eye path along the corridor is a statistically smoothed eighth order polynomial which has been optimized by polynomial regression. Thus, for convergence eye movements, the eye will always follow a path along which the rate of change of mean power is gradual enough to facilitate the appearance of an integrated visual field free of obvious boundaries. Similarly, lateral eye movements will follow along mean power curves which have been statistically smoothed. The directions involved in convergence eye movements and lateral gaze movements have been selected for polynomial regression because they are fundamental to optimal performance of the subject invention. 
     EXAMPLE 1 
     FIGS. 4A, 4B, 4C and 4D represent a lens in accordance with the present invention having a reading addition of 2.00 diopters. FIG. 4A gives the contours of constant surface astigmatism; FIG. 4B gives the contours of constant mean surface power; and FIGS. 4C and 4D provide three-dimensional views of the distribution of surface astigmatism and surface mean power. FIGS. 5A through 5H show horizontal cross-sections of mean power every 5 mm down the lens starting at the top of the progressive zone. The asymmetry becomes more evident as one looks farther down the progressive zone. As the eyes move in lateral gaze, they move parallel to changes in mean power which are gradual. It is also evident that the disparity in mean power between the two eyes is small. FIGS. 6A through 6G profile the mean power curves parallel to the eye path during convergence. These also exhibit smooth gradual undulations in mean power to facilitate easy adaptation and a visual field with no distinct boundaries. FIG. 7 displays the meridional power curves from 0 to 32 mm along the main corridor for a family of progressive lenses of differing addition power but similar characteristics. The levels of astigmatism also remain moderate, below 2.50 diopters, as seen in FIGS. 8A through 8C which show the astigmatism profiles of horizontal sections at 10, 20 and 30 mm into the progressive zone. Binocular imbalance in terms of astigmatic differences between the two eyes is also small. 
     EXAMPLE 2 
     FIGS. 9A, 9B, 9C and 9D represent a lens in accordance with the present invention having a reading addition of 2.50 diopters. FIG. 9A gives the contours of constant surface astigmatism; FIG. 9B gives the contours of constant mean surface power; and FIGS. 9C and 9D provide three-dimensional views of the distribution of surface astigmatism and surface mean power. FIGS. 10A through 10H show horizontal cross-sections of mean power every 5 mm down the lens starting at the top of the progressive zone. FIGS. 11A through 11M show the corresponding curves parallel to the eye path during convergence. These profiles characterize a lens which maintains the important properties of the previous example. Steps have been taken to insure the integrity of the reading area as can be seen in FIG. 12, the family of meridional curves for addition powers of 2.50 and above. Corresponding astigmatism plots of this surface are exhibited in FIGS. 13A through 13C for horizontal sections at 10, 20 and 30 mm into the progressive zone and levels remain moderate for a progressive lens with a 2.50 addition power. 
     When curvatures are converted to corresponding surface heights referenced to a plane tangent to the geometric center of the lens, Tables 1A and 1B and 2A and 2B contain the sagittal values which represent the lenses in examples 1 and 2 respectively. 
     While preferred embodiments have been shown and described, various modifications and substitutions may be made thereto without departing from the spirit and scope of the invention. Accordingly, it is to be understood that the present invention has been described by way of illustration and not limitation. 
     
         TABLE 1A  -  Y(mm)--&gt;  X(mm) -42.5 -40.0 -37.5 -35.0 -32.5 -30.0 -27.5 -25.0 -22.5 -20.0 -17.5 -15.0 -12.5 -10.0 -7.5 -5.0 -2.5 0.0  ----+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+  -42.5 18.589 17.424 16.344 15.345 14.424 13.580 12.810 12.112 11.486 10.928 10.439 10.018 9.6622 9.3724 9.1475 8.9872 8.8912 8.8592  -40.0 17.423 16.273 15.207 14.220 13.310 12.476 11.715 11.026 10.406 9.8550 9.3714 8.9542 8.6026 8.3159 8.0935 7.9349 7.8399 7.8082  -37.5 16.342 15.206 14.151 13.175 12.276 11.451 10.698 10.016 9.4026 8.8573 8.3786 7.9656 7.6175 7.3336 7.1134 6.9564 6.8623 6.8309  -35.0 15.341 14.218 13.174 12.209 11.319 10.502 9.7562 9.0805 8.4732 7.9330 7.4588 7.0496 6.7047 6.4234 6.2052 6.0496 5.9563 5.9253  -32.5 14.420 13.307 12.274 11.318 10.436 9.6261 8.8873 8.2176 7.6156 7.0801 6.6099 6.2042 5.8622 5.5833 5.3668 5.2125 5.1201 5.0893  -30.0 13.575 12.472 11.448 10.500 9.6252 8.8224 8.0897 7.4253 6.8281 6.2968 5.8303 5.4277 5.0883 4.8115 4.5967 4.4436 4.3518 4.3212  -27.5 12.804 11.710 10.694 9.7533 8.8854 8.0887 7.3614 6.7019 6.1090 5.5814 5.1182 4.7185 4.3814 4.1065 3.8932 3.7411 3.6499 3.6196  -25.0 12.105 11.020 10.011 9.0766 8.2148 7.4235 6.7010 6.0458 5.4568 4.9326 4.4723 4.0751 3.7402 3.4670 3.2550 3.1038 3.0132 2.9830  -22.5 11.478 10.399 9.3968 8.4684 7.6119 6.8253 6.1071 5.4558 4.8702 4.3490 3.8914 3.4964 3.1633 2.8916 2.6808 2.5305 2.4404 2.4104  -20.0 10.920 9.8473 8.8506 7.9273 7.0754 6.2930 5.5786 4.9308 4.3481 3.8296 3.3742 2.9812 2.6498 2.3795 2.1697 2.0201 1.9304 1.9006  -17.5 10.430 9.3627 8.3710 7.4522 6.6043 5.8256 5.1145 4.4696 3.8895 3.3733 2.9200 2.5287 2.1987 1.9296 1.7207 1.5717 1.4824 1.4527  -15.0 10.007 8.9447 7.9571 7.0421 6.1977 5.4222 4.7139 4.0714 3.4937 2.9794 2.5278 2.1380 1.8092 1.5410 1.3329 1.1845 1.0956 1.0660  -12.5 9.6507 8.5921 7.6081 6.6963 5.8548 5.0819 4.3759 3.7356 3.1597 2.6471 2.1969 1.8083 1.4807 1.2133 1.0058 0.8579 0.7692 0.7397  -10.0 9.3606 8.3050 7.3237 6.4144 5.5752 4.8043 4.1002 3.4616 2.8872 2.3759 1.9269 1.5393 1.2124 0.9457 0.7388 0.5912 0.5028 0.4733  -7.5 9.1387 8.0851 7.1056 6.1979 5.3600 4.5904 3.8875 3.2499 2.6763 2.1659 1.7175 1.3305 1.0042 0.7380 0.5314 0.3841 0.2958 0.2664  -5.0 8.9893 7.9366 6.9578 6.0506 5.2132 4.4439 3.7411 3.1034 2.5298 2.0191 1.5705 1.1833 0.8567 0.5903 0.3836 0.2363 0.1480 0.1186  -2.5 8.9176 7.8648 6.8857 5.9781 5.1401 4.3699 3.6661 3.0272 2.4521 1.9398 1.4895 1.1004 0.7722 0.5042 0.2963 0.1481 0.0593 0.0298  0.0 8.9286 7.8748 6.8945 5.9855 5.1459 4.3738 3.6677 3.0262 2.4481 1.9325 1.4785 1.0856 0.7534 0.4818 0.2707 0.1202 0.0300 0.0000  2.5 9.0260 7.9702 6.9878 6.0765 5.2343 4.4592 3.7497 3.1043 2.5217 2.0009 1.5422 1.1421 0.8035 0.5256 0.3089 0.1540 0.0610 0.0302  5.0 9.2117 8.1529 7.1674 6.2528 5.4069 4.6277 3.9136 3.2629 2.6743 2.1466 1.6752 1.2717 0.9243 0.6376 0.4129 0.2514 0.1543 0.1220  7.5 9.4859 8.4229 7.4331 6.5139 5.6631 4.8787 4.1586 3.5013 2.9051 2.3691 1.8523 1.4746 1.1165 0.8192 0.5846 0.4150 0.3125 0.2784  10.0 9.8477 8.7790 7.7834 6.8582 6.0011 5.2099 4.4825 3.8171 3.2121 2.6664 2.1790 1.7500 1.3802 1.0713 0.8260 0.6477 0.5394 0.5032  12.5 10.295 9.2192 8.2161 7.2833 6.4183 5.6188 4.8826 4.2079 3.5929 3.0364 2.5377 2.0969 1.7152 1.3948 1.1393 0.9527 0.8391 0.8010  15.0 10.827 9.7415 8.7291 7.7869 6.9123 6.1030 5.3567 4.6714 4.0455 3.4777 2.9575 2.5152 2.1223 1.7916 1.5771 1.3337 1.2157 1.1761  17.5 11.441 10.344 9.3206 8.3673 7.4816 6.6611 5.9035 5.2069 4.5696 3.9905 3.4692 3.0062 2.6034 2.2640 1.9924 1.7940 1.6731 1.6326  20.0 12.137 11.027 9.9899 9.0238 8.1256 7.2927 6.5230 5.8145 5.1658 4.5758 4.0444 3.5723 3.1615 2.8155 2.5389 2.3370 2.2143 2.1733  22.5 12.915 11.789 10.737 9.7568 8.8448 7.9987 7.2164 6.4959 5.8361 5.2360 4.6957 4.2162 3.7996 3.4492 3.1696 2.9657 2.8420 2.8006  25.0 13.775 12.632 11.564 10.568 9.6409 8.7809 7.9856 7.2533 6.5828 5.9736 5.4259 4.9407 4.5202 4.1675 3.8867 3.6824 3.5582 3.5166  27.5 14.720 13.557 12.472 11.459 10.516 9.6417 8.8333 8.0893 7.4089 6.7915 6.2375 5.7480 5.3251 4.9716 4.6911 4.4873 4.3636 4.3220  30.0 15.752 14.569 13.463 12.433 11.474 10.584 9.7626 9.0072 8.3171 7.6921 7.1326 6.6397 6.2153 5.8618 5.5823 5.3797 5.2568 5.2152  32.5 16.875 15.669 14.543 13.493 12.517 11.612 10.777 10.010 9.3105 8.6783 8.1236 7.6177 7.1919 6.8386 6.5600 6.3586 6.2364 6.1945  35.0 18.093 16.863 15.715 14.645 13.650 12.729 11.880 11.101 10.392 9.7527 9.1529 8.6836 8.2564 7.9027 7.6246 7.4239 7.3020 7.2596  37.5 19.412 18.156 16.983 15.891 14.878 13.940 13.076 12.285 11.566 10.919 10.343 9.8401 9.4107 9.0561 8.7777 8.5769 8.4547 8.4116  40.0 20.838 19.553 18.354 17.239 16.205 15.249 14.370 13.565 12.836 12.180 11.598 11.090 10.658 10.301 10.022 9.8201 9.6973 9.6530  42.5 22.378 21.061 19.834 18.694 17.637 16.662 15.766 14.947 14.206 13.540 12.951 12.438 12.001 11.642 11.360 11.157 11.033 10.987 
    
     
         TABLE 1B  -  Y(mm)--&gt; X(mm) 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5  ----+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+ ------+  -42.5 8.8592 8.8912 9.9872 9.1475 9.3724 9.6622 10.018 10.439 10.928 11.486 12.112 12.810 13.580 14.424 15.345 16.344 17.424 18.589  -40.0 7.8082 7.8399 7.9349 8.0935 8.3259 8.6026 8.9542 9.3714 9.8550 10.406 11.026 11.715 12.476 13.310 14.220 15.207 16.273 17.423  -37.5 6.8309 6.8623 6.9564 7.1134 7.3336 7.6175 7.9656 8.3786 8.8573 9.4026 10.016 10.698 11.451 12.276 13.175 14.151 15.206 16.342  -35.0 5.9253 5.9563 6.0496 6.2052 6.4234 6.7047 7.0496 7.4588 7.9330 8.4732 9.0805 9.7562 10.502 11.319 12.209 13.174 14.218 15.341  -32.5 5.0893 5.1201 5.2125 5.3668 5.5833 5.8622 6.2042 6.6099 7.0801 7.6156 8.2176 8.8873 9.6261 10.436 11.318 12.274 13.307 14.420  -30.0 4.3212 4.3518 4.4436 4.5967 4.8115 5.0883 5.4277 5.8303 6.2968 6.8281 7.4253 8.0997 8.8224 9.6252 10.500 11.448 12.472 13.575  -27.5 3.6196 3.6499 3.7411 3.8932 4.1065 4.3814 4.7185 5.1182 5.5814 6.1090 6.7019 7.3614 8.0887 8.8854 9.7533 10.694 11.710 12.804  -25.0 2.9830 3.0132 3.1038 3.2550 3.4670 3.7402 4.0751 4.4723 4.9326 5.4568 6.0458 6.7010 7.4235 8.2148 9.0766 10.011 11.020 12.105  -22.5 2.4104 2.4404 2.5305 2.6808 2.8916 3.1633 3.4964 3.8914 4.3490 4.8702 5.4558 6.1071 6.8253 7.6119 8.4684 9.3968 10.399 11.478  -20.0 1.9006 1.9304 2.0201 2.1697 2.3795 2.6498 2.9812 3.3742 3.8296 4.3481 4.9308 5.5786 6.2930 7.0754 7.9273 8.8506 9.8473 10.920  -17.5 1.4527 1.4824 1.5717 1.7207 1.9296 2.1987 2.5287 2.9200 3.3733 3.8895 4.4696 5.1145 5.8256 6.6043 7.4522 8.3710 9.3627 10.430  -15.0 1.0660 1.0956 1.1845 1.3329 1.5410 1.8092 2.1380 2.5278 2.9794 3.4937 4.0714 4.7139 5.4222 6.1977 7.0421 7.9571 8.9447 10.007  -12.5 0.7397 0.7692 0.8579 1.0058 1.2133 1.4807 1.8083 2.1969 2.6471 3.1597 3.7356 4.3759 5.0819 5.8548 6.6963 7.6081 8.5921 9.6506  -10.0 0.4733 0.5028 0.5912 0.7388 0.9457 1.2124 1.5393 1.9269 2.3759 2.8871 3.4615 4.1001 4.8041 5.5749 6.4141 7.3233 8.3045 9.3599  -7.5 0.2664 0.2958 0.3841 0.5314 0.7379 1.0041 1.3303 1.7171 2.1652 2.6754 3.2486 3.8859 4.5884 5.3576 6.1949 7.1021 8.0811 9.1341  -5.0 0.1186 0.1480 0.2362 0.3833 0.5896 0.8553 1.1811 1.5673 2.0148 2.5242 3.0966 3.7329 4.4344 5.2024 6.0384 6.9442 7.9216 8.9728  -2.5 0.0298 0.0592 0.1476 0.2949 0.5012 0.7669 1.0924 1.4782 1.9252 2.4340 3.0057 3.6413 4.3421 5.1092 5.9444 6.8493 7.8257 8.8759  0.0 0.0000 0.0298 0.1191 0.2674 0.4747 0.7411 1.0669 1.4527 1.8993 2.4076 2.9786 3.6135 4.3135 5.0799 5.9143 6.8185 7.7943 8.8438  2.5 0.0302 0.0608 0.1521 0.3030 0.5129 0.7814 1.1087 1.4953 1.9420 2.4500 3.0203 3.6542 4.3532 5.1186 5.9521 6.8554 7.8305 8.8794  5.0 0.1220 0.1539 0.2486 0.4042 0.6189 0.8917 1.2223 1.6110 2.0587 2.5668 3.1366 3.7695 4.4672 5.2312 6.0634 6.9654 7.9395 8.9876  7.5 0.2784 0.3121 0.4116 0.5738 0.7958 1.0752 1.4111 1.8037 2.2538 2.7629 3.3326 3.9648 4.6612 5.4236 6.2541 7.1546 8.1272 9.1741  10.0 0.5032 0.5390 0.6441 0.8145 1.0455 1.3337 1.6771 2.0754 2.5295 3.0409 3.6116 4.2436 4.9391 5.7002 6.5291 7.4279 8.3969 9.4445  12.5 0.8010 0.8387 0.9496 1.1286 1.3696 1.6679 2.0203 2.4261 2.8858 3.4011 3.9741 4.6072 5.3028 6.0634 6.8912 7.7888 8.7585 9.8030  15.0 1.1761 1.2155 1.3313 1.5182 1.7691 2.0776 2.4399 2.8544 3.3212 3.8421 4.4191 5.0549 5.7522 6.5135 7.3415 8.2388 9.2081 10.252  17.5 1.6326 1.6730 1.7925 1.9858 2.2450 2.5631 2.9351 3.3587 3.8338 4.3617 4.9445 5.5849 6.2856 7.0495 7.8793 8.7780 9.7484 10.793  20.0 2.1733 2.2143 2.3361 2.5338 2.7994 3.1252 3.5058 3.9384 4.4222 4.9582 5.5484 6.1951 6.9013 7.6698 8.5035 9.4056 10.379 11.427  22.5 2.8006 2.8420 2.9650 3.1649 3.4343 3.7655 4.1530 4.5935 5.0860 5.6307 6.2293 6.8841 7.5976 8.3728 9.2127 10.120 11.099 12.152  25.0 3.5166 3.5580 3.6811 3.8812 4.1517 4.4859 4.8783 5.3255 5.8259 6.3795 6.9873 7.6513 8.3739 9.1578 10.006 10.922 11.908 12.969  27.5 4.3220 4.3629 4.4846 4.6832 4.9530 5.2881 5.6837 6.1362 6.6437 7.2059 7.8233 8.4974 9.2304 10.025 10.883 11.809 12.806 13.876  30.0 5.2152 5.2550 5.3746 5.5707 5.8387 6.1735 6.5710 7.0277 7.5416 8.1120 8.7391 9.4240 10.168 10.975 11.846 12.784 13.793 14.877  32.5 6.1945 6.2329 6.3503 6.5437 6.8094 7.1435 7.5422 8.0024 8.5222 9.1006 9.7374 10.433 11.190 12.010 12.895 13.847 14.871 15.970  35.0 7.2596 7.2966 7.4117 7.6027 7.8665 8.1999 8.5997 9.0632 9.5886 10.175 10.821 11.529 12.299 13.133 14.033 15.002 16.043 17.160  37.5 8.4116 8.4472 8.5604 8.7494 9.0118 9.3449 9.7460 10.213 10.744 11.338 11.995 12.714 13.498 14.347 15.264 16.252 17.312 18.449  40.0 9.6530 9.6872 9.7989 9.9865 10.248 10.582 10.985 11.455 11.992 12.595 13.262 13.994 14.793 15.658 16.594 17.601 18.683 19.843  42.5 10.987 11.020 11.131 11.318 11.579 11.914 12.319 12.794 13.338 13.949 14.627 25.373 16.187 17.071 18.026 19.055 20.161 21.347 
    
     
                                           TABLE 2A__________________________________________________________________________Y(mm)--&gt;X(mm)    -42.5   -40.0       -37.5           -35.0               -32.5                   -30.0                       -27.5                           -25.0                               -22.5__________________________________________________________________________-42.5    18.588   17.423       16.343           15.344               14.423                   13.579                       12.809                           12.112                               11.485-40.0    17.422   16.273       15.206           14.219               13.320                   12.475                       11.714                           11.025                               10.405-37.5    16.341   15.205       14.150           13.175               12.275                   11.450                       10.697                           10.035                               9.4020-35.0    15.341   14.217       13.174           12.208               11.318                   10.501                       9.7556                           9.0799                               8.4726-32.5    54.439   13.307       12.273           11.317               10.435                   9.6256                       6.8866                           8.2171                               7.6151-30.0    13.574   12.472       11.447           10.499               9.6247                   8.8219                       8.0892                           7.4249                               6.8277-27.5    12.803   11.710       10.694           9.7528               8.8850                   8.0883                       7.3609                           6.7015                               6.1086-25.0    12.305   11.019       10.011           9.0773               8.2144                   7.4231                       6.7006                           6.0455                               5.4564-22.5    11.477   10.399       9.3965           8.4681               7.6115                   6.8250                       6.1068                           5.4555                               4.8699-20.0    10.919   9.8470       8.8503           7.9270               7.0751                   6.2928                       5.5784                           4.9305                               4.3478-17.5    10.429   9.3625       8.3707           7.4519               6.6041                   5.8254                       5.1143                           4.4693                               3.8893-15.0    10.007   8.9444       7.9569           7.0419               6.1975                   5.4220                       4.7137                           4.0713                               3.4935-12.5    9.6505   8.5920       7.6079           6.6961               5.8546                   5.0817                       4.3758                           3.7355                               3.1596-10.0    9.3601   8.3046       7.3233           6.4141               5.5749                   4.8041                       4.1001                           3.4614                               2.8870-7.5    9.1371   8.0837       7.1043           6.1968               5.3591                   4.5897                       3.8869                           3.2493                               2.6759-5.0    8.9861   7.9338       6.9552           6.0484               5.2113                   4.4422                       3.7396                           3.1022                               2.5287-2.5    8.9142   7.8619       6.8833           5.9761               5.1364                   4.3685                       3.6649                           3.0263                               2.45140.0 8.9289   7.8758       6.8961           5.9877               5.1484                   4.3766                       3.6707                           3.0292                               2.45102.5 9.0363   7.9814       6.9998           6.0891               5.2472                   4.4723                       3.7626                           3.1167                               2.53325.0 9.2394   8.1817       7.1971           6.2829               5.4371                   4.6576                       3.9426                           3.2904                               2.69967.5 9.5387   8.4767       7.4874           6.5682               5.7168                   4.9309                       4.2086                           3.5481                               2.947910.0    9.9320   8.8637       7.8678           6.9415               6.0824                   5.2881                       4.5564                           3.8854                               3.273612.5    10.416   9.3392       8.3342           7.3985               6.5293                   5.7242                       4.9809                           4.2974                               3.672315.0    10.998   9.8993       8.8827           7.9350               7.0533                   6.2352                       5.4783                           4.7808                               4.141117.5    11.644   10.541       9.5102           8.5481               7.6518                   6.8189                       6.0470                           5.3343                               4.679720.0    12.382   11.263       10.215           9.2366               8.3240                   7.4748                       6.6870                           5.9590                               5.289622.5    13.202   12.064       10.998           10.001               9.0706                   8.2045                       7.4005                           6.6572                               5.973925.0    14.106   12.946       11.859           10.843               9.8940                   9.0104                       8.1902                           7.4324                               6.736227.5    15.094   13.911       12.802           11.765               10.797                   9.8961                       9.0600                           9.2882                               7.580230.0    16.171   14.963       13.831           12.772               11.784                   10.865                       10.014                           9.2285                               8.509532.5    17.341   16.106       14.949           13.868               12.860                   11.923                       11.056                           10.257                               9.527735.0    18.609   17.345       16.162           15.057               14.028                   13.073                       12.190                           11.378                               10.63837.5    19.982   18.687       17.476           16.346               15.295                   14.321                       13.421                           12.596                               11.84540.0    21.466   20.137       18.897           17.741               16.666                   15.671                       14.755                           13.915                               13.15342.5    23.070   21.705       20.432           19.247               18.148                   17.132                       16.197                           15.342                               14.566__________________________________________________________________________Y(mm)--&gt;X(mm)    -20.0   -17.5       -15.0           -12.5               -10.0                   -7.5                       -5.0                           -2.5                               0.0__________________________________________________________________________-42.5    10.928   10.439       10.017           9.6615               9.3717                   9.1468                       8.9865                           8.8905                               8.8585-40.0    9.8543   9.3707       8.9536           8.6020               8.3152                   8.0928                       7.9342                           7.8392                               7.8075-37.5    8.8566   8.3780       7.9650           7.6169               7.3330                   7.1128                       6.9558                           6.8617                               6.8303-35.0    7.9324   7.4582       7.0491           6.7042               6.4229                   6.2046                       6.0490                           5.9558                               5.9247-32.5    7.0796   6.6094       6.2037           5.8617               5.5828                   5.3663                       5.2120                           5.1196                               5.0887-30.0    6.2963   5.8298       5.4273           5.0879               4.8110                   4.5962                       4.4431                           4.3513                               4.3207-27.5    5.5810   5.1178       4.7181           4.3810               4.1061                   3.8928                       3.7407                           3.6495                               3.6192-25.0    4.9322   4.4720       4.0747           3.7398               3.4666                   3.2546                       3.1034                           3.0129                               2.9827-22.5    4.3487   3.8910       3.4961           3.1630               2.8913                   2.6805                       2.5302                           2.4401                               2.4101-20.0    3.8293   3.3740       2.9810           2.6496               2.3792                   2.1694                       2.0198                           1.9302                               1.9003-17.5    3.3731   2.9198       2.5285           2.1985               1.9293                   1.7204                       1.5715                           1.4822                               1.4525-15.0    2.9792   2.5276       2.1378           1.8091               1.5409                   1.3327                       1.1843                           1.0954                               1.0658-12.5    2.6470   2.1968       1.8082           1.4805               1.2132                   1.0057                       0.8577                           0.7691                               0.7395-10.0    2.3758   1.9268       1.5392           1.2123               0.9456                   0.7387                       0.5911                           0.5027                               0.4732-7.5    2.1655   1.7173       1.3304           1.0041               0.7379                   0.5313                       0.3840                           0.2957                               0.2663-5.0    2.0183   1.5699       1.1828           0.8564               0.5901                   0.3835                       0.2362                           0.1480                               0.1186-2.5    1.9392   1.4990       1.1001           0.7719               0.5040                   0.2962                       0.1480                           0.0593                               0.02980.0 1.9351   1.4808       1.0874           0.7547               0.4827                   0.2712                       0.1203                           0.0300                               0.00002.5 2.0113   1.5500       1.1492           0.8088               0.5292                   0.3110                       0.1549                           0.0613                               0.03025.0 2.1692   1.6985       1.2873           0.9360               0.6457                   0.4177                       0.2538                           0.1553                               0.12257.5 2.4068   1.9243       1.5004           1.1360               0.8328                   0.5932                       0.4198                           0.3151                               0.280210.0    2.7199   2.2238       1.7858           1.4071               1.0904                   0.8387                       0.6558                           0.5449                               0.507912.5    3.1045   2.5939       2.1412           1.7484               1.4188                   1.1562                       0.9651                           0.8490                               0.810115.0    3.5586   3.0333       2.5665           2.1608               1.8199                   1.5485                       1.3511                           1.2315                               1.191417.5    4.0825   3.5433       3.0638           2.6470               2.2971                   2.0191                       1.8175                           1.6957                               1.655020.0    4.6787   4.1271       3.6369           3.2133               2.8547                   2.5720                       2.3676                           2.2444                               2.203322.5    5.3507   4.7886       4.2898           3.8576               3.4963                   3.2105                       3.0042                           2.8798                               2.836425.0    6.1021   5.5313       5.0259           4.5892               4.2249                   3.9372                       3.7296                           3.6043                               3.562527.5    6.9366   6.3585       5.8481           5.4082               5.0422                   4.7536                       4.5454                           4.4197                               4.377630.0    7.8573   7.2729       6.7585           6.3163               5.9494                   5.6605                       5.4522                           5.3264                               5.284232.5    8.8673   8.2772       7.7591           7.3150               6.9473                   6.6584                       6.4502                           6.3244                               6.281935.0    9.9698   9.3740       8.8522           8.4060               8.0373                   7.7481                       7.5399                           7.4140                               7.370937.5    11.368   10.566       10.040           9.5915               9.2213                   8.9313                       8.7226                           8.5962                               8.552340.0    12.467   11.858       11.327           10.875               10.502                   10.211                       10.001                           9.8735                               9.928742.5    13.870   13.253       12.716           12.259               11.884                   11.590                       11.378                           11.249                               11.203__________________________________________________________________________ 
    
     
                                           TABLE 2B__________________________________________________________________________Y(mm)--&gt;X(mm)    0.0 2.5 5.0 7.5 10.0                   12.5                       15.0                           17.5                               20.0__________________________________________________________________________-42.5    8.8585   8.8905       8.9865           9.1468               9.3717                   9.6615                       10.017                           10.439                               10.928-40.0    7.8075   7.8392       7.9342           8.0928               8.3152                   8.6020                       8.9536                           9.3707                               9.8543-37.5    6.8303   6.8617       6.9558           7.1128               7.3330                   7.6169                       7.9650                           8.3780                               8.8566-35.0    5.9247   5.9558       6.0490           6.2046               6.4229                   6.7042                       7.0491                           7.4582                               7.9324-32.5    5.0887   5.1196       5.2120           5.3663               5.5828                   5.8617                       6.2037                           6.6094                               7.0796-30.0    4.3207   4.3513       4.4431           4.5962               4.8110                   5.0879                       5.4273                           5.8298                               6.2963-27.5    3.6192   3.6495       3.7407           3.8928               4.1061                   4.3810                       4.7181                           5.1178                               5.5810-25.0    2.9827   3.0129       3.1034           3.2546               3.4666                   3.7398                       4.0747                           4.4720                               4.9322-22.5    2.4101   2.4401       2.5302           2.6805               2.8913                   3.1630                       3.4961                           3.8910                               4.3487-20.0    1.9003   1.9302       2.0198           2.1694               2.3792                   2.6496                       2.9810                           3.3740                               3.8293-17.5    1.4525   1.4822       1.5715           1.7204               1.9293                   2.1985                       2.5285                           2.9198                               3.3731-15.0    1.0658   1.0954       1.1843           1.3327               1.5409                   1.8091                       2.1378                           2.5276                               2.9792-12.5    0.7395   0.7691       0.8577           1.0057               1.2132                   1.4805                       1.8082                           2.1968                               2.6470-10.0    0.4732   0.5027       0.5911           0.7387               0.9456                   1.2123                       1.5392                           1.9268                               2.3758-7.5    0.2663   0.2957       0.3840           0.5313               0.7378                   1.0040                       1.3302                           1.7170                               2.1651-9.0    0.1186   0.1480       0.2361           0.3832               0.5895                   0.8553                       1.1810                           1.5673                               2.0147-2.5    0.0298   0.0592       0.1476           0.2948               0.5010                   0.7666                       1.0921                           1.4779                               1.92490.0 0.0000   0.0299       0.1191           0.2674               0.4746                   0.7407                       1.0663                           1.4520                               1.89842.5 0.0302   0.0610       0.1526           0.3039               0.5139                   0.7823                       1.1093                           1.4955                               1.94175.0 0.1225   0.1548       0.2506           0.4074               0.6232                   0.8966                       1.2272                           1.6155                               2.06247.5 0.2802   0.3146       0.4160           0.5808               0.8054                   1.0871                       1.4243                           1.8171                               2.266510.0    0.5079   0.5445       0.6522           0.8265               1.0622                   1.3548                       1.7017                           2.1020                               2.556312.5    0.8101   0.8487       0.9623           1.1460               1.3936                   1.6989                       2.0578                           2.4682                               2.930415.0    1.1914   1.2313       1.3494           1.5412               1.7999                   2.1180                       2.4900                           2.9126                               3.385117.5    1.6550   1.6956       1.8167           2.0145               2.2823                   2.6120                       2.9967                           3.4323                               3.916920.0    2.2033   2.2444       2.3672           2.5690               2.8435                   3.1824                       3.5784                           4.0264                               4.523722.5    2.8384   2.8798       3.0039           3.2081               3.4868                   3.8322                       4.2372                           4.6960                               5.205425.0    3.5625   3.6043       3.7290           3.9343               4.2151                   4.5645                       4.9758                           5.4434                               5.963727.5    4.3776   4.4194       4.5441           4.7493               5.0304                   5.3817                       5.7972                           6.2717                               6.801330.0    5.2842   5.3255       5.4494           5.6536               5.9340                   6.2859                       6.7041                           7.1838                               7.721432.5    6.2819   6.3224       6.4452           6.6478               6.9271                   7.2788                       7.6988                           8.1827                               8.727235.0    7.3709   7.4105       7.5318           7.7329               8.0109                   8.3624                       8.7838                           9.2715                               9.822437.5    8.5523   8.5909       8.7108           8.9105               9.1876                   9.5392                       9.9622                           10.454                               11.01140.0    9.8287   9.8661       8.9849           10.184               10.460                   10.812                       11.237                           11.733                               12.29642.5    11.203   11.240       11.358           11.556               11.832                   12.186                       12.613                           13.114                               13.684__________________________________________________________________________Y(mm)--&gt;X(mm)    22.5   25.0       27.5           30.0               32.5                   35.0                       37.5                           40.0                               42.5__________________________________________________________________________-42.5    11.485   12.112       12.809           13.579               14.423                   15.344                       16.343                           17.423                               18.588-40.0    10.405   11.025       11.714           12.475               13.310                   14.219                       15.206                           16.273                               17.422-37.5    9.4020   10.015       10.697           11.450               12.275                   13.275                       14.150                           15.205                               16.341-35.0    8.4726   9.0799       9.7556           10.501               11.318                   12.208                       13.174                           14.217                               15.341-32.5    7.6151   8.2171       8.8868           9.6256               10.435                   11.317                       12.273                           13.307                               14.419-30.0    6.8277   7.4249       8.0892           8.8219               9.6247                   10.499                       11.447                           12.472                               13.574-27.5    6.1086   6.7015       7.3609           8.0883               8.8850                   9.7528                       10.694                           11.710                               12.803-25.0    5.4564   6.0455       6.7006           7.4231               8.2144                   9.0763                       10.011                           11.019                               12.105-22.5    4.8699   5.4555       6.1068           6.8250               7.6115                   8.4681                       9.3965                           10.399                               11.477-20.0    4.3478   4.9305       5.5784           6.2926               7.0751                   7.9270                       8.8503                           9.8470                               10.919-17.5    3.8893   4.4693       5.1143           5.8254               6.6041                   7.4519                       8.3707                           9.3625                               10.429-15.0    3.4935   4.0713       4.7137           5.4220               6.1975                   7.0419                       7.9569                           8.9444                               10.007-12.5    3.1596   3.7355       4.3758           5.0817               5.8546                   6.6961                       7.6079                           8.5920                               9.6505-10.0    2.8870   3.4614       4.1000           4.8040               5.5748                   6.4140                       7.3232                           8.3044                               9.3597-7.5    2.6754   3.2486       3.8858           4.5884               5.3575                   6.1948                       7.1020                           8.0810                               9.1340-9.0    2.5242   3.0966       3.7329           4.4344               5.2024                   6.0384                       6.9441                           7.9216                               8.9728-2.5    2.4337   3.0855       3.6411           4.3419               5.1091                   5.9443                       6.8492                           7.8257                               8.87590.0 2.4066   2.9776       3.6125           4.3125               5.0790                   5.9136                       6.8179                           7.7938                               8.84342.5 2.4491   3.0190       3.6526           4.3513               5.1166                   5.9501                       6.8535                           7.8286                               8.87785.0 2.5693   3.1379       3.7696           4.4663               5.2295                   6.0609                       6.9626                           7.9363                               8.98437.5 2.7740   3.3427       3.9716           4.6657               5.4260                   6.2546                       7.1535                           8.1247                               9.170710.0    3.0665   3.6348       4.2636           4.9555               5.7130                   6.5383                       7.4337                           8.4018                               9.444912.5    3.4459   4.0171       4.6469           5.3381               6.0936                   6.9162                       7.8086                           8.7734                               9.813315.0    3.9088   4.4858       5.1193           5.8122               6.5679                   7.3895                       8.2800                           9.2425                               10.28017.5    4.4511   5.0369       5.6770           6.3747               7.1334                   7.9566                       8.8475                           9.8097                               10.84620.0    5.0700   5.6667       6.3163           7.0218               7.7867                   8.6146                       9.5090                           10.474                               11.51222.5    5.7643   6.3735       7.0347           7.7507               8.5249                   9.3607                       10.262                           11.232                               12.27525.0    6.5349   7.1570       7.8313           8.5600               9.3461                   10.193                       11.104                           12.083                               13.13527.5    7.3839   8.0189       8.7070           9.4498               10.250                   11.110                       12.035                           13.026                               14.08930.0    8.3144   8.9619       9.6640           10.422               11.237                   12.114                       13.054                           14.061                               15.13932.5    9.3299   9.9894       10.705           11.479               12.311                   13.205                       14.163                           15.188                               16.28535.0    10.434   11.105       11.835           12.625               13.475                   14.388                       15.365                           16.412                               17.53037.5    11.631   12.314       13.058           13.864               14.733                   15.666                       16.665                           17.735                               18.87740.0    12.926   13.620       14.379           15.202               16.090                   17.045                       18.068                           19.213                               20.33342.5    14.323   15.030       15.804           16.645               17.553                   18.531                       19.580                           20.703                               21.903__________________________________________________________________________