Abstract:
This invention includes the derivation of the exact mathematical expressions for the curvature, either cardiocle or expanded cardioid, of the casing of the springless eccentric rotor vane pump, thereby facilitating the precision manufacture of the curved surfaces of the casing using CNC techniques. As a result, the capacity and accuracy of the eccentric rotor vane pump is greatly improved. As the section manufacture and assembly of the casing becomes possible, the mass production of large-sized pumps of 1-meter or larger diameter is now attainable, hitherto regarded as almost impossible, and therefore production cost is also reduced. The unique design which positions the axis of eccentricity in the lower central part of the axis of rotor rotation results in increase in the rotation speed of the rotor, and leads to reduction of friction between the vane ends and the curved surface of the casing as the weight of the vane does not affect the movement of the rotor.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention describes the precision processing of curved surfaces of the cardiocle and expanded cardioid casing in springless eccentric rotor vane pumps. 
     2. Description of the Prior Art 
     In general, vanes used in eccentric rotor vane pumps are fitted with springs so that their length can vary in line with casing surfaces. However, the eccentric rotor vane pump discussed here has a solid vane of constant length. For this type of eccentric rotor vane pump, the key technology is the accuracy of the casing surface curvatures, to allow the edges of a sliding vane match the surface curves as closely as possible no matter what the rotation angle and the eccentricity of the rotor may be. 
     However, the exact mathematical descriptions which accurately represent the curves drawn by the movements of the vane edges in an eccentric rotor vane pump have not been found until now. Thus processing of curved casing surfaces has been possible only via the recopy method. This method has several significant weaknesses: (1) Curved surfaces have to be retraced and remodelled each time eccentricity or casing size needs to be changed. (2) Precision processing is not quite possible, especially for large-sized casings. (3) The entire surface of the casing has to be processed at one time. (4) The edges of scraping, sliding vanes make poor contact with casing surfaces. 
     Moreover, with this recopy method, the accuracy of casing surface processing varies with the eccentricity of the pump, the angle of rotation of the vane, and the distance the vane travels. As there have been no geometrical equations which exactly describe the curves drawn by the vane rotation, such advanced manufacturing techniques as CNC, and processing in sections, have not been available. The only possible manufacturing method was the recopy method, using a prototype curved action. 
     SUMMARY OF THE INVENTION 
     In this invention, however, the following equations (A) and (B), which represent the curves drawn by the movement of vanes of fixed length in eccentric rotor vane pumps, are derived on the basis of these curves always falling into two categories, cardiocle and expanded cardioid curves, regardless of rotor eccentricity and vane length:              P   =     2      a        {     1   +       (     R   -   r     )            sin                 θ       2      a         -           R   2     -         (     R   -   r     )     2                     cos   2        θ           2      a         }        for                                cardiocles             (   A   )                                            P   =     2      a        {     1   +       (     R   -   r     )            sin                 θ       2      a           }                   for                                expanded                 cardioids             (   B   )                                 
     Nomenclature in the equations will be discussed in detail later, in reference to FIGS. 1,  3 ,  5  and  6 . 
     These two equations represent in terms of analytic geometry the curved surfaces of eccentric rotor pump casings, and thereby alow the precision processing of casings using CNC techniques. As the equations do not depend on rotor eccentricity and vane length, casings of any size can be manufactured to the highest levels of accuracy current engineering technology permits; and even further, more processing in sections is now possible. 
     As a result, not only precision processing, but also mass production, of large-sized springless eccentric rotor vane pumps of 1-meter or larger diameter is now possible, thus making feasible the supply to customers of eccentric rotor vane pumps at more reasonable prices. 
     In other current eccentric rotor vane pumps, the center of eccentricity of the rotor is set at the upper section or sides of the casing center for better ventilation and smooth valve movement. But the movement of a vane causes friction with the casing surfaces, as the centrifugal force generated by the rotating vane is in the same direction as the gravitation force exerted on the rotor. Therefore the rotation speed of the rotor has to be kept low. However, the vane of the eccentric rotor vane pump being discussed here makes large-area contact with the casing surfaces when sliding on surfaces; and thus the center of eccentricity of the rotor can be placed in the lower section of the casing center. Additionally, the centrifugal force produced by the rotation of the vane is reduced by the weight of the vane. Therefore the rotation speed of the rotor can be sped up. 
     In particular, as shown in FIG. 10, existing thrust bearings may be used for the processing of large-sized casings of 1-meter or greater diameter, so that the rotor axis can be designed vertically, reducing gravitational pull due to the weight of the rotating vane and increasing operational life. 
     As the casing diameter increases, the weight of the vane increases and so, too, does the friction produced by the vane when sliding and scraping along the casing surface. For this reason the manufacture of large-sized eccentric rotor vane pumps was regarded as impractical in the past. 
     By positioning the rotor shaft vertically, it is possible to reduce the friction between the ends of the vane and the casing surface, and thus to increase the size of eccentric rotor pumps. Furthermore the mathematical descriptions of cardiocle and expanded cardioid curves derived and shown in this invention allows the implementation of CNC techniques in the manufacture of casings, and subsequent increase in casing surface accuracy. CNC processing makes possible both mass production and cost reduction. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a geometric representation of the movement of an eccentric rotor as contained in the invention referred to in this invention. 
     FIG. 2 compares a cardiocle with a simple cardioid. 
     FIG. 3 shows the operation of an eccentric rotor vane pump with a cardiocle casing. 
     FIG. 4 is the actual description of an eccentric rotor vane pump with a cardiocle casing. 
     FIG. 5 compares the curvatures of cardiocle and expanded cardioid casings. 
     FIG. 6 shows the relationship between the size of an eccentric rotor and an expanded cardioid. 
     FIG. 7 shows the operation of an eccentric rotor vane pump with an expanded cardioid casing. 
     FIG. 8 describes section processing of a pump casing using the methodology introduced in this invention. 
     FIG. 9 describes an eccentric rotor vane pump of horizontal design. 
     FIG. 10 describes an eccentric rotor vane pump of vertical design. 
     FIG. 11 displays the components of the eccentric rotor vane pump described in this invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The derivation of the two equations for cardiocles and expanded cardioids, in reference to the figures and in terms of analytic geometry, are shown below. 
     FIG. 1 shows a cross-section of an eccentric rotor pump in Cartesian coordinates, for geometric analysis of the casing surfaces of the pump. The surface of circular rotor {circle around ( 2 )} touches basic circle {circle around ( 1 )} at point internally Ĉ. When rotor {circle around ( 2 )} rotates anticlockwise by θ° around the axis of eccentricity, which goes through point Oe, vane {circle around ( 3 )}, which is inserted in rotor {circle around ( 2 )}, also rotates in the same direction as the vane, sliding and scraping along the casing surface. One end of vane {circle around ( 3 )}, P 1  (X 1 , Y 1 ), then moves along the arc of basic circle {circle around ( 1 )}, i.e. J 1 →Ĉ→J 2 . Vane {circle around ( 3 )} moves in the direction of the diameter along the two guides between the two crescent halves of the assembled rotor {circle around ( 2 )}, passing through the eccentricity center Oe. The other end, P 2  (X 2 , Y 2 ), describes the dotted curve {circle around ( 4 )}. 
     The length of vane {circle around ( 3 )} is constant; ie., the distance between P 1  (X 1 , Y 1 ) and P 2  (X 2 , Y 2 ), 2{square root over ( r +L (2 R −r +L ))}=2 a , is also constant. This means that the distance between the two points J 1  and J 2  on the x-axis, and the distance between the two points on the y-axis, Ĉ of the perigee and {circle around (m)} of the apogee, are constant. Here, an idealized curve {circle around ( 4 )} is produced, where the distance between any two points on the curve passing through the center is always constant. If the radius of basic circle {circle around ( 1 )}, R, and the radius of rotor {circle around ( 2 )}, r, are determined, a mathematical equation describing the motion of the two ends of vane {circle around ( 3 )}, P 1  and P 2 , can be derived, with the angle of rotation, θ°, as the only variable. 
     Then the equation which describes the curve {circle around ( 4 )} is written in Cartesian coordinates as: 
     
       
         X 2 +Y 2   ={2{square root over ( r +L ( 2   R−r +L ))}+(   R−r )sin θ−{square root over ( R   2 +L −( R−r +L ) 2  +L cos 2 +L θ)}} 2   (1), 
       
     
     where 0°≦θ≦180°. 
     In this equation, r denotes the radius of rotor {circle around ( 2 )}, R denotes the radius of basic circle {circle around ( 1 )}, and θ is the angle of rotation of vane {circle around ( 3 )}. This equation, in polar coordinates, is: 
     
       
         P= 2{square root over ( r +L ( 2   R−r +L ))}+(   R−r )sin θ−{square root over ( R   2 +L −( R−r +L ) 2 +L cos 2 +L θ)}  (2) 
       
     
     The equation describing the basic circle {circle around ( 1 )} can be written as: 
     
       
         X 2 +Y 2 ={{square root over ( R   2 +L −( R−r +L ) 2  +L cos 2 +L θ)}−( R−r )sin θ} 2   (3) 
       
     
     in Cartesian coordinates, and 
     
       
           P={square root over (R 2 +L −( R−r +L ) 2 +L cos 2 +L θ)}−(   R−r )sin θ  (4) 
       
     
     in polar coordinates. 
     If half of the length of the vane, {square root over (r(2+L R−r),)} is replaced with a into Equations (1) or (2), the equation becomes:              P   =     2      a        {     1   +       (     R   -   r     )            sin                   θ   .         2      a         -           R   2     -         (     R   -   r     )     2                     cos   2        θ           2      a         }               (   5   )                                
     This equation is equivalent to Equations (2) and (4) for curve {circle around ( 1 )} and {circle around ( 4 )}, i.e., the equation for cardiocles. Equation (5) resembles the equation for a simple cardioid,  P=a (1+sin θ), for dotted curve  4 ′ in FIG.  2 . But, Equation (5) is smaller by its third term, {square root over (R 2 +L −( R−r +L ) 2  +L cos 2 +L θ)}, than that describing curve  4 ′. In other words, equation (5) shows a curve 4′ as a cardioid flattened by the amount {square root over ( R   2 +L −( R−r +L ) 2  +L cos 2 +L θ)} in comparison with an ordinary cardioid  4 ′ in the range, 0°≦θ≦180°. And this cardioid curve connects at the two points J 1  and J 2  with the arc of circle {circle around ( 1 )} in the range 180°≦θ≦360°. This composite curve describes the curve drawn by the full rotation of vane {circle around ( 3 )}. It is named “cardiocle” for being a flattened cardioid in the range, 0°≦θ≦180°, and for being a circle in the range, 180°≦θ≦360°. 
     FIG. 2 gives graphical comparison of the composite cardiocle curve {circle around ( 4 )} with an ordinary cardioid  4 ′, calculated and drawn using a computer in accordance with the widely-known cardioid equation and the cardiocle equation (5) derived here. As shown in FIG. 2, the distance between the y-intercept of the cardioid  4 ′ and the lower point Oe is 2 a =2 r {square root over (r(2+L R−r))}; and thus dotted cardiocle curve {circle around ( 4 )} is the flattened down by r, the radius of the rotor {circle around ( 2 )}, along the y-axis in the range y≧0; and expanded below Oe, also by the amount r. Along the y-axis in the range of yso--; Curve {circle around ( 4 )}, a cardiocle, has the composition of a cardioid in the J 1 -m-J 2  section and of a circular arc in the J 1 -C-J 2  section. 
     FIG. 3 is a mechanical drawing, which describes the movement of an eccentric rotor pump with a cardiocle casing. An exact equation, in which the only variable is θ, the angle of rotation of vane {circle around ( 3 )} or rotor {circle around ( 2 )}, can be derived to represent the above-mentioned cardiocle curve drawn by rotation of the vane. Using this equation, accurate casing surfaces can not be processed through CNC techniques. 
     As shown in FIGS. 3 and 4, the casing is fitted with an inlet, {circle around ( 13 )}, and an outlet, {circle around ( 14 )}, for the flow of liquid into and out of the pump. The inlet and outlet are shown in the fourth and third quadrangles in FIGS.  3 . The outer periphery of the casing is surrounded by a cooling chamber, to the outer side of which water jackets are attached. 
     When the vane mounted on the rotor, as in FIGS. 3 and 4, is rotated anticlockwise, suction force is produced in the casing section containing inlet {circle around ( 13 )}, due to pressure decrease, and drainage force in the section containing outlet {circle around ( 14 )}, due to pressure increase. Fluid inflow and outflow are repeated in tandem with the rotation of the rotor. 
     In addition to the heat generated by friction between rotating rotor {circle around ( 2 )} and vane {circle around ( 3 )} and the casing surface {circle around ( 4 )}, additional heat is generated due to the continuous kinetic movement of fluid molecules during the repeated inflow and outflow of the liquid. This problem can be solved by applying current water-cooling or air-cooling techniques. Other current eccentric rotor vane pumps require substantial amounts of high-viscosity sealing oil, as their vane ends do not closely or uniformly scrape along tne casing surfaces due to their inaccurately processed casings. However, the equations for curve {circle around ( 4 )} derived in this invention make possible the processing of casing surfaces to the highest possible degree of accuracy, thus requiring only small amounts of low-viscosity sealing oil and making operations more economical. 
     In order to acquire different curvatures, a curve was drawn using Equation (5) minus the last term, {square root over (R 2 +L −(R−r) 2  +L cos 2 +L θ)}. This new curve also shows that the length of the vane, or casing diameter, remains constant during full rotations. From this, a new equation (6), for what we will call an “expanded cardioid” from now on, is derived.                P   =     2      a        {     1   +         (     R   -   r     )       2      a          sin                 θ       }                            (   6   )                                
     This new equation is represented by curve  4 ″ in FIG.  5 . This curve is not defined as an ellipse by mathematical definition, although it looks like one. Equation (6) shows that it is an expanded form of the ordinary cardioid ( P=a (1+sin θ)); and is thus named an “expanded cardioid”. As shown in FIG. 5, the expanded cardioid curve  4 ″ is an enlargement, by R the radius of basic circle {circle around ( 1 )}, of the cardiocle curve {circle around ( 4 )}, in both directions along the y-axis. The length of the vane for this curve, as shown in FIG. 6, is exactly twice that for the cardiocles as shown in FIGS. 1 and 2. This equation can be effectively and ideally applied in the precision processing of another type of eccentric rotor vane pump with expanded cardioid casing. As this expanded cardioid curve is closer to a circle than a cardiocle, rotor movement is expected to be smoother. 
     In the case of the expanded cardioid curve  4 ″ shown in FIG. 6, the radius of the rotor is 2{square root over (r(2+L R−r))}−R+r. The rotor is positioned symmetrically, (2{square root over (r(2+L R−r))}−R+r) above the lower y-intercept and (2{square root over (r(2+L R−r))}+R−r) below the upper y-intercept, on the y-axis. Thus the center of the rotor can be exactly determined. 
     An interesting comparison can be made here; Equation (6) for the expanded cardioid suffices for the range 0°≦θ≦360°, while Equation(5) for the cardiocle suffices only for the range 0≦θ≦180°. 
     The equations (1) through (6) derived in this invention form a mathematical basis for computer numerical controlled manufacturing of casings of eccentric rotor vane pumps. On the basis of these equations, part processing and assembly of casings of sizes far surpassing the limits set by currently available machine tool technology is now possible for any R and r, the respective radii of any arbitrary primary circle and any eccentric rotor. As CNC techniques become used instead of the tradtional recopy method, mass production becomes possible, thus reducing production costs and allowing the production good quality pumps at reasonable prices. Furthermore, as manufacturing in sections becomes possible, no additional processing equipment is required for large-size casings. 
     As one practical example of this, invention, FIG. 7 illustrates the operation of a springless eccentric rotor vane pump with an expanded cardioid casing. FIG. 8 describes section processing of a pump casing where the radius R of the basic circle {circle around ( 1 )} is 1,000 mm and the radius r of the eccentric rotor {circle around ( 2 )} is 600 mm. The shaded areas in sectors A, B and C are the parts to be processed in sections using the methodology introduced in this invention. The following table 1 shows the coordinates (x, y) calculated with the equations which describe the two-dimensional cross section of the casing (FIG.  8 ), over the range 0≦θ≦90°. 
     
       
         
               
               
               
             
               
             
               
               
               
             
               
             
               
               
               
             
               
             
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 X 
                 Y 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 0° ≦ θ ≦ 30° 
               
             
          
           
               
                   
                 0.327692 
                 0.120531 
               
               
                   
                 0.655831 
                 0.240369 
               
               
                   
                 0.984415 
                 0.359512 
               
               
                   
                 1.313441 
                 0.477958 
               
               
                   
                 1.642910 
                 0.595706 
               
               
                   
                 1.972818 
                 0.712755 
               
               
                   
                 2.303164 
                 0.829101 
               
               
                   
                 2.633947 
                 0.944745 
               
               
                   
                 2.965165 
                 1.059685 
               
               
                   
                 3.296817 
                 1.173918 
               
               
                   
                 3.628899 
                 1.287444 
               
               
                   
                 3.961412 
                 1.400260 
               
               
                   
                 4.294353 
                 1.512366 
               
               
                   
                 4.627720 
                 1.623759 
               
               
                   
                 4.961512 
                 1.734439 
               
               
                   
                 5.295727 
                 1.844403 
               
               
                   
                 5.630364 
                 1.953650 
               
               
                   
                 5.965420 
                 2.062180 
               
               
                   
                 6.300894 
                 2.169989 
               
               
                   
                 6.636784 
                 2.277076 
               
               
                   
                 6.973088 
                 2.383441 
               
               
                   
                 7.309805 
                 2.489081 
               
               
                   
                 7.646933 
                 2.593996 
               
               
                   
                 7.984470 
                 2.698183 
               
               
                   
                 8.322415 
                 2.801641 
               
               
                   
                 8.660765 
                 2.904369 
               
               
                   
                 8.999519 
                 3.006365 
               
               
                   
                 9.338676 
                 3.107628 
               
               
                   
                 9.678233 
                 3.208156 
               
               
                   
                 10.018189 
                 3.307948 
               
               
                   
                 10.358541 
                 3.407003 
               
               
                   
                 10.699289 
                 3.505318 
               
               
                   
                 11.040429 
                 3.602893 
               
               
                   
                 11.381962 
                 3.699726 
               
               
                   
                 11.723883 
                 3.795816 
               
               
                   
                 12.066193 
                 3.891162 
               
               
                   
                 12.408889 
                 3.985761 
               
               
                   
                 12.751969 
                 4.079612 
               
               
                   
                 13.095432 
                 4.172715 
               
               
                   
                 13.439275 
                 4.265068 
               
               
                   
                 13.783497 
                 4.356669 
               
               
                   
                 14.128096 
                 4.447516 
               
               
                   
                 14.473070 
                 4.537610 
               
               
                   
                 14.818417 
                 4.626948 
               
               
                   
                 15.164135 
                 4.715528 
               
               
                   
                 15.510223 
                 4.803350 
               
               
                   
                 15.856679 
                 4.890413 
               
               
                   
                 16.203500 
                 4.976714 
               
               
                   
                 16.550685 
                 5.062252 
               
               
                   
                 16.898233 
                 5.147027 
               
               
                   
                 17.246140 
                 5.231037 
               
               
                   
                 17.594406 
                 5.314281 
               
               
                   
                 17.943028 
                 5.396756 
               
               
                   
                 18.292005 
                 5.478463 
               
               
                   
                 18.641335 
                 5.559399 
               
               
                   
                 18.991015 
                 5.639564 
               
               
                   
                 19.341044 
                 5.718956 
               
               
                   
                 19.691420 
                 5.797573 
               
               
                   
                 20.042140 
                 5.875415 
               
               
                   
                 20.393204 
                 5.952481 
               
               
                   
                 20.744610 
                 6.028768 
               
               
                   
                 21.096354 
                 6.104277 
               
               
                   
                 21.448436 
                 6.179005 
               
               
                   
                 21.800853 
                 6.252951 
               
               
                   
                 22.153604 
                 6.326114 
               
               
                   
                 22.506686 
                 6.398494 
               
               
                   
                 22.860097 
                 6.470088 
               
               
                   
                 23.213837 
                 6.540895 
               
               
                   
                 23.567901 
                 6.610914 
               
               
                   
                 23.922290 
                 6.680145 
               
               
                   
                 24.277000 
                 6.748586 
               
               
                   
                 24.632031 
                 6.816235 
               
               
                   
                 24.987379 
                 6.883092 
               
               
                   
                 25.343042 
                 6.949155 
               
               
                   
                 25.699020 
                 7.014423 
               
               
                   
                 26.055310 
                 7.078895 
               
               
                   
                 26.411909 
                 7.142570 
               
               
                   
                 26.766816 
                 7.205447 
               
               
                   
                 27.126030 
                 7.267524 
               
               
                   
                 27.483547 
                 7.328801 
               
               
                   
                 27.841366 
                 7.389276 
               
               
                   
                 28.199487 
                 7.448948 
               
               
                   
                 28.557905 
                 7.507817 
               
               
                   
                 28.916620 
                 7.565881 
               
               
                   
                 29.275628 
                 7.623138 
               
               
                   
                 29.634929 
                 7.679589 
               
               
                   
                 29.994519 
                 7.735231 
               
               
                   
                 30.354397 
                 7.790063 
               
               
                   
                 30.714561 
                 7.844085 
               
               
                   
                 31.075008 
                 7.897296 
               
               
                   
                 31.435738 
                 7.949694 
               
               
                   
                 31.796747 
                 8.001279 
               
               
                   
                 32.158033 
                 8.052049 
               
               
                   
                 32.519596 
                 8.102003 
               
               
                   
                 32.881432 
                 8.151140 
               
               
                   
                 33.243539 
                 8.199460 
               
               
                   
                 33.605916 
                 8.246961 
               
               
                   
                 33.968560 
                 8.293642 
               
               
                   
                 34.331470 
                 8.339502 
               
               
                   
                 34.694643 
                 8.384541 
               
               
                   
                 35.058077 
                 8.428756 
               
               
                   
                 35.421770 
                 8.472148 
               
               
                   
                 35.785720 
                 8.514715 
               
               
                   
                 36.149926 
                 8.556457 
               
               
                   
                 36.514384 
                 8.597372 
               
               
                   
                 36.879093 
                 8.637459 
               
               
                   
                 37.244051 
                 8.676718 
               
               
                   
                 37.609255 
                 8.715147 
               
               
                   
                 37.974704 
                 8.752745 
               
               
                   
                 38.340396 
                 8.789513 
               
               
                   
                 38.706327 
                 8.825448 
               
               
                   
                 39.072498 
                 8.860550 
               
               
                   
                 39.438904 
                 8.894817 
               
               
                   
                 39.805544 
                 8.928250 
               
               
                   
                 40.172416 
                 8.960846 
               
               
                   
                 40.539519 
                 8.992606 
               
               
                   
                 40.906848 
                 9.023528 
               
               
                   
                 41.274404 
                 9.053612 
               
               
                   
                 41.642183 
                 9.082856 
               
               
                   
                 42.010183 
                 9.111260 
               
               
                   
                 42.378403 
                 9.138823 
               
               
                   
                 42.746839 
                 9.165543 
               
               
                   
                 43.115491 
                 9.191421 
               
               
                   
                 43.484355 
                 9.216455 
               
               
                   
                 43.853431 
                 9.240645 
               
               
                   
                 44.222714 
                 9.263989 
               
               
                   
                 44.592205 
                 9.286458 
               
               
                   
                 44.961899 
                 9.308139 
               
               
                   
                 45.331796 
                 9.328943 
               
               
                   
                 45.701892 
                 9.348898 
               
               
                   
                 46.072187 
                 9.368003 
               
               
                   
                 46.442677 
                 9.386259 
               
               
                   
                 46.813361 
                 9.403664 
               
               
                   
                 47.184236 
                 9.420217 
               
               
                   
                 47.555300 
                 9.435918 
               
               
                   
                 47.926551 
                 9.450766 
               
               
                   
                 48.297987 
                 9.464759 
               
               
                   
                 48.669606 
                 9.477899 
               
               
                   
                 49.041406 
                 9.490183 
               
               
                   
                 49.413384 
                 9.501610 
               
               
                   
                 49.785638 
                 9.512181 
               
               
                   
                 50.157866 
                 9.521895 
               
               
                   
                 50.530366 
                 9.530751 
               
               
                   
                 50.903036 
                 9.538747 
               
               
                   
                 51.275873 
                 9.545884 
               
               
                   
                 51.648875 
                 9.552161 
               
               
                   
                 52.022041 
                 9.557577 
               
               
                   
                 52.395368 
                 9.562131 
               
               
                   
                 52.768853 
                 9.565823 
               
               
                   
                 53.142495 
                 9.568652 
               
               
                   
                 53.516291 
                 9.570618 
               
               
                   
                 53.890239 
                 9.571719 
               
               
                   
                 54.264338 
                 9.571956 
               
               
                   
                 54.638584 
                 9.571327 
               
               
                   
                 55.012976 
                 9.569833 
               
               
                   
                 55.387511 
                 9.567471 
               
               
                   
                 55.762187 
                 9.564243 
               
               
                   
                 56.137002 
                 9.560146 
               
               
                   
                 56.511954 
                 9.555181 
               
               
                   
                 56.887041 
                 9.549347 
               
               
                   
                 57.262260 
                 9.542644 
               
               
                   
                 57.637609 
                 9.535070 
               
               
                   
                 58.013086 
                 9.526625 
               
               
                   
                 58.388688 
                 9.517310 
               
               
                   
                 58.764415 
                 9.507122 
               
               
                   
                 59.140262 
                 9.496062 
               
               
                   
                 59.516228 
                 9.484130 
               
               
                   
                 59.892312 
                 9.471323 
               
               
                   
                 60.268509 
                 9.457643 
               
               
                   
                 60.644820 
                 9.443089 
               
               
                   
                 61.021240 
                 9.427659 
               
               
                   
                 61.397763 
                 9.411354 
               
               
                   
                 61.774402 
                 9.394173 
               
               
                   
                 62.151139 
                 9.376116 
               
               
                   
                 62.527978 
                 9.357182 
               
               
                   
                 62.904916 
                 9.337370 
               
               
                   
                 63.281950 
                 9.316681 
               
               
                   
                 63.659079 
                 9.295113 
               
               
                   
                 64.036300 
                 9.272667 
               
               
                   
                 64.413612 
                 9.249341 
               
               
                   
                 64.791011 
                 9.225136 
               
               
                   
                 65.168495 
                 9.200050 
               
               
                   
                 65.546063 
                 9.174085 
               
               
                   
                 65.923712 
                 9.147238 
               
               
                   
                 66.301440 
                 9.119510 
               
               
                   
                 66.679245 
                 9.090900 
               
               
                   
                 67.057124 
                 9.061409 
               
               
                   
                 67.435075 
                 9.031035 
               
               
                   
                 67.813096 
                 8.999778 
               
               
                   
                 68.191184 
                 8.967638 
               
               
                   
                 68.569338 
                 8.934614 
               
               
                   
                 68.947555 
                 8.900706 
               
               
                   
                 69.325833 
                 8.865914 
               
               
                   
                 69.704170 
                 8.830238 
               
               
                   
                 70.082563 
                 8.793676 
               
               
                   
                 70.461010 
                 8.756229 
               
               
                   
                 70.839508 
                 8.717897 
               
               
                   
                 71.218057 
                 8.678678 
               
               
                   
                 71.596653 
                 8.638574 
               
               
                   
                 71.975294 
                 8.597583 
               
               
                   
                 72.353978 
                 8.555705 
               
               
                   
                 72.732702 
                 8.512939 
               
               
                   
                 73.111465 
                 8.469287 
               
               
                   
                 73.490263 
                 8.424747 
               
               
                   
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                 8.379318 
               
               
                   
                 74.247960 
                 8.333002 
               
               
                   
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                 8.285797 
               
               
                   
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                 8.237704 
               
               
                   
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                 8.188721 
               
               
                   
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                 8.138849 
               
               
                   
                 76.142675 
                 8.088088 
               
               
                   
                 76.521681 
                 8.036438 
               
               
                   
                 76.900703 
                 7.983897 
               
               
                   
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                 7.930467 
               
               
                   
                 77.658784 
                 7.876146 
               
               
                   
                 78.037840 
                 7.820934 
               
               
                   
                 78.416902 
                 7.764833 
               
               
                   
                 78.795968 
                 7.707840 
               
               
                   
                 79.175036 
                 7.649956 
               
               
                   
                 79.554105 
                 7.591181 
               
               
                   
                 79.933171 
                 7.531515 
               
               
                   
                 80.312232 
                 7.470958 
               
               
                   
                 80.691287 
                 7.409509 
               
               
                   
                 81.070332 
                 7.347168 
               
               
                   
                 81.449366 
                 7.283935 
               
               
                   
                 81.828386 
                 7.219811 
               
               
                   
                 82.207390 
                 7.154794 
               
               
                   
                 82.586376 
                 7.088885 
               
               
                   
                 82.965341 
                 7.022084 
               
               
                   
                 83.344283 
                 6.954391 
               
               
                   
                 83.723201 
                 6.885804 
               
               
                   
                 84.102091 
                 6.816326 
               
               
                   
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                 6.745955 
               
               
                   
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                 6.674691 
               
               
                   
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                 6.602534 
               
               
                   
                 85.617333 
                 6.529485 
               
               
                   
                 85.996053 
                 6.455542 
               
               
                   
                 86.374732 
                 6.380707 
               
               
                   
                 86.753367 
                 6.304979 
               
               
                   
                 87.131957 
                 6.228358 
               
               
                   
                 87.510500 
                 6.150843 
               
               
                   
                 87.888992 
                 6.072436 
               
               
                   
                 88.267432 
                 5.993136 
               
               
                   
                 88.645818 
                 5.912943 
               
               
                   
                 89.024147 
                 5.831857 
               
               
                   
                 89.402417 
                 5.749877 
               
               
                   
                 89.780626 
                 5.667005 
               
               
                   
                 90.158771 
                 5.583240 
               
               
                   
                 90.536850 
                 5.498582 
               
               
                   
                 90.914861 
                 5.413031 
               
               
                   
                 91.292802 
                 5.326588 
               
               
                   
                 91.670670 
                 5.239251 
               
               
                   
                 92.048464 
                 5.151022 
               
               
                   
                 92.426180 
                 5.061900 
               
               
                   
                 92.803817 
                 4.971886 
               
               
                   
                 93.181372 
                 4.880979 
               
               
                   
                 93.558844 
                 4.789180 
               
               
                   
                 93.336229 
                 4.696488 
               
               
                   
                 94.313526 
                 4.602904 
               
               
                   
                 94.690732 
                 4.508428 
               
               
                   
                 95.067845 
                 4.413060 
               
               
                   
                 95.444863 
                 4.316801 
               
               
                   
                 95.821784 
                 4.219649 
               
               
                   
                 96.198605 
                 4.121606 
               
               
                   
                 96.575324 
                 4.022671 
               
               
                   
                 96.951938 
                 3.922846 
               
               
                   
                 97.328447 
                 3.822128 
               
               
                   
                 97.704847 
                 3.720520 
               
               
                   
                 98.081135 
                 3.618021 
               
               
                   
                 98.457311 
                 3.514632 
               
               
                   
                 98.833371 
                 3.410352 
               
               
                   
                 99.209313 
                 3.305181 
               
               
                   
                 99.585136 
                 3.199121 
               
               
                   
                 99.960836 
                 3.092170 
               
               
                   
                 100.336412 
                 2.984330 
               
               
                   
                 100.711861 
                 2.875600 
               
               
                   
                 101.087181 
                 2.765981 
               
               
                   
                 101.462370 
                 2.655473 
               
               
                   
                 101.837426 
                 2.544076 
               
               
                   
                 102.212346 
                 2.431791 
               
               
                   
                 102.587128 
                 2.318617 
               
               
                   
                 102.961771 
                 2.204555 
               
               
                   
                 103.336270 
                 2.089605 
               
               
                   
                 103.710625 
                 1.973768 
               
               
                   
                 104.084834 
                 1.857043 
               
               
                   
                 104.458893 
                 1.739431 
               
               
                   
                 104.832801 
                 1.620933 
               
               
                   
                 105.206555 
                 1.501548 
               
               
                   
                 105.580154 
                 1.381277 
               
               
                   
                 105.953595 
                 1.260120 
               
               
                   
                 106.326875 
                 1.138077 
               
               
                   
                 106.699993 
                 1.015149 
               
               
                   
                 107.072946 
                 0.891337 
               
               
                   
                 107.445733 
                 0.766639 
               
               
                   
                 107.818350 
                 0.641058 
               
               
                   
                 108.190796 
                 0.514592 
               
               
                   
                 108.563069 
                 0.387243 
               
               
                   
                 108.935165 
                 0.259011 
               
               
                   
                 109.307084 
                 0.129896 
               
             
          
           
               
                 30° ≦ θ ≦ 60° 
               
             
          
           
               
                   
                 0.371910 
                 0.129871 
               
               
                   
                 0.744227 
                 0.258937 
               
               
                   
                 1.116948 
                 0.387199 
               
               
                   
                 1.490071 
                 0.514655 
               
               
                   
                 1.863596 
                 0.641303 
               
               
                   
                 2.237519 
                 0.767143 
               
               
                   
                 2.611839 
                 0.892174 
               
               
                   
                 2.986553 
                 1.016395 
               
               
                   
                 3.361661 
                 1.139804 
               
               
                   
                 3.737159 
                 1.262401 
               
               
                   
                 4.113046 
                 1.384184 
               
               
                   
                 4.489319 
                 1.505153 
               
               
                   
                 4.865978 
                 1.625307 
               
               
                   
                 5.243019 
                 1.744643 
               
               
                   
                 5.620441 
                 1.863163 
               
               
                   
                 5.998241 
                 1.980864 
               
               
                   
                 6.376419 
                 2.097745 
               
               
                   
                 6.754971 
                 2.213805 
               
               
                   
                 7.133896 
                 2.329045 
               
               
                   
                 7.513192 
                 2.443461 
               
               
                   
                 7.892849 
                 2.557052 
               
               
                   
                 8.272873 
                 2.669819 
               
               
                   
                 8.653262 
                 2.781760 
               
               
                   
                 9.034014 
                 2.892875 
               
               
                   
                 9.415126 
                 3.003162 
               
               
                   
                 9.796596 
                 3.112621 
               
               
                   
                 10.178424 
                 3.221251 
               
               
                   
                 10.560605 
                 3.329051 
               
               
                   
                 10.943140 
                 3.436020 
               
               
                   
                 11.326025 
                 3.542157 
               
               
                   
                 11.709258 
                 3.647461 
               
               
                   
                 12.092838 
                 3.751931 
               
               
                   
                 12.476762 
                 3.855566 
               
               
                   
                 12.861028 
                 3.958365 
               
               
                   
                 13.245635 
                 4.060329 
               
               
                   
                 13.630580 
                 4.161454 
               
               
                   
                 14.015861 
                 4.261742 
               
               
                   
                 14.401477 
                 4.361190 
               
               
                   
                 14.787424 
                 4.459798 
               
               
                   
                 15.173701 
                 4.557565 
               
               
                   
                 15.560307 
                 4.654490 
               
               
                   
                 15.947238 
                 4.750573 
               
               
                   
                 16.334493 
                 4.845812 
               
               
                   
                 16.722070 
                 4.940207 
               
               
                   
                 17.109967 
                 5.033756 
               
               
                   
                 17.498181 
                 5.126460 
               
               
                   
                 17.886711 
                 5.218317 
               
               
                   
                 18.275554 
                 5.309326 
               
               
                   
                 18.664709 
                 5.399487 
               
               
                   
                 19.054173 
                 5.488798 
               
               
                   
                 19.443945 
                 5.577259 
               
               
                   
                 19.834021 
                 5.664870 
               
               
                   
                 20.224401 
                 5.751628 
               
               
                   
                 20.615082 
                 5.837535 
               
               
                   
                 21.006062 
                 5.922588 
               
               
                   
                 21.397338 
                 6.006786 
               
               
                   
                 21.788910 
                 6.090131 
               
               
                   
                 22.180774 
                 6.172619 
               
               
                   
                 22.572928 
                 6.254252 
               
               
                   
                 22.965372 
                 6.335027 
               
               
                   
                 23.358101 
                 6.414945 
               
               
                   
                 23.751115 
                 6.494004 
               
               
                   
                 24.144411 
                 6.572203 
               
               
                   
                 24.537987 
                 6.649543 
               
               
                   
                 24.931841 
                 6.726022 
               
               
                   
                 25.325971 
                 6.801640 
               
               
                   
                 25.720375 
                 6.876395 
               
               
                   
                 26.115050 
                 6.950288 
               
               
                   
                 26.509996 
                 7.023317 
               
               
                   
                 26.905208 
                 7.095482 
               
               
                   
                 27.300686 
                 7.166782 
               
               
                   
                 27.696427 
                 7.237216 
               
               
                   
                 28.092429 
                 7.306784 
               
               
                   
                 28.488691 
                 7.375485 
               
               
                   
                 28.885209 
                 7.443319 
               
               
                   
                 29.281982 
                 7.510284 
               
               
                   
                 29.679007 
                 7.576380 
               
               
                   
                 30.076283 
                 7.641607 
               
               
                   
                 30.473808 
                 7.705964 
               
               
                   
                 30.871579 
                 7.769450 
               
               
                   
                 31.269593 
                 7.832064 
               
               
                   
                 31.667850 
                 7.893806 
               
               
                   
                 32.066347 
                 7.954676 
               
               
                   
                 32.465082 
                 8.014672 
               
               
                   
                 32.864052 
                 8.073795 
               
               
                   
                 33.263256 
                 8.132042 
               
               
                   
                 33.662691 
                 8.189415 
               
               
                   
                 34.062356 
                 8.245912 
               
               
                   
                 34.462247 
                 8.301533 
               
               
                   
                 34.862364 
                 8.356277 
               
               
                   
                 35.262703 
                 8.410144 
               
               
                   
                 35.663263 
                 8.463133 
               
               
                   
                 36.064042 
                 8.515243 
               
               
                   
                 36.465037 
                 8.566474 
               
               
                   
                 36.866246 
                 8.616825 
               
               
                   
                 37.267667 
                 8.666297 
               
               
                   
                 37.669299 
                 8.714887 
               
               
                   
                 38.071138 
                 8.762597 
               
               
                   
                 38.473183 
                 8.809424 
               
               
                   
                 38.875431 
                 8.855370 
               
               
                   
                 39.277881 
                 8.900432 
               
               
                   
                 39.680530 
                 8.944612 
               
               
                   
                 40.083376 
                 8.987907 
               
               
                   
                 40.486417 
                 9.030319 
               
               
                   
                 40.889651 
                 9.071845 
               
               
                   
                 41.293076 
                 9.112486 
               
               
                   
                 41.696689 
                 9.152242 
               
               
                   
                 42.100488 
                 9.191111 
               
               
                   
                 42.504472 
                 9.229094 
               
               
                   
                 42.908637 
                 9.266190 
               
               
                   
                 43.312983 
                 9.302398 
               
               
                   
                 43.717506 
                 9.337718 
               
               
                   
                 44.122205 
                 9.372149 
               
               
                   
                 44.527077 
                 9.405692 
               
               
                   
                 44.932120 
                 9.438345 
               
               
                   
                 45.337332 
                 9.470109 
               
               
                   
                 45.742711 
                 9.500983 
               
               
                   
                 46.148255 
                 9.530965 
               
               
                   
                 46.553962 
                 9.560057 
               
               
                   
                 46.959829 
                 9.588258 
               
               
                   
                 47.365854 
                 9.615567 
               
               
                   
                 47.772035 
                 9.641983 
               
               
                   
                 48.178370 
                 9.667507 
               
               
                   
                 48.584857 
                 9.692138 
               
               
                   
                 48.991494 
                 9.715876 
               
               
                   
                 49.398278 
                 9.738720 
               
               
                   
                 49.805207 
                 9.760671 
               
               
                   
                 50.212279 
                 9.781726 
               
               
                   
                 50.619492 
                 9.801887 
               
               
                   
                 51.026844 
                 9.821153 
               
               
                   
                 51.434333 
                 9.839524 
               
               
                   
                 51.841955 
                 9.856998 
               
               
                   
                 52.249710 
                 9.873577 
               
               
                   
                 52.657595 
                 9.889259 
               
               
                   
                 53.065608 
                 9.904045 
               
               
                   
                 53.473747 
                 9.917933 
               
               
                   
                 53.882009 
                 9.930925 
               
               
                   
                 54.290393 
                 9.943018 
               
               
                   
                 54.698896 
                 9.954214 
               
               
                   
                 55.107515 
                 9.964511 
               
               
                   
                 55.516250 
                 9.973910 
               
               
                   
                 55.925097 
                 9.982410 
               
               
                   
                 56.334055 
                 9.990012 
               
               
                   
                 56.743121 
                 9.996714 
               
               
                   
                 57.152293 
                 10.002516 
               
               
                   
                 57.561569 
                 10.007418 
               
               
                   
                 57.970947 
                 10.011421 
               
               
                   
                 58.380425 
                 10.014523 
               
               
                   
                 58.790000 
                 10.016725 
               
               
                   
                 59.199670 
                 10.018025 
               
               
                   
                 59.609433 
                 10.018425 
               
               
                   
                 60.019288 
                 10.017924 
               
               
                   
                 60.429231 
                 10.016521 
               
               
                   
                 60.839260 
                 10.014217 
               
               
                   
                 61.249374 
                 10.011011 
               
               
                   
                 61.659570 
                 10.006903 
               
               
                   
                 62.069846 
                 10.001892 
               
               
                   
                 62.480200 
                 9.995979 
               
               
                   
                 62.890629 
                 9.989164 
               
               
                   
                 63.301132 
                 9.981446 
               
               
                   
                 63.711707 
                 9.972825 
               
               
                   
                 64.122350 
                 9.963300 
               
               
                   
                 64.533060 
                 9.952873 
               
               
                   
                 64.943836 
                 9.941542 
               
               
                   
                 65.354673 
                 9.929308 
               
               
                   
                 65.765572 
                 9.916170 
               
               
                   
                 66.176528 
                 9.902128 
               
               
                   
                 66.587540 
                 9.887182 
               
               
                   
                 66.998607 
                 9.871332 
               
               
                   
                 67.409725 
                 9.854578 
               
               
                   
                 67.820892 
                 9.836919 
               
               
                   
                 68.232107 
                 9.818357 
               
               
                   
                 68.643367 
                 9.798889 
               
               
                   
                 69.054670 
                 9.778517 
               
               
                   
                 69.466014 
                 9.757241 
               
               
                   
                 69.877396 
                 9.735059 
               
               
                   
                 70.288815 
                 9.711973 
               
               
                   
                 70.700268 
                 9.687982 
               
               
                   
                 71.111754 
                 9.663086 
               
               
                   
                 71.523269 
                 9.637284 
               
               
                   
                 71.934812 
                 9.610578 
               
               
                   
                 72.346381 
                 9.582966 
               
               
                   
                 72.757972 
                 9.554450 
               
               
                   
                 73.169586 
                 9.525027 
               
               
                   
                 73.581218 
                 9.494700 
               
               
                   
                 73.992867 
                 9.463467 
               
               
                   
                 74.404531 
                 9.431329 
               
               
                   
                 74.816207 
                 9.398286 
               
               
                   
                 75.227894 
                 9.364337 
               
               
                   
                 75.639589 
                 9.329483 
               
               
                   
                 76.051290 
                 9.293723 
               
               
                   
                 76.462994 
                 9.257058 
               
               
                   
                 76.874701 
                 9.219488 
               
               
                   
                 77.286407 
                 9.181012 
               
               
                   
                 77.698110 
                 9.141631 
               
               
                   
                 78.109808 
                 9.101344 
               
               
                   
                 78.521499 
                 9.060152 
               
               
                   
                 78.933182 
                 9.018055 
               
               
                   
                 79.344852 
                 8.975053 
               
               
                   
                 79.756510 
                 8.931145 
               
               
                   
                 80.168151 
                 8.886333 
               
               
                   
                 80.579775 
                 8.840615 
               
               
                   
                 80.991379 
                 8.793993 
               
               
                   
                 81.402960 
                 8.746465 
               
               
                   
                 81.814517 
                 8.698032 
               
               
                   
                 82.226048 
                 8.648695 
               
               
                   
                 82.637550 
                 8.598453 
               
               
                   
                 83.049021 
                 8.547307 
               
               
                   
                 83.460459 
                 8.495255 
               
               
                   
                 83.871862 
                 8.442300 
               
               
                   
                 84.283227 
                 8.388440 
               
               
                   
                 84.694553 
                 8.333676 
               
               
                   
                 85.105837 
                 8.278008 
               
               
                   
                 85.517078 
                 8.221436 
               
               
                   
                 85.928272 
                 8.163960 
               
               
                   
                 86.339419 
                 8.105580 
               
               
                   
                 86.750515 
                 8.046297 
               
               
                   
                 87.161558 
                 7.986110 
               
               
                   
                 87.572547 
                 7.925020 
               
               
                   
                 87.983480 
                 7.863027 
               
               
                   
                 88.394353 
                 7.800131 
               
               
                   
                 88.805165 
                 1.736332 
               
               
                   
                 89.215914 
                 7.671630 
               
               
                   
                 89.626597 
                 7.606026 
               
               
                   
                 90.037214 
                 7.539520 
               
               
                   
                 90.447760 
                 7.472112 
               
               
                   
                 90.858234 
                 7.403801 
               
               
                   
                 91.268635 
                 7.334589 
               
               
                   
                 91.678959 
                 7.264476 
               
               
                   
                 92.089205 
                 7.193461 
               
               
                   
                 92.499371 
                 7.121545 
               
               
                   
                 92.909454 
                 7.048728 
               
               
                   
                 93.319452 
                 6.975011 
               
               
                   
                 93.729363 
                 6.900393 
               
               
                   
                 94.139186 
                 6.824875 
               
               
                   
                 94.548917 
                 6.748457 
               
               
                   
                 94.958555 
                 6.671139 
               
               
                   
                 95.368097 
                 6.592922 
               
               
                   
                 95.777542 
                 6.513805 
               
               
                   
                 96.186888 
                 6.433790 
               
               
                   
                 96.596131 
                 6.352876 
               
               
                   
                 97.005270 
                 6.271063 
               
               
                   
                 97.414303 
                 6.188352 
               
               
                   
                 97.823228 
                 6.104744 
               
               
                   
                 98.232043 
                 6.020237 
               
               
                   
                 98.640745 
                 5.934834 
               
               
                   
                 99.049332 
                 5.848533 
               
               
                   
                 99.457803 
                 5.761336 
               
               
                   
                 99.866155 
                 5.673243 
               
               
                   
                 100.274385 
                 5.584253 
               
               
                   
                 100.682493 
                 5.494367 
               
               
                   
                 101.090475 
                 5.403586 
               
               
                   
                 101.498330 
                 5.311910 
               
               
                   
                 101.906055 
                 5.219339 
               
               
                   
                 102.313648 
                 5.125874 
               
               
                   
                 102.721108 
                 5.031514 
               
               
                   
                 103.128432 
                 4.936261 
               
               
                   
                 103.535618 
                 4.840114 
               
               
                   
                 103.942664 
                 4.743074 
               
               
                   
                 104.349567 
                 4.645142 
               
               
                   
                 104.756326 
                 4.546317 
               
               
                   
                 105.162939 
                 4.446600 
               
               
                   
                 105.569403 
                 4.345991 
               
               
                   
                 105.975716 
                 4.244492 
               
               
                   
                 106.381876 
                 4.142101 
               
               
                   
                 106.787882 
                 4.038820 
               
               
                   
                 107.193730 
                 3.934649 
               
               
                   
                 107.599420 
                 3.829588 
               
               
                   
                 108.004948 
                 3.723638 
               
               
                   
                 108.410312 
                 3.616799 
               
               
                   
                 108.815512 
                 3.509072 
               
               
                   
                 109.220543 
                 3.400456 
               
               
                   
                 109.625406 
                 3.290954 
               
               
                   
                 110.030096 
                 3.180563 
               
               
                   
                 110.434612 
                 3.069287 
               
               
                   
                 110.838953 
                 2.957124 
               
               
                   
                 111.243116 
                 2.844075 
               
               
                   
                 111.647098 
                 2.730141 
               
               
                   
                 112.050898 
                 2.615321 
               
               
                   
                 112.454514 
                 2.499618 
               
               
                   
                 112.857944 
                 2.383030 
               
               
                   
                 113.261185 
                 2.265559 
               
               
                   
                 113.664235 
                 2.147205 
               
               
                   
                 114.067093 
                 2.027968 
               
               
                   
                 114.469756 
                 1.907849 
               
               
                   
                 114.872223 
                 1.786849 
               
               
                   
                 115.274490 
                 1.664967 
               
               
                   
                 115.676556 
                 1.542205 
               
               
                   
                 116.078420 
                 1.418563 
               
               
                   
                 116.480078 
                 1.294041 
               
               
                   
                 116.881529 
                 1.168640 
               
               
                   
                 117.282771 
                 1.042361 
               
               
                   
                 117.683802 
                 0.915203 
               
               
                   
                 118.084619 
                 0.787168 
               
               
                   
                 118.485221 
                 0.658256 
               
               
                   
                 118.885605 
                 0.528468 
               
               
                   
                 119.285769 
                 0.397804 
               
               
                   
                 119.685712 
                 0.266264 
               
               
                   
                 120.085432 
                 0.133850 
               
               
                   
                 120.484925 
                 0.000561 
               
             
          
           
               
                 60° ≦ θ ≦ 90° 
               
             
          
           
               
                   
                 0.400229 
                 0.131263 
               
               
                   
                 0.800795 
                 0.261688 
               
               
                   
                 1.201698 
                 0.391276 
               
               
                   
                 1.602934 
                 0.520024 
               
               
                   
                 2.004502 
                 0.647934 
               
               
                   
                 2.406401 
                 0.775002 
               
               
                   
                 2.808627 
                 0.901230 
               
               
                   
                 3.211179 
                 1.026617 
               
               
                   
                 3.614056 
                 1.151160 
               
               
                   
                 4.017254 
                 1.274861 
               
               
                   
                 4.420772 
                 1.397717 
               
               
                   
                 4.824608 
                 1.519729 
               
               
                   
                 5.228761 
                 1.640895 
               
               
                   
                 5.633227 
                 1.761215 
               
               
                   
                 6.038005 
                 1.880689 
               
               
                   
                 6.443093 
                 1.999314 
               
               
                   
                 6.848490 
                 2.117092 
               
               
                   
                 7.254192 
                 2.234021 
               
               
                   
                 7.660198 
                 2.350099 
               
               
                   
                 8.066506 
                 2.465328 
               
               
                   
                 8.473114 
                 2.579706 
               
               
                   
                 8.880020 
                 2.693232 
               
               
                   
                 9.287221 
                 2.805905 
               
               
                   
                 9.694717 
                 2.917726 
               
               
                   
                 10.102504 
                 3.028693 
               
               
                   
                 10.510582 
                 3.138805 
               
               
                   
                 10.918947 
                 3.248063 
               
               
                   
                 11.327598 
                 3.356465 
               
               
                   
                 11.736532 
                 3.464010 
               
               
                   
                 12.145749 
                 3.570699 
               
               
                   
                 12.555245 
                 3.676530 
               
               
                   
                 12.965019 
                 3.781503 
               
               
                   
                 13.375069 
                 3.885617 
               
               
                   
                 13.785392 
                 3.988871 
               
               
                   
                 14.195987 
                 4.091266 
               
               
                   
                 14.606851 
                 4.192800 
               
               
                   
                 15.017983 
                 4.293473 
               
               
                   
                 15.429380 
                 4.393284 
               
               
                   
                 15.841041 
                 4.492232 
               
               
                   
                 16.252964 
                 4.590317 
               
               
                   
                 16.665145 
                 4.687539 
               
               
                   
                 17.077585 
                 4.783896 
               
               
                   
                 17.490279 
                 4.879389 
               
               
                   
                 17.903227 
                 4.974016 
               
               
                   
                 18.316426 
                 5.067778 
               
               
                   
                 18.729874 
                 5.160673 
               
               
                   
                 19.143569 
                 5.252701 
               
               
                   
                 19.557510 
                 5.343861 
               
               
                   
                 19.971694 
                 5.434153 
               
               
                   
                 20.386118 
                 5.523577 
               
               
                   
                 20.800782 
                 5.612131 
               
               
                   
                 21.215682 
                 5.699816 
               
               
                   
                 21.630818 
                 5.786630 
               
               
                   
                 22.046186 
                 5.872573 
               
               
                   
                 22.461785 
                 5.957646 
               
               
                   
                 22.877613 
                 6.041846 
               
               
                   
                 23.293667 
                 6.125174 
               
               
                   
                 23.709947 
                 6.207629 
               
               
                   
                 24.126448 
                 6.289211 
               
               
                   
                 24.543170 
                 6.369919 
               
               
                   
                 24.960111 
                 6.449753 
               
               
                   
                 25.377268 
                 6.528712 
               
               
                   
                 25.794640 
                 6.606795 
               
               
                   
                 26.212223 
                 6.684003 
               
               
                   
                 26.630017 
                 6.760335 
               
               
                   
                 27.048019 
                 6.835790 
               
               
                   
                 27.466228 
                 6.910368 
               
               
                   
                 27.884640 
                 6.984068 
               
               
                   
                 28.303254 
                 7.056891 
               
               
                   
                 28.722068 
                 7.128835 
               
               
                   
                 29.141080 
                 7.199899 
               
               
                   
                 29.560288 
                 7.270085 
               
               
                   
                 29.979690 
                 7.339391 
               
               
                   
                 30.399283 
                 7.407816 
               
               
                   
                 30.819066 
                 7.475361 
               
               
                   
                 31.239036 
                 7.542025 
               
               
                   
                 31.659192 
                 7.607807 
               
               
                   
                 32.079531 
                 7.672708 
               
               
                   
                 32.500052 
                 7.736726 
               
               
                   
                 32.920751 
                 7.799862 
               
               
                   
                 33.341628 
                 7.862114 
               
               
                   
                 33.762681 
                 7.923484 
               
               
                   
                 34.183906 
                 7.983969 
               
               
                   
                 34.605302 
                 8.043570 
               
               
                   
                 35.026867 
                 8.102287 
               
               
                   
                 35.448598 
                 8.160118 
               
               
                   
                 35.870495 
                 8.217065 
               
               
                   
                 36.292554 
                 8.273125 
               
               
                   
                 36.714774 
                 8.328300 
               
               
                   
                 37.137152 
                 8.382588 
               
               
                   
                 37.559687 
                 8.435990 
               
               
                   
                 37.982376 
                 8.488505 
               
               
                   
                 38.405217 
                 8.540132 
               
               
                   
                 38.828209 
                 8.590872 
               
               
                   
                 39.251349 
                 8.640723 
               
               
                   
                 39.674635 
                 8.689686 
               
               
                   
                 40.098064 
                 8.737761 
               
               
                   
                 40.521636 
                 8.784947 
               
               
                   
                 40.945347 
                 8.831243 
               
               
                   
                 41.369196 
                 8.876650 
               
               
                   
                 41.793181 
                 8.921167 
               
               
                   
                 42.217300 
                 8.964794 
               
               
                   
                 42.641549 
                 9.007530 
               
               
                   
                 43.065929 
                 9.049376 
               
               
                   
                 43.490435 
                 9.090330 
               
               
                   
                 43.915067 
                 9.130394 
               
               
                   
                 44.339822 
                 9.169566 
               
               
                   
                 44.764698 
                 9.207846 
               
               
                   
                 45.189693 
                 9.245234 
               
               
                   
                 45.614805 
                 9.281730 
               
               
                   
                 46.040031 
                 9.317333 
               
               
                   
                 46.465371 
                 9.352044 
               
               
                   
                 46.890821 
                 9.385861 
               
               
                   
                 47.316379 
                 9.418785 
               
               
                   
                 47.742044 
                 9.450816 
               
               
                   
                 48.167813 
                 9.481953 
               
               
                   
                 48.593685 
                 9.512197 
               
               
                   
                 49.019657 
                 9.541546 
               
               
                   
                 49.445727 
                 9.570000 
               
               
                   
                 49.871893 
                 9.597561 
               
               
                   
                 50.298153 
                 9.624226 
               
               
                   
                 50.724505 
                 9.649997 
               
               
                   
                 51.150946 
                 9.674872 
               
               
                   
                 51.577475 
                 9.698852 
               
               
                   
                 52.004090 
                 9.721937 
               
               
                   
                 52.430788 
                 9.744126 
               
               
                   
                 52.857568 
                 9.765419 
               
               
                   
                 53.284427 
                 9.785817 
               
               
                   
                 53.711363 
                 9.805318 
               
               
                   
                 54.138375 
                 9.823923 
               
               
                   
                 54.565459 
                 9.841631 
               
               
                   
                 54.992614 
                 9.858443 
               
               
                   
                 55.419839 
                 9.874358 
               
               
                   
                 55.847130 
                 9.889376 
               
               
                   
                 56.274485 
                 9.903497 
               
               
                   
                 56.701903 
                 9.916721 
               
               
                   
                 57.129382 
                 9.929048 
               
               
                   
                 57.556919 
                 9.940478 
               
               
                   
                 57.984513 
                 9.951010 
               
               
                   
                 58.412161 
                 9.960644 
               
               
                   
                 58.839860 
                 9.969381 
               
               
                   
                 59.267610 
                 9.977220 
               
               
                   
                 59.695408 
                 9.984161 
               
               
                   
                 60.123252 
                 9.990204 
               
               
                   
                 60.551139 
                 9.995349 
               
               
                   
                 60.979068 
                 9.999596 
               
               
                   
                 61.407037 
                 10.002945 
               
               
                   
                 61.835043 
                 10.005395 
               
               
                   
                 62.263085 
                 10.006947 
               
               
                   
                 62.691160 
                 10.007601 
               
               
                   
                 63.119266 
                 10.007356 
               
               
                   
                 63.547402 
                 10.006213 
               
               
                   
                 63.975564 
                 10.004171 
               
               
                   
                 64.403751 
                 10.001231 
               
               
                   
                 64.831962 
                 9.997392 
               
               
                   
                 65.260193 
                 9.992654 
               
               
                   
                 65.688442 
                 9.987018 
               
               
                   
                 66.116709 
                 9.980483 
               
               
                   
                 66.544990 
                 9.973049 
               
               
                   
                 66.973283 
                 9.964717 
               
               
                   
                 67.401586 
                 9.955486 
               
               
                   
                 67.829898 
                 9.945356 
               
               
                   
                 68.258216 
                 9.934327 
               
               
                   
                 68.686538 
                 9.922409 
               
               
                   
                 69.114862 
                 9.909574 
               
               
                   
                 69.543186 
                 9.895849 
               
               
                   
                 69.971508 
                 9.881226 
               
               
                   
                 70.399825 
                 9.865704 
               
               
                   
                 70.828136 
                 9.849283 
               
               
                   
                 71.256439 
                 9.831964 
               
               
                   
                 71.684730 
                 9.813746 
               
               
                   
                 72.113010 
                 9.794630 
               
               
                   
                 72.541274 
                 9.774615 
               
               
                   
                 72.969522 
                 9.753702 
               
               
                   
                 73.397751 
                 9.731890 
               
               
                   
                 73.825959 
                 9.709181 
               
               
                   
                 74.254144 
                 9.685573 
               
               
                   
                 74.682304 
                 9.661067 
               
               
                   
                 75.110436 
                 9.635663 
               
               
                   
                 75.538539 
                 9.609360 
               
               
                   
                 75.966611 
                 9.582160 
               
               
                   
                 76.394649 
                 9.554063 
               
               
                   
                 76.822652 
                 9.525067 
               
               
                   
                 77.250617 
                 9.495174 
               
               
                   
                 77.678542 
                 9.464384 
               
               
                   
                 78.106425 
                 9.432696 
               
               
                   
                 78.534265 
                 9.400111 
               
               
                   
                 78.962058 
                 9.366628 
               
               
                   
                 79.389804 
                 9.332249 
               
               
                   
                 79.817499 
                 9.296973 
               
               
                   
                 80.245142 
                 9.260800 
               
               
                   
                 80.672731 
                 9.223730 
               
               
                   
                 81.100263 
                 9.185764 
               
               
                   
                 81.527737 
                 9.146902 
               
               
                   
                 81.955150 
                 9.107143 
               
               
                   
                 82.382501 
                 9.066489 
               
               
                   
                 82.809787 
                 9.024939 
               
               
                   
                 83.237006 
                 8.982493 
               
               
                   
                 83.664156 
                 8.939151 
               
               
                   
                 84.091236 
                 8.894914 
               
               
                   
                 84.518242 
                 8.849782 
               
               
                   
                 84.945173 
                 8.803755 
               
               
                   
                 85.372028 
                 8.756833 
               
               
                   
                 85.798803 
                 8.709017 
               
               
                   
                 86.225496 
                 8.660306 
               
               
                   
                 86.652106 
                 8.610702 
               
               
                   
                 87.078631 
                 8.560203 
               
               
                   
                 87.505069 
                 8.508810 
               
               
                   
                 87.931416 
                 8.456524 
               
               
                   
                 88.357672 
                 8.403345 
               
               
                   
                 88.783835 
                 8.349272 
               
               
                   
                 89.209901 
                 8.294307 
               
               
                   
                 89.635870 
                 8.238449 
               
               
                   
                 90.061738 
                 8.181699 
               
               
                   
                 90.487505 
                 8.124056 
               
               
                   
                 90.913168 
                 8.065522 
               
               
                   
                 91.338724 
                 8.006096 
               
               
                   
                 91.764173 
                 7.945779 
               
               
                   
                 92.189511 
                 7.884570 
               
               
                   
                 92.614736 
                 7.822471 
               
               
                   
                 93.039848 
                 7.759482 
               
               
                   
                 93.464843 
                 7.695602 
               
               
                   
                 93.889719 
                 7.630832 
               
               
                   
                 94.314475 
                 7.565172 
               
               
                   
                 94.739108 
                 7.498623 
               
               
                   
                 95.163616 
                 7.431185 
               
               
                   
                 95.587998 
                 7.362858 
               
               
                   
                 96.012251 
                 7.293642 
               
               
                   
                 96.436373 
                 7.223539 
               
               
                   
                 96.860362 
                 7.152547 
               
               
                   
                 97.284216 
                 7.080668 
               
               
                   
                 97.707933 
                 7.007902 
               
               
                   
                 98.131511 
                 6.934249 
               
               
                   
                 98.554948 
                 6.859709 
               
               
                   
                 98.978242 
                 6.784283 
               
               
                   
                 99.401391 
                 6.707971 
               
               
                   
                 99.824392 
                 6.630774 
               
               
                   
                 100.247244 
                 6.552691 
               
               
                   
                 100.669944 
                 6.473724 
               
               
                   
                 101.092491 
                 6.393872 
               
               
                   
                 101.514883 
                 6.313136 
               
               
                   
                 101.937117 
                 6.231517 
               
               
                   
                 102.359191 
                 6.149014 
               
               
                   
                 102.781104 
                 6.065628 
               
               
                   
                 103.202853 
                 5.981359 
               
               
                   
                 103.624437 
                 5.896208 
               
               
                   
                 104.045853 
                 5.810176 
               
               
                   
                 104.467099 
                 5.723262 
               
               
                   
                 104.888173 
                 5.635467 
               
               
                   
                 105.309073 
                 5.546792 
               
               
                   
                 105.729798 
                 5.457236 
               
               
                   
                 106.150344 
                 5.366801 
               
               
                   
                 106.570711 
                 5.275486 
               
               
                   
                 106.990895 
                 5.183292 
               
               
                   
                 107.410896 
                 5.090220 
               
               
                   
                 107.830710 
                 4.996270 
               
               
                   
                 108.250337 
                 4.901442 
               
               
                   
                 108.669773 
                 4.805737 
               
               
                   
                 109.089017 
                 4.709155 
               
               
                   
                 109.508067 
                 4.611698 
               
               
                   
                 109.926921 
                 4.513364 
               
               
                   
                 110.345576 
                 4.414155 
               
               
                   
                 110.764031 
                 4.314071 
               
               
                   
                 111.182284 
                 4.213112 
               
               
                   
                 111.600333 
                 4.111280 
               
               
                   
                 112.018175 
                 4.008574 
               
               
                   
                 112.435809 
                 3.904996 
               
               
                   
                 112.853233 
                 3.800545 
               
               
                   
                 113.270444 
                 3.695222 
               
               
                   
                 113.687441 
                 3.589027 
               
               
                   
                 114.104222 
                 3.481961 
               
               
                   
                 114.520784 
                 3.374025 
               
               
                   
                 114.937125 
                 3.265219 
               
               
                   
                 115.353244 
                 3.155544 
               
               
                   
                 115.769139 
                 3.044999 
               
               
                   
                 116.184807 
                 2.933587 
               
               
                   
                 116.600247 
                 2.821306 
               
               
                   
                 117.015456 
                 2.708158 
               
               
                   
                 117.430433 
                 2.594143 
               
               
                   
                 117.845175 
                 2.479262 
               
               
                   
                 118.259681 
                 2.363516 
               
               
                   
                 118.673948 
                 2.246904 
               
               
                   
                 119.087975 
                 2.129427 
               
               
                   
                 119.501759 
                 2.011087 
               
               
                   
                 119.915299 
                 1.891883 
               
               
                   
                 120.328592 
                 1.771816 
               
               
                   
                 120.741637 
                 1.650887 
               
               
                   
                 121.154431 
                 1.529096 
               
               
                   
                 121.566973 
                 1.406444 
               
               
                   
                 121.979261 
                 1.282931 
               
               
                   
                 122.391291 
                 1.158559 
               
               
                   
                 122.803064 
                 1.033327 
               
               
                   
                 123.214576 
                 0.907236 
               
               
                   
                 123.625825 
                 0.780287 
               
               
                   
                 124.036811 
                 0.652481 
               
               
                   
                 124.447529 
                 0.523817 
               
               
                   
                 124.857980 
                 0.394298 
               
               
                   
                 125.268160 
                 0.263922 
               
               
                   
                 125.678067 
                 0.132692 
               
               
                   
                 126.087701 
                 0.000607 
               
               
                   
                   
               
             
          
         
       
     
     A pump casing can be divided into convenient sizes and manufactured in sections. Finished parts can be assembled with nuts and bolts provided in the package, following instructions, to form a casing of the desired curvature. 
     FIG. 9 describes the disassembled parts of an eccentric rotor vane pump of horizontal design, and FIG. 10 describes the disassembled parts of an eccentric rotor vane pump of vertical design. FIG. 11 shows the components of the eccentric rotor vane pump described in this invention. In the manufacture of large-sized casings using the existing manufacturing method, the entire casing is manufactured as a single piece and the size of the rotor increases in proportion to the size of the casing. In this case the processing of the accurate guide surface which meets with the sliding, scraping vane is severely disabled. In order to overcome this limitation, two semi-circular rotors ( 5  and  5 ′) are separately manufactured, as shown in FIG.  11 . On the inside of each semi-circular rotor, guide grooves ( 7 ′) are formed to match the projecting parts {circle around ( 7 )} on both sides of vane {circle around ( 3 )}, so that the projecting parts can move along the grooves when the vane slides back and forth. The casing parts ( 1  and  6 ) are held together with bolts and side covers ( 9  and  9 ′) are tightly placed on the open sides of the casing also using bolts. The rotating discs ( 8  and  8 ′) drives the eccentric rotor ( 2 ) to otates in close contact with the inner surface of the casing. The sealing parts ( 10  and  10 ′) are fitted inside the side covers ( 9  and  9 ′), and sealing liquid is applied to the contacting surfaces between the sealing parts and the rotating discs ( 8  and  8 ′) and shafts ( 12  and  12 ′). The bearing boxes ( 11  and  11 ′) are attached to the sealing parts using bolts, to support the rotating shafts ( 12  and  12 ′). 
     The reference number  13  denotes the fluid inlet and the number  14 , the fluid outlet. The number  16 ,  17  and  18  in the figures refer to bolts and nuts provided in the package. The number  15  in FIG. 10 denotes the thrust bearing which is used to support the weight of an eccentric rotor oI vertical shaft. 
     In an eccentric rotor vane pump of vertical shaft as shown in FIG. 10, the rotor experiences increasing weight as casing size increases. In addition to the lower shaft and the bearing in the bearing box, therefore, a large-sized pump as an in-built thrust bearing to support the weight and thus allow smooth rotations regardless of the rotor weight. As casing size increases, weight of the vane also increases. For this reason, vane {circle around ( 3 )} is designed to reciprocate horizontally, along the guide faces of the vertical axial rotor. So the vane can slide and scrape the inner surface of the casing in close contact, no matter how large casing size and vane weight may be. Friction and centrifugal force generated by the rotating vane of a large-sized pump can also be greatly reduced. The weight of vane {circle around ( 3 )} still affects the horizontal movement of the vane, while due to horizontal rotations the two ends of the vane, sliding and scraping in contact with the curved surface of the casing, can no longer affect the gravitational pull on the vane. Therefore vane {circle around ( 3 )} is designed to contain the appropriate number of convex parts ( 7 ), and the semi-circular rotors, the same number of grooves ( 7 ′) as convex parts. Or a suitable device such as beating is installed at the center of mass on the upper or bottom side of the vane, so as to absorb and reduce the weight of vane {circle around ( 3 )}. As a result, the eccentric rotor vane pump of this design can undertake smooth horizontal movement, which is one of the major purports of this invention. 
     Springless eccentric rotor vane pumps (of either horizontal or vertical shaft) with cardiocle and expanded cardioid casings derived from Equations (5) and (6), as explained above, solve the limitations of, and problems posed by, current eccentric rotor vane pumps. Processing of large-size pumps is now possible with mathematical formation of casing curatures, hitherto regarded as impossible. In addition, as these pumps can perform more revolutions per unit time, pump size can be greatly reduced; pumps one-fifth the size of curtent large-size, large-output pumps can produce the same amounts of output. Moreover the achievement of exact mathematical descriptions of the cardiocle and expanded cardioid is opening a new chapter in pump technology in terms of analytic geometry. 
     The following section on ‘what is claimed’ merely suggests a few applications of this invention. Further changes or corrections are still possible, but these are conceptually part of the invention.