Abstract:
The present invention relates to a method of recognizing and tracking a spatial point, and more particularly to a method of using a point light source and a spatial point recognition device to measure the coordinates of the point light source and the coordinates of the convergent point of the spatial point recognition device based on the principle of parallax of human eyes, as to achieve the purpose of recognizing the position of a spatial point. Further, the spatial point recognition device is capable of moving the convergent point, such that the coordinates of the convergent point are superimposed onto the coordinates of the point light source, so as to achieve the purpose of tracking a spatial point automatically. At the same time, the spatial point recognition device can also receive the coordinates of a new convergent point to reset the position of the convergent point, so as to achieve the purpose of resetting the convergent point.

Description:
CROSS REFERENCE TO RELATED PATENT APPLICATION 
     This patent application is based on Taiwan, R.O.C. patent application No. 096108692 filed on Mar. 14, 2007. 
     FIELD OF THE INVENTION 
     The present invention relates to a method of recognizing and tracking a spatial point, and more particularly to a method of using a point light source and a spatial point recognition device to measure the coordinates of the point light source and the coordinates of the convergent point of the spatial point recognition device based on the principle of parallax of human eyes, as to achieve the purpose of recognizing the position of a spatial point. Further, the spatial point recognition device is capable of moving the convergent point, such that the coordinates of the convergent point are superimposed onto the coordinates of the point light source, so as to achieve the purpose of tracking a spatial point automatically. At the same time, the spatial point recognition device can also receive the coordinates of a new convergent point to reset the position of the convergent point, so as to achieve the purpose of resetting the convergent point. 
     BACKGROUND OF THE INVENTION 
       FIG. 1  shows a schematic view of imaging of a 2D optical lens. 
     A general 2D optical lens is formed by a spherical lens  1 . After an object point o of a paraxial point light source goes through a 2D optical lens  1 , a characteristic of the image includes a point-shaped image point i xy , and the relation of its image position follows the principle of geometric optics as shown in Equation (1) as listed below. 
                       1     l   o       +     1     l   i         =     1   f             (   1   )               
where, l o  is an object distance of the point light source o, and l i  is an image distance, and f is a focal length of the optical lens  1 . In addition, another characteristic of the geometric optical image resides on that a non-deviated light exists between the point light source o and the image point i xy , and the light passes through a geometric center O lens  of the 2D optical lens  1 .
 
       FIG. 2  is a schematic view of forming an image by a 1D optical lens. 
     A general 1D optical lens is formed by semi-cylindrical lenses  2 ,  3 , and its imaging principle follows the theory of geometric optics, but the 1D optical lens has the 1D imaging ability only. Therefore, 1D longitudinal focusing optical lens  2  transforms a point light source into a transverse line image i y , and a 1D transverse focusing optical lens  3  transforms a point light source into a longitudinal line image i x , and the image transformation also follows the principle of geometric optics as listed in Equation (1). 
       FIG. 3  is a schematic view of a general optical camera. The camera  5  can be a general optical camera, a digital camera, or a camcorder, comprising an adjustable aperture  6 , a variable focusing lens module  7 , an image sensing and recording device  8 . For the object point o 1 , an appropriate adjustment of the focal length f of the variable focusing lens module  7  is made to obtain an image point i 1 , and form an image on the image recording device  8 . In general, a relation of l i ≈f can be obtained due to the object distance l 0  being larger than the focal length f. 
     For an object point o 2  at a different distance, the variable focusing lens module  7  is used for obtaining another image point i 2 . By an appropriate adjustment of the size of the adjustable aperture  6 , another similar image point i 2  can be obtained on the image recording device  8 . 
       FIG. 4(   a ) is a schematic view of a structure of a human eyeball. 
     Human eyeball  10  is similar to an optical camera, and mainly comprises an iris  11 , an eye crystalline lens  13  and a retina  15 . Compared with the optical camera, the function of the iris  11  is to adjust the size of the pupil  12  and can be considered as an adjustable aperture; the eye crystalline lens  13  serves as a lens, such that the ciliaris muscle  14  can be moved to change the curvature (or focal length) of the eye crystalline lens  13 , and thus can be considered as a variable focusing lens; and the retina  15  can be considered as an image sensor for transmitting an obtained image to a brain through the optic nerve  16 , and the brain processes, stores and recognizes the visual space. In general, the viewing direction of the eyeball  10  (or known as the visual axis  17 , which is the optical axis of the optical system of the eyeball) bases on the up-and-down and left-and-right movements of the eyeball  10  to change the direction of the visual axis  17  in a limited extent. Further, the left-and-right and up-and-down rotation of the neck can change the direction of the visual axis  17  is a great extent. 
       FIG. 4(   b ) is a schematic view of a structure of a visual space. 
     As to the left and right eyes  21 ,  22 , the visual space refers to the space existed and observed by the left and right eyes  21 ,  22 . Firstly, a world coordinate system O(X, Y, Z) is defined, such that î, ĵ, {circumflex over (k)} are unit vectors of the coordinate axes in the coordinate system. The coordinate axes of the world coordinate system are fixed, and the world coordinate system constitutes the space to define the visual space. Further, another rotating but immoving neck coordinate system O N (X N , Y N , Z N ) is set at the origin (0, 0, 0) of the world coordinate system O(X, Y, Z) and î N , ĵ N , {circumflex over (k)} N  are set to be the unit vectors of the coordinate axes in the coordinate system, and the coordinate system O N (X N , Y N , Z N ) is rotated such that the axis Z N  can be rotated to an angle Θ with respect to the axis Y N , and the axis X N  can be rotated to an angle Φ as shown in  FIG. 4(   c ). 
     Further, a rotating but immoving left eye coordinate system O L (X L , Y L , Z L ) is set at the position (S/2, H, 0) of the neck coordinate system O N (X N , Y N , Z N ), and û LX , û LY , û LZ  are set to be the unit vectors of the coordinate axes, and a rotating but immoving right eye coordinate system O R (X R , Y R , Z R ) is set at the position (−S/2, H, 0) of O N (X N , Y N , Z N ), and û RX , û RY , û RZ  are set to be the unit vectors of the coordinate axes. Therefore O N (X N , Y N , Z N ) can be rotated to an angle of Θ or Φ to drive the rotation of O L (X L , Y L , Z L ), O R (X R , Y R , Z R ) and the point O H  respectively, and Θ and Φ can be considered as angles of left-and-right (horizontal) and up-and-down (vertical) rotations of the neck. S is defined as the stereo base; O H  is defined as the center of stereo base; and the length of H is considered as the height of cervical (neck) spine as shown in  FIGS. 4(   f ) and  4 ( g ). 
     In  FIG. 4(   b ), a point F L  at the position (0, 0, f) on the left eye coordinate system O L (X L , Y L , Z L ) is defined as the center position of the left eye crystalline lens. The axis Z L  of the coordinate system O L (X L , Y L , Z L ) can be rotated to an angle θ L  with respect to the axis Y L , and an angle φ L  with respect to the axis X L  as shown in  FIG. 4(   d ). Therefore, the angles θ L , φ L  can be considered as the angles of left-and-right (horizontal) and up-and-down (vertical) rotations similar to the rotations of the left eyeball as shown in  FIG. 4(   f ) and  FIG. 4(   g ) respectively. Since the O L (X L , Y L , Z L ) of the left eye coordinate system is set on the retina, therefore the retina disposed proximate to the origin of the coordinates can be considered to be disposed on the plane X L -Y L , and the plane X L -Y L  is defined as the plane of the left image. 
     In  FIG. 4(   b ), a point F R  at the position (0, 0, f) on the right eye coordinate system O R (X R , Y R , Z R ) is defined as of the center position of the right eye crystalline lens. The axis Z R  of the coordinate system O R (X R , Y R , Z R ) can be rotated to an angle θ R  with respect to the axis Y R , and an angle of φ R  with respect to the axis the axis X R  as shown in  FIG. 4(   e ). Therefore, the angles θ R , φ R  can be considered as the angles of the rotations similar to left-and-right (horizontal) and up-and-down (vertical) rotations of the right eyeball as shown in  FIGS. 4(   f ) and  4 ( g ). Since the right eye coordinate system O R (X R , Y R , Z R ) is set on the retina therefore the retina disposed on the origin of the coordinates can be considered to be disposed on the plane X R -Y R , and thus the plan X R -Y R  is defined as the plant of the right image. 
       FIG. 4(   h ) is a schematic geometric view of a (convergent point), an object point and each coordinate when two eyes are staring at a point. If two eyes are staring at a point in the front, the visual axes of two eyes are intersected at a point which is called a point of view or convergent point V. In other words, the axes Z L , Z R  are the visual axes of the left and right eyes  21 ,  22  respectively, and the two visual axes must be intersected at a point which is the convergent point V. In a world coordinate system O(X, Y, Z), the coordinates of the convergent point V are (X V , Y V , Z V ). In the neighborhood of the convergent point V, there is an object point P, and in the world coordinate system O(X, Y, Z), the coordinates of the object point P are (X P , Y P , Z P ). Further, the position of a point image I L  of the object point P formed by the left eye crystalline lens at the coordinate system of the left eye retina O L (X L , Y L , Z L ) are I L (x L , y L , 0); and the position of a point image I R  of the object point P formed by the right eye crystalline lens at the O R (X R , Y R , Z R ) coordinate system of the right eye retina are I R (x R , y R , 0). In general, the angles (θ L , θ R , φ L , φ R , Θ, Φ) of the visual axes of both left and right eyes are adjusted appropriately to superimpose the convergent point V with the object point P to achieve the purpose of tracking and staring at an object. 
     As to human eyes, the up-and-down movements of the left and right eyeballs are limited to the same angle due to the evolution of human vision to recognize the 3D space effectively. In other words, φ L =φ R =φ. The condition of this limitation gives rise to a very important result which illustrates the vertical coordinate of point image I L , I R  on the retina is consistent. In other words, y L =y R . Another necessary condition of causing such result is that the focal length of both left and right eyes must be equal to assure the consistency of the size of an image formed at the left and right eyes. The so-called parallax phenomenon refers to the difference between both images of an object point P, other than the convergent point V, formed on the retinas of the left and right eyes. In other words, x L ≠x R . With the parallax, a human eye is capable of recognizing the distance of the space. Further, the value of a stereo base S is the most fundamental factor of determining the magnitude of the parallax. For the recognition of an object at a farther end, the parallax effect can be improved by increasing the stereo base S. As to the space recognition of a human vision, the angles (θ L , θ R , φ, Θ, Φ, S, f) are defined as a group of convergent parameters, and the parameters determine the parallax effect to achieve the purpose of recognizing the space. 
     SUMMARY OF THE INVENTION 
     Therefore, it is a primary objective of the present invention to overcome the shortcomings of the prior art by using the current existing prior art to achieve the effect of human eyes of recognizing the space distance. The invention provides a method of recognizing a spatial point based on the principle of parallax of human eyes to the space recognition, and uses a point light source and a spatial point recognition device to measure the coordinates of an object point of the point light source, and the coordinates of the convergent point of the spatial point recognition device to achieve the effect of recognizing the position of a spatial point light source. Further, the spatial point recognition device can move the convergent point to superimpose the coordinates of the convergent point with the coordinates of the object point to achieve the purpose of tracking the object point automatically. In the meantime, the spatial point recognition device can receive the coordinates of an external new convergent point to achieve the purpose of resetting the coordinates of a convergent point. 
     To make it easier for our examiner to understand the description of the present invention, we use preferred embodiments and related drawings with respective numerals to illustrate the invention. 
     The calculating method based on the theory of the parallax is described first, and then the preferred embodiments will be described as follows. 
       FIG. 5(   d ) is a schematic view of a geometric relation of each coordinate system, convergent point, object point and image point of a known visual axis parameters of visual axes. According to the principle of parallax, the coordinates of the convergent point V(X V , Y V , Z V ) and the object point P(X P , Y P , Z P ) can be calculated by using the known conditions such as the convergent parameters visual axis (θ L , θ R , φ, Θ, Φ, S, f) and the point image coordinates I L (x L , y L , 0), I R (x R , y R , 0) of the left and right eyes. Also, the new convergent parameters can be obtained to achieving the tracking purpose. 
     1. Define each coordinate system at an initial state. 
       FIG. 4(   b ) is a schematic view of a visual space at an initial state. 
     In the figure showing the initial geometric relation of each coordinate system, a neck coordinate system O N (X N , Y N , Z N ), a left eye coordinate system O L (X L , Y L , Z L ), and a right eye coordinate system O R (X R , Y R , Z R ) in the world coordinate system O(X, Y, Z) are set before the angles θ L , θ R , φ, Θ, Φ are rotated, and the origin of each coordinate system is defined as follows: 
     
       
         
           
             
               
                 
                   
                     
                       O 
                       N 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       O 
                       L 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               S 
                               2 
                             
                           
                         
                         
                           
                             H 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       O 
                       R 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               - 
                               
                                 S 
                                 2 
                               
                             
                           
                         
                         
                           
                             H 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In addition, the unit vector of each coordinate axis of the world coordinate system O(X, Y, Z) is defined as follows:
 
î=(1, 0, 0) , ĵ=(0, 1, 0) , {circumflex over (k)}=(0, 0, 1)  (3)
 
     The unit vector of each coordinate axis on each coordinate system with respect to the world coordinate system O(X, Y, Z), is defined as follows:
 
î N =(1, 0, 0), ĵ N =(0, 1, 0), {circumflex over (k)} N =(0, 0, 1)  (4)
 
û LX =(1, 0, 0), û LY =(0, 1, 0), û LZ =(0, 0, 1)  (5)
 
û RX =(1, 0, 0), û RY =(0, 1, 0), û RZ =(0, 0, 1)  (6)
 
     2. Rotation of the Neck Parameter Θ and Φ 
       FIG. 5(   a ) shows a relative geometric relation of each coordinate system, when the neck coordinate system O N (X N , Y N , Z N ) rotates angle Θ about the axis Y N  first.  FIG. 5(   b ) shows a relative geometric relation of each coordinate system after the next rotation of angle Φ about the axis X N . The positive and negative values of the angles Θ, Φ are defined according to the right-hand rule. 
     Since the transformation of coordinate rotations relates to the order of rotations, therefore the angle Θ is rotated first, and then the angle Φ is rotated, and the result will be different from that obtained by rotating the angle Φ first and then the angle of Θ angle later. Therefore, the present invention includes the order of rotations of the left and right eye point image coordinates θ L , θ R , φ L , φ R , and one of the order of rotations is discussed for the purpose of illustrating the invention only, and other orders of rotations will not be described here, since their calculations are the same. 
     To clearly highlight the change of coordinate axes and unit vectors, the neck coordinate system is rotated to an angle of +Θ which means the neck turns to the left side during the rotation about the vertical axis. For the rotation about the horizontal axis, the neck coordinate system is rotated to an angle of +Φ, which means that the neck turns downward. Therefore, the transformation of the left eye coordinate system O L (X L , Y L , Z L ) and the right eye coordinate system O R (X R , Y R , Z R ) in the world coordinate system O(X, Y, Z) can be defined as follows: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           X 
                         
                       
                       
                         
                           Y 
                         
                       
                       
                         
                           Z 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 X 
                                 L 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 L 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 L 
                                 ″ 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       O 
                       L 
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           X 
                         
                       
                       
                         
                           Y 
                         
                       
                       
                         
                           Z 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 X 
                                 R 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 R 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 R 
                                 ″ 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       O 
                       R 
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where, 
     
       
         
           
             
               
                 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       
                         Θ 
                         , 
                         Φ 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Θ 
                           
                         
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Φ 
                           
                         
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Φ 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Φ 
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Φ 
                           
                         
                       
                       
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Θ 
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Φ 
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Φ 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     O 
                     L 
                     ″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                         
                         
                           
                             
                               Y 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                         
                         
                           
                             
                               Z 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Φ 
                               
                               + 
                               
                                 
                                   S 
                                   2 
                                 
                                 ⁢ 
                                 c0s 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                               
                             
                           
                         
                         
                           
                             
                               H 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Φ 
                               
                               - 
                               
                                 
                                   S 
                                   2 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     O 
                     R 
                     ″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                         
                         
                           
                             
                               Y 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                         
                         
                           
                             
                               Z 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Φ 
                               
                               - 
                               
                                 
                                   S 
                                   2 
                                 
                                 ⁢ 
                                 c0s 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                               
                             
                           
                         
                         
                           
                             
                               H 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Φ 
                               
                               + 
                               
                                 
                                   S 
                                   2 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Θ 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     For the world coordinate system O(X, Y, Z), left eye coordinate system O L (X L , Y L , Z L ) and right eye coordinate system O R (X R , Y R , Z R ), after the neck coordinate system O N (X N , Y N , Z N ) is rotated through an angle Θ or Φ, the unit vector of each coordinate axis is changed as follows: 
     
       
         
           
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     LX 
                     ″ 
                   
                   = 
                   
                     
                       
                         u 
                         ^ 
                       
                       RX 
                       ″ 
                     
                     = 
                     
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     LY 
                     ″ 
                   
                   = 
                   
                     
                       
                         u 
                         ^ 
                       
                       RY 
                       ″ 
                     
                     = 
                     
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               0 
                             
                           
                           
                             
                               1 
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     LZ 
                     ″ 
                   
                   = 
                   
                     
                       
                         u 
                         ^ 
                       
                       RZ 
                       ″ 
                     
                     = 
                     
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Equations (12) to (14) are expanded to obtain:
 
 û″   LX   =û″   RX =cos Θ î −sin Θ {circumflex over (k)}   (15)
 
 û″   LY   =û″   RY =sin Θ sin Φ î +cos Φ ĵ +cos Θ sin Φ {circumflex over (k)}   (16)
 
 û″   LZ   =û″   RZ =sin Θ cos Φ{circumflex over ( i )}−sin Φ ĵ +cos Θ cos Φ {circumflex over (k)}   (17)
 
     3. Rotation of Eye Parameters θ L , θ R , φ 
       FIG. 5(   c ) shows a geometric variation relation of the left and right eye coordinate systems being rotated to an angle θ L , θ R , φ. The positive or negative values of the angles θ L , θ R , φ of the left and right eye coordinate systems are defined by the right-hand rule. To clearly highlight the change of the coordinate axes and the unit vectors, the left eye coordinate system is rotated to an angle θ L  when the vertical axis is rotated. In this embodiment, θ L &lt;0 which is a negative value. The right eye coordinate system is rotated to an angle θ R . In this embodiment, θ R &gt;0 which is a positive value. In other words, both left and right eyes are staring at the front side. Further, the horizontal axis of the two are rotated to an angle φ angle. In this embodiment, φ&lt;0 which is a negative value. In other words, both left and right eyes are staring upward. 
     Therefore, the coordinate transformation between the left eye coordinate system O L (X L , Y L , Z L ) and the world coordinate system O(X, Y, Z) is given below: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           X 
                         
                       
                       
                         
                           Y 
                         
                       
                       
                         
                           Z 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         
                           R 
                           L 
                         
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                             , 
                             
                               θ 
                               L 
                             
                             , 
                             φ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 X 
                                 L 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 L 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 L 
                                 ″ 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       O 
                       L 
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     where, 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       L 
                     
                     ⁡ 
                     
                       ( 
                       
                         Θ 
                         , 
                         Φ 
                         , 
                         
                           θ 
                           L 
                         
                         , 
                         φ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 L 
                               
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 L 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 L 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 L 
                               
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 L 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 L 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     The matrix is simplified to obtain: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       L 
                     
                     ⁡ 
                     
                       ( 
                       
                         Θ 
                         , 
                         Φ 
                         , 
                         
                           θ 
                           L 
                         
                         , 
                         φ 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               11 
                             
                           
                         
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               12 
                             
                           
                         
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                       
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               21 
                             
                           
                         
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               22 
                             
                           
                         
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               22 
                             
                           
                         
                       
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               31 
                             
                           
                         
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               32 
                             
                           
                         
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Equation (19) is expanded and substituted into Equation (20) to obtain:
 
 R   L11 =cos Θ cos θ L −sin Θ cos Φ sin θ L   (21)
 
 R   L12 =cos Θ sin θ L  sin φ+sin Θ sin Φ cos φ+sin Θ cos Φ cos θ L  sin φ  (22)
 
 R   L13 =cos Θ sin θ L  cos φ−sin Θ sin Φ sin φ+sin Θ cos Φ cos θ L  cos φ  (23)
 
R L21 =sin Φ sin θ L   (24)
 
 R   L22 =cos Φ cos φ−sin Φ cos θ L  sin φ  (25)
 
 R   L23 =−cos Φ sin φ−sin Φ cos θ L  cos φ  (26)
 
 R   L31 =−sin Θ cos θ L −cos Θ cos Φ sin θ L   (27)
 
 R   L32 =−sin Θ sin θ L  sin φ+cos Θ sin Φ cos φ+cos Θ cos Φ cos θ L  sin φ  (28)
 
 R   L33 =sin Θ sin θ L  cos φ−cos Θ sin Φ sin φ+cos Θ cos Φ cos θ L  cos φ  (29)
 
     Further, the coordinate transformation between the right eye coordinate system O R (X R , Y R , Z R ) and the world coordinate system O(X, Y, Z) is given below: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           X 
                         
                       
                       
                         
                           Y 
                         
                       
                       
                         
                           Z 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         
                           R 
                           R 
                         
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                             , 
                             
                               θ 
                               R 
                             
                             , 
                             φ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 X 
                                 R 
                                 ″″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 R 
                                 ″″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 R 
                                 ″″ 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       O 
                       R 
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     where, 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       R 
                     
                     ⁡ 
                     
                       ( 
                       
                         Θ 
                         , 
                         Φ 
                         , 
                         
                           θ 
                           R 
                         
                         , 
                         φ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Φ 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 R 
                               
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 R 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 R 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 R 
                               
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 R 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 R 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     The matrix is simplified to obtain: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       R 
                     
                     ⁡ 
                     
                       ( 
                       
                         Θ 
                         , 
                         Φ 
                         , 
                         
                           θ 
                           R 
                         
                         , 
                         φ 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               11 
                             
                           
                         
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               12 
                             
                           
                         
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                       
                       
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               21 
                             
                           
                         
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               22 
                             
                           
                         
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               22 
                             
                           
                         
                       
                       
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               31 
                             
                           
                         
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               32 
                             
                           
                         
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     Equation (31) is expanded and substituted into Equation (32) to obtain:
 
 R   R11 =cos Θ cos θ R −sin Θ cos Φ sin θ R   (33)
 
 R   R12 =cos Θ sin θ R  sin φ+sin Θ sin Φ cos φ+sin Θ cos Φ cos θ R  sin φ  (34)
 
 R   R13 =cos Θ sin θ R  cos φ−sin Θ sin Φ sin φ+sin Θ cos Φ cos θ R  cos φ  (35)
 
R R21 =sin Φ sin θ R   (36)
 
 R   R22 =cos Φ cos φ−sin Φ cos θ R  sin φ  (37)
 
 R   R23 =−cos Φ sin φ−sin Φ cos θ R  cos φ  (38)
 
 R   R31 =−sin Θ cos θ R −cos Θ cos Φ sin θ R   (39)
 
 R   R32 =−sin Θ sin θ R  sin φ+cos Θ sin Φ cos φ+cos Θ cos Φ cos θ R  sin φ  (40)
 
 R   R33 =sin Θ sin θ R  cos φ−cos Θ sin Φ sin φ+cos Θ cos Φ cos θ R  cos φ  (41)
 
     Further, the change of unit vectors of the coordinate axes of the left eye coordinate system with respect to the world coordinate system is given below: 
     
       
         
           
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     LX 
                     ″″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               LXi 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               LXj 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               LXk 
                               ″″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             L 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 L 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   21 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   31 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     LY 
                     ″″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               LYi 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               LYj 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               LYk 
                               ″″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             L 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 L 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     LZ 
                     ″″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               LZi 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               LZj 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               LZk 
                               ″″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             L 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 L 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   13 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   23 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   33 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     Further, the change of unit vectors of the coordinate axes of the right eye coordinate system with respect to the world coordinate system O(X, Y, Z) is given below: 
     
       
         
           
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     RX 
                     ″″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               RXi 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               RXj 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               RXk 
                               ″″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 R 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   21 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   31 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     RY 
                     ″″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               RYi 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               RYj 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               RYk 
                               ″″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 R 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     RZ 
                     ″″ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               RZi 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               RZj 
                               ″″ 
                             
                           
                         
                         
                           
                             
                               u 
                               RZk 
                               ″″ 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 R 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   13 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   23 
                                 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   33 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     4. Calculation of Convergent Point 
       FIG. 5(   c ) shows that after the left and right eye coordinate systems O L (X L , Y L , Z L ), O R (X R , Y R , Z R ) is rotated and fixed according to convergent point parameter (θ L , θ R , φ, Θ, Φ, S) with respect to the world coordinate system O(X, Y, Z) (wherein θ L &lt;0, θ R &gt;0 and φ&lt;0), and the intersection point of the left visual axis Z′″ L  and the right visual axis Z′″ R  is the convergent point V(X V , Y V , Z V ). Therefore, the position (X V , Y V , Z V ) of the convergent point V is determined by the convergent point parameter (θ L , θ R , φ, Θ, Φ, S), and calculated as follows: 
     Along the left visual axis Z″″ L , a left convergent point vector  O″ L V′  parallel to the left visual axis is defined as: 
     
       
         
           
             
               
                 
                   = 
                   
                     ( 
                     
                       
                         X 
                         - 
                         
                           X 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                       , 
                       
                         Y 
                         - 
                         
                           Y 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                       , 
                       
                         Z 
                         - 
                         
                           Z 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     Along the right visual axis Z″″ R , a right convergent point vector  O″ R V″  parallel to the right visual axis is defined as: 
     
       
         
           
             
               
                 
                   = 
                   
                     ( 
                     
                       
                         X 
                         - 
                         
                           X 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                       , 
                       
                         Y 
                         - 
                         
                           Y 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                       , 
                       
                         Z 
                         - 
                         
                           Z 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
     Since the left and right convergent point vectors  O″ L V′ ,  O″ R V″  are parallel to the unit vectors û″″ LZ , û″″ RZ  each visual axis, therefore 
     
       
         
           
             
               
                 
                   
                     
                       
                         u 
                         ^ 
                       
                       LZ 
                       ″″ 
                     
                     × 
                   
                   = 
                   
                     
                        
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       i 
                                       ^ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       R 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         13 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 X 
                                 - 
                                 
                                   X 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                   ″ 
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 j 
                                 ^ 
                               
                             
                             
                               
                                 k 
                                 ^ 
                               
                             
                           
                           
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   23 
                                 
                               
                             
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   33 
                                 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 - 
                                 
                                   Y 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                   ″ 
                                 
                               
                             
                             
                               
                                 Z 
                                 - 
                                 
                                   Z 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                   ″ 
                                 
                               
                             
                           
                         
                       
                        
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         u 
                         ^ 
                       
                       RZ 
                       ″″ 
                     
                     × 
                   
                   = 
                   
                     
                        
                       
                         
                           
                             
                               i 
                               ^ 
                             
                           
                           
                             
                               j 
                               ^ 
                             
                           
                           
                             
                               k 
                               ^ 
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                         
                           
                             
                               X 
                               - 
                               
                                 X 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               Y 
                               - 
                               
                                 Y 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               Z 
                               - 
                               
                                 Z 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                         
                       
                        
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
     Equation (50) is expanded to obtain: 
     
       
         
           
             
               
                 
                   Z 
                   = 
                   
                     
                       
                         
                           R 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             33 
                           
                         
                         
                           R 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             23 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Y 
                           - 
                           
                             Y 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       Z 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
             
               
                 
                   X 
                   = 
                   
                     
                       
                         
                           R 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             13 
                           
                         
                         
                           R 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             33 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Z 
                           - 
                           
                             Z 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       X 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       
                         
                           R 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             23 
                           
                         
                         
                           R 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             13 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           X 
                           - 
                           
                             X 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       Y 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     Equation (51) is expanded to obtain: 
     
       
         
           
             
               
                 
                   Z 
                   = 
                   
                     
                       
                         
                           R 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             33 
                           
                         
                         
                           R 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             23 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Y 
                           - 
                           
                             Y 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       Z 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
             
               
                 
                   X 
                   = 
                   
                     
                       
                         
                           R 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             13 
                           
                         
                         
                           R 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             33 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Z 
                           - 
                           
                             Z 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       X 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       
                         
                           R 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             23 
                           
                         
                         
                           R 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             13 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           X 
                           - 
                           
                             X 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       Y 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
     Since the left and right visual axes are intersected at the convergent point V(X V , X V , Z V ), Equations (52) to (54) and Equations (55) to (57) have a common solution X=X V ; Y=Y V ; Z=Z V . 
     Since Equation (52)=Equation (55), therefore 
     
       
         
           
             
               
                 
                   
                     Y 
                     V 
                   
                   = 
                   
                     
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         Y 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                         ″ 
                       
                     
                     - 
                     
                       
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         Y 
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                         ″ 
                       
                     
                     + 
                     
                       
                         
                           Z 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                         - 
                         
                           Z 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                         - 
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
     Since Equation (53)=Equation (56), therefore 
     
       
         
           
             
               
                 
                   
                     Z 
                     V 
                   
                   = 
                   
                     
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         Z 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                         ″ 
                       
                     
                     - 
                     
                       
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         Z 
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                         ″ 
                       
                     
                     + 
                     
                       
                         
                           X 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                         - 
                         
                           X 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                         - 
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
     Since Equation (54)=Equation (57) 
     
       
         
           
             
               
                 
                   
                     X 
                     V 
                   
                   = 
                   
                     
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         X 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                         ″ 
                       
                     
                     - 
                     
                       
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         X 
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                         ″ 
                       
                     
                     + 
                     
                       
                         
                           Y 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                         - 
                         
                           Y 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                       
                       
                         
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                           
                             R 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                         - 
                         
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                           
                             R 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
     Further a central visual axis vector  O″ H V  is defined, and its angle is defined below: 
     
       
         
           
             
               
                 
                   
                     θ 
                     M 
                   
                   = 
                   
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     ⁢ 
                                     g 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         j 
                                         ^ 
                                       
                                       N 
                                       ″ 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     j 
                                     ^ 
                                   
                                   N 
                                   ″ 
                                 
                               
                             
                             
                                
                               
                                 - 
                                 
                                   
                                     ( 
                                     
                                       ⁢ 
                                       g 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           j 
                                           ^ 
                                         
                                         N 
                                         ″ 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     j 
                                     N 
                                     ″ 
                                   
                                 
                               
                                
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               i 
                               ^ 
                             
                             N 
                             ″ 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       π 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
     Further the visual axis convergence angle is defined below:
 
θ c =θ R −θ L   (62)
 
     When two eyes are staring at the right front, the following geometric relations exist:
 
θ M =0; −θ L =θ R =θ 0 &gt;0; θ C =2θ R =2θ 0   (63)
 
     5. Calculation of Object Point 
     In  FIG. 5(   d ), an object point P(X P , Y P , Z P ) exists in the neighborhood of the convergent point V(X V , Y V , Z V ). Let the imaging positions of the object point on the left right eye coordinate system be the left point image I L (x L , y i , 0) and the right point image I R (x R , y i , 0) respectively, and thus the position (X P , Y P , Z P ) of the object point P can be determined by the convergent point parameter (θ L , θ R , φ, Θ, Φ, S, f) and the image position (x L , x R , y i ), and calculated as follows: 
     According to the coordinate transformation in Equations (10) and (18), the coordinates of the left point image on the world coordinate system can be obtained as follows: 
     
       
         
           
             
               
                 
                   
                     I 
                     L 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               X 
                               IL 
                             
                           
                         
                         
                           
                             
                               Y 
                               IL 
                             
                           
                         
                         
                           
                             
                               Z 
                               IL 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             L 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 L 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   x 
                                   L 
                                 
                               
                             
                             
                               
                                 
                                   y 
                                   i 
                                 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
     The coordinate transformation of Equations (11) and (30) can be used for calculating the coordinates of the right point image on the world coordinate system as follows: 
     
       
         
           
             
               
                 
                   
                     I 
                     R 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               X 
                               IR 
                             
                           
                         
                         
                           
                             
                               Y 
                               IR 
                             
                           
                         
                         
                           
                             
                               Z 
                               IR 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 R 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   x 
                                   L 
                                 
                               
                             
                             
                               
                                 
                                   y 
                                   i 
                                 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
     where, Y IL =Y IR    
     Let the coordinates of the center F L , F R  of the left right eye crystalline lens on the left right eye coordinate system be (0, 0, f), and the coordinates of F L , F R  on the world coordinate system are given below: 
     
       
         
           
             
               
                 
                   
                     F 
                     L 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               X 
                               FL 
                             
                           
                         
                         
                           
                             
                               Y 
                               FL 
                             
                           
                         
                         
                           
                             
                               Z 
                               FL 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             L 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 L 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 f 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     R 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               X 
                               FR 
                             
                           
                         
                         
                           
                             
                               Y 
                               FR 
                             
                           
                         
                         
                           
                             
                               Z 
                               FR 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             R 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               Θ 
                               , 
                               Φ 
                               , 
                               
                                 θ 
                                 R 
                               
                               , 
                               φ 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 f 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                                 ″ 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     The world coordinates (X P , Y P , Z P ) of the object point P is calculated by utilizing the properties of linear geometric optics of the object point P, the center F L , F R  of the left right eye crystalline lens, and the left right point image I L , I R  as follows: 
     Equation (66) subtracts Equation (64) to obtain the left point image vector  I L F L    as follows: 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         
                           R 
                           L 
                         
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                             , 
                             
                               θ 
                               L 
                             
                             , 
                             φ 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 - 
                                 
                                   x 
                                   L 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   y 
                                   i 
                                 
                               
                             
                           
                           
                             
                               f 
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     R 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   x 
                                   L 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     12 
                                   
                                 
                                 ⁢ 
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 fR 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   13 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     R 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       21 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   x 
                                   L 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     22 
                                   
                                 
                                 ⁢ 
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 fR 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   23 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     R 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       31 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   x 
                                   L 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     32 
                                   
                                 
                                 ⁢ 
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 fR 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   33 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
     Along the direction  F L P , a left object point vector  F L P′  parallel to  F L P  is defined as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   X 
                                 
                               
                               
                                 
                                   Y 
                                 
                               
                               
                                 
                                   R 
                                 
                               
                             
                             ] 
                           
                           - 
                           
                             
                               
                                 R 
                                 L 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   Θ 
                                   , 
                                   Φ 
                                   , 
                                   
                                     θ 
                                     L 
                                   
                                   , 
                                   φ 
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     f 
                                   
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             [ 
                             
                               
                                 
                                   
                                     X 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                               
                                 
                                   
                                     Y 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                               
                                 
                                   
                                     Z 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   X 
                                   - 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       13 
                                     
                                   
                                   - 
                                   
                                     X 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                             
                             
                               
                                 
                                   Y 
                                   - 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       23 
                                     
                                   
                                   - 
                                   
                                     Y 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                             
                             
                               
                                 
                                   Z 
                                   - 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       33 
                                     
                                   
                                   - 
                                   
                                     Z 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
           
         
       
     
     Since the image is limited by the linear geometric optics, the left point image vector  I L F L    and the left object point vector  F L P′  must be in the same direction, and
 
    I   L   F   L   ×    F   L   P′   =
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           × 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   i 
                                   ^ 
                                 
                               
                               
                                 
                                   j 
                                   ^ 
                                 
                               
                               
                                 
                                   k 
                                   ^ 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         R 
                                         
                                           L 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           11 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       x 
                                       L 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         12 
                                       
                                     
                                     ⁢ 
                                     
                                       y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       13 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         R 
                                         
                                           L 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           21 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       x 
                                       L 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         22 
                                       
                                     
                                     ⁢ 
                                     
                                       y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       23 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         R 
                                         
                                           L 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           31 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       x 
                                       L 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         32 
                                       
                                     
                                     ⁢ 
                                     
                                       y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       33 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   X 
                                   - 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       13 
                                     
                                   
                                   - 
                                   
                                     X 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                               
                                 
                                   Y 
                                   - 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       23 
                                     
                                   
                                   - 
                                   
                                     Y 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                               
                                 
                                   Z 
                                   - 
                                   
                                     fR 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       33 
                                     
                                   
                                   - 
                                   
                                     Z 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
           
         
       
     
     Each component of the three axes is zero after equation (70) is expanded, and thus obtaining following three equations: 
     
       
         
           
             
               
                 
                   Z 
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   31 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 32 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   21 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 22 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Y 
                           - 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                           - 
                           
                             Y 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       fR 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         33 
                       
                     
                     + 
                     
                       Z 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
             
               
                 
                   X 
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 12 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   31 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 32 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Z 
                           - 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                           - 
                           
                             Z 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       fR 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         13 
                       
                     
                     + 
                     
                       X 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   21 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 22 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 12 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           X 
                           - 
                           
                             fR 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                           - 
                           
                             X 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       fR 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         23 
                       
                     
                     + 
                     
                       Y 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   73 
                   ) 
                 
               
             
           
         
       
     
     Further equation (67) subtracts Equation (65) to obtain a right image point vector  I R F R    as follows: 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         
                           R 
                           R 
                         
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             , 
                             Φ 
                             , 
                             
                               θ 
                               R 
                             
                             , 
                             φ 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 - 
                                 
                                   x 
                                   R 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   y 
                                   i 
                                 
                               
                             
                           
                           
                             
                               f 
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     R 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   x 
                                   R 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     12 
                                   
                                 
                                 ⁢ 
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 fR 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   13 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     R 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       21 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   x 
                                   R 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     22 
                                   
                                 
                                 ⁢ 
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 fR 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   23 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     R 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       31 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   x 
                                   R 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     32 
                                   
                                 
                                 ⁢ 
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 fR 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   33 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
     Along the direction of  F R P , a right object point vector  F R P″  parallel to  F R P  is defined as follows: 
     
       
         
           
             
               
           
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
           
         
       
     
     Since the image is limited by the linear geometric optics, the right point image vector  I R F R    and the right object point vector  F R P″  must be in the same direction, and 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           × 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   i 
                                   ^ 
                                 
                               
                               
                                 
                                   j 
                                   ^ 
                                 
                               
                               
                                 
                                   k 
                                   ^ 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         R 
                                         
                                           R 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           11 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       x 
                                       R 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       
                                         R 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         12 
                                       
                                     
                                     ⁢ 
                                     
                                       y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     fR 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       13 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         R 
                                         
                                           R 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           21 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       x 
                                       R 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       
                                         R 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         22 
                                       
                                     
                                     ⁢ 
                                     
                                       y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     fR 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       23 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         R 
                                         
                                           R 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           31 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       x 
                                       R 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       
                                         R 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         32 
                                       
                                     
                                     ⁢ 
                                     
                                       y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     fR 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       33 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   X 
                                   - 
                                   
                                     fR 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       13 
                                     
                                   
                                   - 
                                   
                                     X 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                               
                                 
                                   Y 
                                   - 
                                   
                                     fR 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       23 
                                     
                                   
                                   - 
                                   
                                     Y 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                     ″ 
                                   
                                 
                               
                               
                                 
                                   Z 
                                   - 
                                   
                                     fR 
                                     
                                       R 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       33 
                                     
                                   
                                   - 
                                   
                                     Z 
                                     LR0 
                                     ″ 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
           
         
       
     
     Each component of the three axes is zero after equation (76) is expanded, and thus obtaining following three equations: 
     
       
         
           
             
               
                 
                   Z 
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   31 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               R 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 32 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   21 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               R 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 22 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Y 
                           - 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                           - 
                           
                             Y 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       fR 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         33 
                       
                     
                     + 
                     
                       Z 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
             
               
                 
                   X 
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               R 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 12 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   31 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               R 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 32 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           Z 
                           - 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               33 
                             
                           
                           - 
                           
                             Z 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       fR 
                       
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           13 
                         
                         ⁢ 
                         
                             
                         
                       
                     
                     + 
                     
                       X 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   21 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               R 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 22 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               23 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                             
                             ⁢ 
                             
                               x 
                               R 
                             
                           
                           - 
                           
                             
                               R 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 12 
                               
                             
                             ⁢ 
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           X 
                           - 
                           
                             fR 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               13 
                             
                           
                           - 
                           
                             X 
                             
                               R 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ″ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       fR 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         23 
                       
                     
                     + 
                     
                       Y 
                       
                         R 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
           
         
       
     
     Since the left object point vector  F L P′  and the right object point vector  f R P″  are intersected at the object point P(X P , Y P , Z P ), Equations (71)˜(73) and Equations (77)˜(79) have common solutions X=X P ; Y=Y P ; Z=Z P . 
     From Equation (71)=Equation (77), we obtain 
     
       
         
           
             
               
                 
                   
                     Y 
                     P 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                             + 
                             
                               Y 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                             + 
                             
                               Y 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             + 
                             
                               Z 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                         - 
                         
                           ( 
                           
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             + 
                             
                               Z 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
           
         
       
     
     From Equation (72)=Equation (78), we obtain 
     
       
         
           
             
               
                 
                   
                     Z 
                     p 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             + 
                             
                               Z 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                             + 
                             
                               Z 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
                           fR 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             13 
                           
                         
                         + 
                         
                           X 
                           
                             R 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           ″ 
                         
                         - 
                         
                           ( 
                           
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                             + 
                             
                               X 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     31 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   32 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 33 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
     From Equation (73)=Equation (79), we obtain 
     
       
         
           
             
               
                 
                   
                     X 
                     p 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                             + 
                             
                               X 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                               ″ 
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 L 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   12 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 13 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     21 
                                   
                                 
                               
                               ⁢ 
                               
                                 x 
                                 R 
                               
                             
                             - 
                             
                               
                                 R 
                                 
                                   R 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   22 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               fR 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 23 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   R 
                                   
                                     R 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
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                                 ⁢ 
                                 
                                     
                                 
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                                     ⁢ 
                                     
                                         
                                     
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                                   ⁢ 
                                   
                                       
                                   
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                               ⁢ 
                               
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                                 13 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
           
         
       
     
     6. Calculation of the Tracking of an Object Point 
     To clearly recognize spaces and perceive the distance and position of the object point, we must move the convergence of our eyes constantly to achieve the purpose of tracking the object point. The tracking can be divided into a small-angle tracking and a large-angle tracking according to the tracking range. The small-angle tracking refers to the rotation of eyeballs for changing (θ L , θ R , φ) when the object point is closer to the convergent point. In other words, a smaller change of angle is used for changing the convergent point, and moving the position of the convergent point towards the object point. The large-angle object refers to the rotations of neck and eyeballs for the object point farther from the convergent point. In other words, a proper convergent point parameter (Θ, Φ, θ L , θ R , φ) is reset to achieve the purpose of tracking the object point. In general, we usually adjust the angle of our eyeballs and neck at the same time unconsciously. In other words, (Θ, Φ, θ L , θ R , φ) keeps the convergent point at the most attentive object point all the time to achieve the purpose of perceiving the distance and position of the object point. 
     Calculation of Small-Angle Tracking 
       FIG. 5(   e ) shows a schematic view of small-angle tracking. 
     As described before, changing the angle (θ L , θ R , φ) of the eyeballs or rotating the visual axes of the left and right eyes is the way to achieve the purpose of small-angle tracking when an object point that is closer to the convergent point, so that the difference between the direction û LZ   new , û RZ   new  of the new visual axes of the left and right eyes and the direction û″ LZ , û″ RZ  of the original visual axes can be used for finding the change of angle Δθ L , Δθ R , Δφ required for the small-angle tracking visual axis. 
     Firstly, the unit vectors û LZ   new , û RZ   new  of the new visual axes of the left and right eyes are calculated as follows: 
     
       
         
           
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     LZ 
                     new 
                   
                   = 
                   
                     
                        
                        
                     
                   
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     RZ 
                     new 
                   
                   = 
                   
                     
                        
                        
                     
                   
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
           
         
       
     
     To simplify the calculation and description of the equations, û LZ   new , û RZ   new  are decomposed into vertical and horizontal components, and defined below:
 
 û   LZ   new | ⊥ =( û   LZ   new    û″″   LY ) û″″   LY   ; û   LZ   new   |=û   LZ   new −( û   LZ   new    û″″   LY ) û″″   LY   (85)
 
 û   RZ   new | ⊥ =( û   RZ   new    û″″   RY ) û″″   RY   ; û   RZ   new   |=û   RZ   new −( û   RZ   new    û″″   RY ) û″″   RY   (86)
 
     Track a Convergent Point in the Vertical Direction. 
     Tracking a convergent point in the vertical direction refers to the calculation of Δφ along the vertical direction, compared with the old visual axis and the new visual axis û LZ   new . As described above, the characteristics of human eyes parallax indicate that the angles of the left and right visual axis in the vertical direction are nearly the same, and thus it is necessary to calculate the angle for one of the eyes only, and the calculation is given below:
 
Δφ=cos −1 ( û   LZ   new    û″″   LY )−π/2  (87)
 
     (1) Track the Convergent Point Upward (Δφ&lt;0) 
     For the world coordinate system, if the object point P(X P , Y P , Z P ) is higher than the convergent point V(X V , Y V , Z V ) or Z P &gt;Z V , then cos(û LZ   new  û″″ LY )&gt;0. Therefore, the included angle between û LZ   new  and û″″ LY  is smaller than π/2, and Δφ&lt;0 
     (2) Track a Convergent Point Downward (Δφ&gt;0) 
     If the object point P(X P , Y P , Z P ) is lower than the convergent point V(X V , Y V , Z V ) or Z P &lt;Z V , then cos(û LZ   new  û″″ LY )&lt;0. Therefore, the included angle between û LZ   new  and û″″ LY  is greater than π/2, and Δφ&gt;0. 
     Track a Convergent Point in the Horizontal Direction. 
     The change Δθ L  of convergent point in the horizontal direction can be calculated from the included angle between and û LZ   new | and û″″ LY , and the change Δθ R  can be calculated from the included angle between û LZ   new | and û″″ LY  below: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       L 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             
                               
                                 u 
                                 ^ 
                               
                               LZ 
                               new 
                             
                             ⁢ 
                             
                               | 
                               P 
                             
                           
                           ) 
                         
                         ⁢ 
                         g 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             ^ 
                           
                           LZ 
                           ′′′′ 
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 
                                   u 
                                   ^ 
                                 
                                 LZ 
                                 new 
                               
                               ⁢ 
                               
                                 | 
                                 P 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   LZ 
                                   new 
                                 
                                 ⁢ 
                                 
                                   | 
                                   P 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   88 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       R 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             
                               
                                 u 
                                 ^ 
                               
                               RZ 
                               new 
                             
                             ⁢ 
                             
                               | 
                               P 
                             
                           
                           ) 
                         
                         ⁢ 
                         g 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             ^ 
                           
                           RZ 
                           ′′′′ 
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 
                                   u 
                                   ^ 
                                 
                                 RZ 
                                 new 
                               
                               ⁢ 
                               
                                 | 
                                 P 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   RZ 
                                   new 
                                 
                                 ⁢ 
                                 
                                   | 
                                   P 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
     
     Assumed that the tracking is a small-angle tracking, which means that the change of angle of Δθ L  and Δθ R  is relatively small, the value of the trigonometric function cos Δθ L  or cos Δθ R  is eventually positive, regardless of the angle Δθ L , Δθ R  being positive or negative. Therefore, the positive or negative value of Δθ L  and Δθ R  is determined by the relative position of the object point P(X P , Y P , Z P ) and the original convergent point V(X V , Y V , Z V ) as shown in  FIG. 5(   e ). The determination is divided into the following four types: 
     (3) Track a Convergent Point on the Right Side (Δθ L &lt;0 and Δθ R &lt;0). 
     For the world coordinate system, if the geometric relations  O″ L V ×  O″ L P û″″ LY &lt;0 and  O″ R V ×  O″ R P û″″ RY &lt;0 exist, then the object point P(X P , Y P , Z P ) is considered to be on the right side of the convergent point V(X V , Y V , Z V ), and thus Δθ L &lt;0 and Δθ R &lt;0. 
     (4) Track a Convergent Point on the Left Side (Δθ L &gt;0 and Δθ R &gt;0) 
     For the world coordinate system, if the geometric relations  O″ L V ×  O″ L P û″″ LY &gt;0 and  O″ V V ×  O″ R P û″″ RY &gt;0 exist, then the object point P(X P , Y P , Z P ) is considered to be on the left side of the convergent point V(X V , Y V , Z V ) left, and thus Δθ L &gt;0 and Δθ R &gt;0. 
     (5) Track a Convergent Point at a Near Side (Δθ L &lt;0 and Δθ R &gt;0). 
     For the world coordinate system, if the geometric relations  O″ L V ×  O″ L P û″″ LY &lt;0 and  O″ R V ×  O″ R P û″″ RY &gt;0, then the object point P(X P , Y P , Z P ) is considered to be at the front side of the convergent point V(X V , Y V , Z V ), and thus Δθ L &lt;0 and Δθ R &gt;0. 
     (6) Track a Convergent Point at a Far End (Δθ L &gt;0 and Δθ R &lt;0) 
     For the world coordinate system, if the geometric relations  O″ L V ×  O″ L P û″″ LY &gt;0 and  O″ R V ×  O″ R P û″″ RY &lt;0, exist, then the object point P(X P , Y P , Z P ) is considered to be at the rear side of the convergent point V(X V , Y V , Z V ), and thus Δθ L &gt;0 and Δθ R &lt;0. 
     According to Equations (87)˜(89), the new convergent point parameter is
 
(θ L   New , θ R   New , φ New , Ω New , Φ New   , S   New )=(θ L +Δθ L , θ R +Δθ R   , φ+Δφ, Θ, Φ, S )  (90)
 
     Further, the vector of the central visual axis is changed from  O″ H V  to  O″ H P , and the angle of the new central visual axis is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     θ 
                     M 
                     New 
                   
                   = 
                   
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     ⁢ 
                                     g 
                                     ⁢ 
                                     
                                       
                                         j 
                                         ^ 
                                       
                                       N 
                                       ″ 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     j 
                                     ^ 
                                   
                                   N 
                                   ″ 
                                 
                               
                             
                             
                                
                               
                                 - 
                                 
                                   
                                     ( 
                                     
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       g 
                                       ⁢ 
                                       
                                         
                                           j 
                                           ^ 
                                         
                                         N 
                                         ″ 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       j 
                                       ^ 
                                     
                                     N 
                                     ″ 
                                   
                                 
                               
                                
                             
                           
                           ⁢ 
                           g 
                           ⁢ 
                           
                             
                               i 
                               ^ 
                             
                             N 
                             ″ 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       π 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
           
         
       
     
     Calculation of Large-Angle Tracking 
       FIGS. 5(   f ) and  5 ( g ) show schematic views of a large-angle tracking. 
     As described above, the rotations of a neck and eyeballs change (Θ, Φ, θ L , θ R , φ) and can track an object point at a farther end from the convergent point to achieve the purpose of large-angle tracking. In general, the method of tracking an object point at a farther end from the convergent point usually adjusts the neck parameter (Θ, Φ) to align the central visual axis with the object point by aright posture of  O″ H P ⊥ĵ″ N  first. Therefore, the angle Θ New  of the neck coordinate system is rotated horizontally, such that after the object point P(X P , Y P , Z P ) falls on the plane Y′ N -Z′ N  of the neck coordinate system, then the angle Φ New  of the neck coordinate system is rotated vertically, such that the central visual axis is aligned with the object point P(X P , Y P , Z P ) by the posture of  O″ H P ⊥ĵ″ N . Finally in the neck coordinate system O″ N , the coordinates of the object point P can be (x P , y P , z P ), and thus the coordinate of the object point P along the axis X″ N  is x P =0, and the coordinate of the object point P along the axis Y″ N  is y P =H. Θ New  and Φ New  are calculated as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   x 
                                   P 
                                 
                               
                             
                             
                               
                                 
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                           ] 
                         
                         = 
                           
                         ⁢ 
                         
                           
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                           ⁡ 
                           
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                         = 
                           
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                   ( 
                   92 
                   ) 
                 
               
             
             
               
                 
                   
                     ∴ 
                     
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                   = 
                   
                     
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                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           X 
                           P 
                         
                         
                           Z 
                           P 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   93 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               X 
                               P 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               Θ 
                               New 
                             
                           
                           + 
                           
                             
                               Z 
                               P 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               Θ 
                               New 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
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                         New 
                       
                     
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                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Φ 
                         New 
                       
                     
                   
                   = 
                   H 
                 
               
               
                 
                   ( 
                   94 
                   ) 
                 
               
             
           
         
       
     
     After the new parameter (Θ New , Φ New ) of the neck is decided, the new parameter (θ L   New , θ R   New , φ New ) of the rotation of eyeballs can be determined. Since the central visual axis  O″ H P  has aligned with the object point P(X P , Y P , Z P ), therefore the new parameter of the rotation of eyeballs comes with the following relations: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         θ 
                         L 
                         New 
                       
                     
                     = 
                     
                       
                         θ 
                         R 
                         New 
                       
                       = 
                       
                         θ 
                         New 
                       
                     
                   
                   , 
                   
                     
                       φ 
                       New 
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   95 
                   ) 
                 
               
             
             
               
                 
                   
                     θ 
                     New 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       
                         sin 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             × 
                           
                           
                             
                                
                                
                             
                             ⁢ 
                             
                                
                                
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   96 
                   ) 
                 
               
             
           
         
       
     
     According to Equation (63), the angle of the new central visual axis is
 
θ M   New =0  (97)
 
     The above and other objects, features and advantages of the present invention will become apparent from the following detailed description taken with the accompanying drawing. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic view of forming an image by a 2D optical lens; 
         FIG. 2  is a schematic view of forming an image by a 1D optical lens; 
         FIG. 3  is a schematic view of a structure of a general optical camera; 
         FIG. 4(   a ) is a schematic view of a structure of a human eyeball; 
         FIG. 4(   b ) is a schematic view of an initial status of a visual space; 
         FIG. 4(   c ) is a schematic view of rotating coordinate axes of a neck coordinate system. 
         FIG. 4(   d ) is a schematic view of rotating coordinate axes of a left eye coordinate system; 
         FIG. 4(   e ) is a schematic view of rotating coordinate axes of a right eye coordinate system; 
         FIG. 4(   f ) is a schematic view of rotating left and right eyes in a horizontal direction; 
         FIG. 4(   g ) is a schematic view of rotating an eyeball in a vertical direction; 
         FIG. 4(   h ) is a schematic geometric view of a convergent point, an object point and each coordinate when two eyes are staring at an object; 
         FIG. 5(   a ) shows a relative geometric relation of each coordinate system, when a neck coordinate system rotates in an angle Θ with respect to a Y N , axis; 
         FIG. 5(   b ) shows a relative geometric relation of each coordinate system, when a neck coordinate system rotates in an angle Φ with respect to a X N  axis; 
         FIG. 5(   c ) a schematic view of a geometric variation relation of left and right eye coordinate systems when the left and right eye coordinate systems rotate at an angle of θ L , θ R  or φ; 
         FIG. 5(   d ) is a schematic view of a geometric relation of each coordinate system, convergent point, object point and image point of a known visual axis visual parameter; 
         FIG. 5(   e ) is a schematic view of tracking a small-angle object point; 
         FIG. 5(   f ) is a schematic planar view of tracking a large-angle object point; 
         FIG. 5(   g ) is a schematic view of tracking a large-angle object point; 
         FIG. 6(   a ) is a schematic view of a method in accordance with a first preferred embodiment of the present invention; 
         FIG. 6(   b ) is a schematic view of a spatial point recognition device in accordance with a first preferred embodiment of the present invention; 
         FIG. 6(   c ) is a schematic view of a parallax imaging device in accordance with a first preferred embodiment of the present invention; 
         FIG. 6(   d ) is a schematic view of left and right cameras in accordance with a first preferred embodiment of the present invention; 
         FIG. 6(   e ) is a schematic view of a spatial point computing device in accordance with a first preferred embodiment of the present invention; 
         FIG. 7(   a ) is a schematic view of a method in accordance with a second preferred embodiment of the present invention; 
         FIG. 7(   b ) is a schematic view of a spatial point recognition device in accordance with a second preferred embodiment of the present invention; 
         FIG. 7(   c ) is a schematic view of a parallax imaging device in accordance with a second preferred embodiment of the present invention; 
         FIG. 7(   d ) is a schematic view of left, middle and right cameras in accordance with a second preferred embodiment of the present invention; and 
         FIG. 7(   e ) is a schematic view of a spatial point computing device in accordance with a second preferred embodiment of the present invention. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring to  FIG. 6(   a ) for a schematic view of a method or recognizing a spatial point in accordance with a first preferred embodiment of the present invention, the method  100  comprises a point light source  101  and a spatial point recognition device  105 . The point light source  101  is formed by an active point light source capable of emitting scattered visible light or non-visible light  102  or a passive point light source which is a light reflecting body for reflecting light from other light sources. The spatial point recognition device  105  receives the scattered light  102  emitted by the point light source  101  and output a set of convergent point coordinates signals (X V , Y V , Z V )  190  of the spatial point recognition device  105  and a set of object point coordinates signals (X P , Y P , Z P )  191  of the point light source  101 . Further, the spatial point recognition device is capable of moving the convergent point, such that the convergent point is moved to superimpose the coordinates of the convergent point with the coordinates of the object points to achieve the purpose of tracking the object point automatically. At the same time, the spatial point recognition device  105  can receive the coordinates signal (X V   New , Y V   New , Z V   New )  192  of a new convergent point provided by other external devices to reset the position of the convergent point, so as to and achieve the purpose of resetting the convergent point of the spatial point recognition device  105 . 
       FIG. 6(   b ) shows a schematic view of a spatial point recognition device in accordance with a first preferred embodiment of the present invention, the spatial point recognition device  105  comprises a parallax imaging device  110  and a spatial point computing device  170 . The parallax imaging device  110  receives a light  102  emitted by the point light source  101  and output a set of convergent point parameter signals (θ L , θ R , φ, Ω, Φ, S, f),  150  and a pair of left and right 2D image signals  158  with a parallax signal (x L , x R , y i ). In addition, the parallax imaging device  110  also receives a set of new convergent point parameter signal (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  153  outputted by the spatial point computing device  170  to change the position of the convergent point of the parallax imaging device  110 . The spatial point computing device  170  receives the convergent point parameter signal (θ L , θ R , φ, Ω, Φ, S, f)  150 , and a pair of left and right 2D image signals  158  with a parallax signal (x L , x R , y i ) to calculate and output the convergent point coordinates signal (X V , Y V , Z V )  190  of the parallax imaging device according to Equations (58), (59) and (60), and calculate and output the object point coordinates signal (X P , Y P , Z P )  191  of the point light source  101  according to Equations (80), (81) and (82), and also calculate and output a set of new convergent point parameter signals (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  153  according to Equations (87), (88), (89) and (90) to achieve the purpose of small-angle tracking. Further, a new convergent point parameter signal (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  153  is calculated and outputted according to Equations (93), (94), (95) and (96) to achieve the purpose of large-angle tracking. The spatial point computing device  170  also can receive an external new convergent point coordinates signal (X V   New , Y V   New , Z V   New )  192 , and after (X V   New , Y V   New , Z V   New ) is substituted into (X P , Y P , Z P ) of Equations (93), (94), (95) and (96) to compute another set of new convergent point parameter signals  153  (θ L   New , θ R   New , φ New , Ω New , Φ New , S New ) the signal is outputted compulsorily to achieve the purpose of resetting a convergent point. 
       FIG. 6(   c ) shows a schematic view of a parallax imaging device in accordance with a first preferred embodiment of the present invention, the parallax imaging device  110  comprises a pair of left and right cameras  120 ,  130 , a set of convergent point positioning mechanical device  140 , a convergent point reading device  151 , a convergent point setting device  154  and a pair of left and right 2D image memory and output devices  157 . The left and right cameras  120 ,  130  are installed on the convergent point positioning mechanical device  140  for receiving a scattered light  102  emitted by the point light source  101  and outputting a pair of left and right 2D image signals  156  respectively. After the left and right 2D image memory and output devices  157  receive the left and right 2D image signals  156 , a pair of left and right 2D image signals  158  are stored and outputted. The convergent point positioning mechanical device  140  is a structure having a plurality of rotating and displacing mechanical structures, actuators, and positioning sensors for fixing, carrying, rotating and moving the left and right cameras  120 ,  130 . Since the structure is too complicated to be shown in the figure, and it is generally a prior art of the automation, and thus will not be described here. By receiving a convergent point driving control signal  155 , all actuators can be driven to change and reset the direction of the visual axes of the left and right cameras  120 ,  130 , so as to achieve the purpose of changing and resetting the position of the convergent point of a parallax imaging device. 
     The convergent point reading device  151  can first read the signals of all positioning sensor on the convergent point positioning mechanical device  140  and the signal of the focal length of the camera, and then convert these signals into the convergent point parameter and finally output a convergent point parameter signal (θ L , θ R , φ, Ω, Φ, S, f)  150 . In addition, the convergent point setting device  154  receives a new convergent point parameter signal (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  153  and converts the signal  153  into the convergent point driving control signal  155  to output the signal  155 . Therefore, the convergent point positioning mechanical device  140  receives the new convergent point driving control signal  155  outputted by the convergent point setting device  154  to achieve the purpose of driving and controlling all actuators (or changing the convergent point). 
       FIG. 6(   d ) shows a schematic view of left and right cameras in accordance with a first preferred embodiment of the present invention, each of the left and right cameras  120 ,  130  comprise an optical filter  121 , an automatic fine-tuning 2D aperture  122 , an automatic focusing 2D lens module  123 , and a 2D image sensor  124 . The optical filter  121  is provided for filtering a light source with a wavelength other than the wavelength of the point light source; the automatic fine-tuning 2D aperture  122  can be a crevice in the shape of a circular hole, and the hole diameter of the crevice can be controlled and tuned automatically; the automatic focusing 2D lens module  123  is a lens module having the horizontal and vertical direction 2D focusing ability for automatically focusing and outputting the focal length; and the 2D image sensor  124  of the left camera  120  is a general prior art CCD or CMOS for outputting a left 2D image signal  156  with a parallax signal (x L , y i ). Further, the 2D image sensor  124  of the right camera  130  can be a general prior art CCD or CMOS for outputting a right 2D image signal  156  with a parallax signal (x R , y i ). 
     Referring to  FIG. 6(   e ) for a schematic view of a spatial point computing device in accordance with a first preferred embodiment of the present invention, the spatial point computing device  170  comprises two electronic interfaces  171 ,  181 , and a computing procedure device  172 . The electronic interface  171  is formed by plurality of general electronic components including digital I/O ports, ADCs and DACs to serve as an interface of connecting the parallax imaging device to read the convergent point parameter signal (θ L , θ R , φ, Ω, Φ, S, f)  150  and a pair of left and right 2D image signal  158  with a parallax signal (x L , x R , y i ) and outputting a set of new convergent point parameter signals (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  153 . The electronic interface  181  is a standard transmission interface formed by a general cable or a wireless RS232, USB or network to serve as an interface of connecting other external devices. With the standard transmission interface, a new convergent point coordinates signal (X V   New , Y V   New , Z V   New )  192  can be read, the convergent point coordinates signal (X V , Y V , Z V )  190  of the parallax imaging device and the object point coordinates signal (X P , Y P , Z P )  191  of the point light source can be outputted. The computing procedure device  172  is a microcomputer comprising a general microprocessor, a DSP, and a memory device. The memory device includes a computing logic procedure  173  for processing the 2D image signals  158  to obtain the coordinates (x L , y i ) and (x R , y i ) of the left and right point images by finding the brightest point of the image on the 2D image sensor  124 . According to the inputted convergent point parameter signal (θ L , θ R , φ, Ω, Φ, S, f)  150  and the obtained parallax signal (x L , x R , y i ), the computing logic procedure  173  computes and outputs a convergent point coordinates signal (X V , Y V , Z V )  190  and an object point coordinates signal (X P , Y P , Z P )  191 , and calculate and output a set of new convergent point parameter signals (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  153  to achieve the purpose of tracking the object point. Further, the computing logic procedure  173  calculates and outputs a set of new convergent point parameter signals (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  153  according to the inputted new convergent point coordinates signals (X V   New , Y V   New , Z V   New )  192  to achieve the purpose of resetting the new convergent point. 
     Referring to  FIG. 7(   a ) for a method of recognizing a spatial point in accordance with a second preferred embodiment of the present invention, the method  200  comprises a point light source  201  and a spatial point recognition device  205 . The point light source  201  can be an active point light source capable of emitting scattered visible light or non-visible light  202  or a passive point light source which is a light reflecting body for reflecting the light of other light sources. The spatial point recognition device  205  receives the scattered light  202  emitted by the point light source  201  and outputs the convergent point coordinates signal (X V , Y V , Z V )  290  of the spatial point recognition device and the object point coordinates signal (X P , Y P , Z P )  291  of the point light source  201 . The spatial point recognition device  205  is capable of moving the convergent point, such that the convergent point is moved to superimpose the coordinates of the convergent point with the coordinates of the object point to achieve the purpose of tracking the object point automatically. At the same time, the spatial point recognition device  205  also receives a new coordinates signal (X V   New , Y V   New , Z V   New )  292  of the convergent point provided by other external devices to reset the convergent point of the spatial point recognition device  205 , so as to and achieve the purpose of resetting the convergent point of the spatial point recognition device  205 . 
     Referring to  FIG. 7(   b ) for a schematic view of a spatial point recognition device in accordance with a second preferred embodiment of the present invention, the spatial point recognition device  205  comprises a parallax imaging device  210 , and a spatial point computing device  270 . After the parallax imaging device  210  receives the light  202  emitted by the point light source  201 , a set of convergent point parameter signals (θ L , θ M , θ R , φ, Ω, Φm, S, f)  250  and a set of left, middle and right 1D image signals  258  are outputted. In addition, the parallax imaging device  210  also receives a set of new convergent point parameter signals (θ L   New , θ M   New , θ R   New , φ New , Ω New , Φ New , S New )  253  outputted by the spatial point computing device  270  to change the convergent point of the parallax imaging device  210 . After the spatial point computing device  270  receives the convergent point parameter signal (θ L , θ M , θ R , φ, Ω, Φ, S, f)  250 , and a set of 1D image signal  258  with a parallax signal (x L , x R y i ), Equations (58), (59) and (60) are used for calculating and outputting the convergent point coordinates signal (X V , Y V , Z V )  290  of the parallax imaging device. Further, Equations (80), (81) and (82) can be used for calculating and outputting of the object point coordinates signal (X P , Y P , Z P )  291  of the point light source  201 . In the meantime, Equations (87), (88), (89), (90) and (91) can be used for calculating and outputting a set of new convergent point parameter signals (θ L   New , θ M   New , θ R   New , φ New , Ω New , Φ New , S New )  253  to achieve the purpose of small-angle tracking. Equations (93), (94), (95), (96) and (97) can be used for calculating and outputting a set of new convergent point parameter signals (θ L   New , θ M   New , θ R   New , φ New , Ω New , Φ New , S New )  253  to achieve the purpose of large-angle tracking. In addition, the spatial point computing device  270  receives an external set of new convergent point coordinates signals (X V   New , Y V   New , Z V   New )  292 , and substitutes (X V   New , Y V   New , Z V   New ) to the (X P , Y P , Z P ) in Equations (93), (94), (95), (96) and (97) to compute another set of new convergent point parameter signals  253  (θ L   New , θ M   New , θ R   New , φ New , Ω New , Φ New , S New ), and then output this signal compulsorily to achieve the purpose of resetting the convergent point. 
     Referring to  FIG. 7(   c ) for a parallax imaging device in accordance with a second preferred embodiment of the present invention, the parallax imaging device  210  comprises a left camera  220 , a middle camera  225 , a right camera  230 , a set of convergent point positioning mechanical device  240 , a convergent point reading device  251 , a convergent point setting device  254  and a left, a middle and a right 1D image memory and output devices  257 . After receiving scattered light  202  emitted by the point light source  201 , the left, middle and right cameras  220 ,  225 ,  230  output a left, middle and right 1D image signals  256  respectively. After the left, middle and right 1D image memory and output devices  257  receive the left, middle and right 1D image signals  256 , the left, middle and right 1D image signals  258  are stored and outputted respectively. Further, the left, middle and right cameras  220 ,  225 ,  230  are installed onto the convergent point positioning mechanical device  240 . The convergent point positioning mechanical device  240  comprises a plurality of rotating and displacing mechanical structures, actuators and positioning sensors for fixing, carrying, rotating and moving the left, middle and right cameras  220 ,  225 ,  230 . Since the structure is too complicated to be shown in the figure, and it is generally a prior art of the automation, and thus will not be described here. With the device having these automated structures and elements, the parameters (θ L , θ M , θ R , φ, Ω, Φ, S, f) can be changed to achieve the purpose of changing the direction of the visual axis of the left, middle and right cameras  220 ,  225 ,  230 . Further, the convergent point positioning mechanical device  240  receives a convergent point driving control signal  255  for driving all actuators to change and reset the direction of the visual axis of the left, middle and right cameras  220 ,  225 ,  230 , so as to achieve the purpose of changing and resetting the position of the convergent point of the parallax imaging device  210 . 
     The convergent point reading device  251  first reads the signals of all positioning sensors on the convergent point positioning mechanical device  240  and the signal of the focal length of the cameras, and then convert these signals into the convergent point parameter and finally output a convergent point parameter signal (θ L , θ M , θ R , φ, Ω, Φ, S, f)  250 . Further, the convergent point setting device  254  receives a new convergent point parameter signal (θ L   New , θ M   New , θ R   New , φ New , Ω New , Φ New , S New )  253  and converts the signal  253  into the convergent point driving control signal  255  to output the signal  255 . Therefore, the convergent point positioning mechanical device  240  can receive the new convergent point driving control signal  255  outputted by the convergent point setting device  254  to achieve the purpose of driving and controlling all actuators or changing the convergent point. 
     Referring to  FIG. 7(   d ) for a schematic view of left, middle and right cameras in accordance with a second preferred embodiment of the present invention, each of the left, middle and right cameras  220 ,  225 ,  230  comprises an optical filter  221 , an automatic fine-tuning 1D aperture  222 , an automatic focusing 1D lens module  223 , and a 1D image sensor  224 . The optical filter is provided for filtering a light source with a wavelength other than the wavelength of the point light source. 
     In the left camera  220 , the 1D aperture  222  is a rectangular slit, whose width can be controlled and tuned automatically, and the direction of the slit is aligned in a vertical direction, which means the longer side of the slit is installed in the vertical direction. The 1D lens module  223  is a lens module with a horizontal focusing ability for automatically focusing and outputting the focal length. The 1D image sensor  224  is formed by a general CCD or CMOS, and the direction of the sensor is aligned in the horizontal direction, which means the longer side of the sensor is installed in the horizontal direction for outputting a left 1D image signal with a parallax signal (x L ). 
     In the middle camera  225 , the 1D aperture  222  is a rectangular slit whose width can be controlled and tuned automatically, and the direction of the slit is aligned in the horizontal direction, which means the longer side of the slit is installed in the horizontal direction; the 1D lens module  223  is a lens module having the vertical focusing ability for automatically focusing and outputting the focal length; and the 1D image sensor is formed by a general traditional CCD or CMOS, and the direction of the sensor is aligned in the vertical direction, which means the longer side of the sensor is installed in the vertical direction, for outputting a 1D image signal with a parallax signal (y i ). 
     In the right camera  230 , the 1D aperture  222  is a rectangular slit whose width can be controlled and tuned automatically, and the direction of the slit is aligned in the vertical direction, which means the longer side of the slit is installed in the vertical direction; the 1D lens module  223  is a lens module having the horizontal focusing ability, for automatically focusing and outputting the focal length; and the 1D image sensor  224  is formed by a general traditional CCD or CMOS, and the direction of the sensor is aligned in the horizontal direction, which means the longer side of the sensor is installed in the horizontal direction, for outputting a right 1D image signal with a parallax signal (x R ). 
     Referring to  FIG. 7(   e ) for a schematic view of a spatial point computing device in accordance with a second preferred embodiment of the present invention, the spatial point computing device  270  comprises two electronic interfaces  271 ,  281  and a computing procedure device  272 . 
     The electronic interface  271  comprises plurality of general electronic components including digital I/O ports, ADCs and DACs to serve as an interface of connecting the parallax imaging device  210  to read the convergent point parameter signal  250  (θ L , θ M , θ R , φ, Ω, Φ, S, f) and a set of left, middle and right 1D image signals  258  with a parallax signal (x L , x R , y i ) and outputting a set of new convergent point parameter signals  253  (θ L   New , θ M   New , θ R   New , φ New , Ω New , Φ New , S New ). 
     The electronic interface  281  is a standard transmission interface formed by a general cable or wireless RS232, USB or network to serve as an interface of connecting other external devices. With the standard transmission interface, a new convergent point coordinates signal (X V   New , Y V   New , Z V   New )  292  can be read, the convergent point coordinates signal  290  (X V , Y V , Z V ) of the parallax imaging device, and output the object point coordinates signal  291  (X P , Y P , Z P ) of the point light source can be outputted 
     The computing procedure device  272  is a microcomputer comprising a general microprocessor, a DSP and a memory device. The memory device includes a computing logic procedure  273  for processing the 1D image signal  258  to obtain the coordinates (X L , X R , y i ) of the left, middle and right point images by finding the brightest point of the image on the 1D image sensor. According to the inputted convergent point parameter signal (θ L , θ R , φ, Ω, S, f)  250  and the obtained parallax signal (x L , x R , y i ), the computing logic procedure  273  computes and outputs a convergent point coordinates signal (X V , Y V , Z V )  290  and an object point coordinates signal (X P , Y P , z P )  191 , and calculate and output a set of new convergent point parameter signals (θ L   New , θ R   New , φ New , Ω New , Φ New , S New )  253  to achieve the purpose of tracking the object point. Further, the computing logic procedure  273  calculates and outputs a set of new convergent point parameter signal (θ L   New , θ M   New , θ R   New , φ New , Ω New , Φ New , S New )  253  according to the inputted new convergent point coordinates signal (X V   New , Y V   New , Z V   New )  292  to output the signal compulsorily and reset the position of the convergent point to achieve the purpose of resetting the new convergent point. 
     In summation of the description above, the characteristics of the method in accordance with the present invention has been fully disclosed, and the invention improves over the prior art and complies with the requirements of patent application, and is thus duly filed for patent application. 
     While the invention has been described by means of specific embodiments, numerous modifications and variations could be made thereto by those skilled in the art without departing from the scope and spirit of the invention set forth in the claims.