Edit detail for SandBoxRotationMatrix revision 1 of 1

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Rotation matrix

rotationX(alpha) == [[1,0,0],[0,cos(alpha),-sin(alpha)],[0,sin(alpha),cos(alpha)]]

rotationY(beta) == [[cos(beta),0,sin(beta)],[0,1,0],[-sin(beta),0,cos(beta)]]

rotationZ(gamma) == [[cos(gamma),-sin(gamma),0],[sin(gamma),cos(gamma),0],[0,0,1]]

rotation(alpha,beta,gamma) == rotationX(alpha)*rotationY(beta)*rotationZ(gamma)

rotation(alpha,beta,gamma)

Compiling function rotationX with type Variable(alpha) -> List(List(
      Expression(Integer)))

Compiling function rotationY with type Variable(beta) -> List(List(
      Expression(Integer))) 
   There are 31 exposed and 40 unexposed library operations named * 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op *
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named * 
      with argument type(s) 
                       List(List(Expression(Integer)))
                       List(List(Expression(Integer)))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.

Compiling function rotationZ with type Variable(gamma) -> List(List(
      Expression(Integer))) 

   (5)
   [[cos(beta)cos(gamma), - cos(beta)sin(gamma), sin(beta)],

     [cos(alpha)sin(gamma) + cos(gamma)sin(alpha)sin(beta),
      - sin(alpha)sin(beta)sin(gamma) + cos(alpha)cos(gamma),
      - cos(beta)sin(alpha)]
     ,

     [sin(alpha)sin(gamma) - cos(alpha)cos(gamma)sin(beta),
      cos(alpha)sin(beta)sin(gamma) + cos(gamma)sin(alpha),
      cos(alpha)cos(beta)]
     ]

Transformation for a 3d point (x1,x2,x3) by rotation and translation (t1,t2,t3)

transform(x1,x2,x3,alpha,beta,gamma,t1,t2,t3) == rotation(alpha,beta,gamma) * [[x1],[x2],[x3]]+[[t1],[t2],[t3]]

m := transform(x1,x2,x3,alpha,beta,gamma,t1,t2,t3);

   Cannot compile map: rotation 
   We will attempt to interpret the code.

p := [[(m(1,1)/m(3,1))-y1,m(2,1)/m(3,1)-y2]]*[[(m(1,1)/m(3,1))-y1],[m(2,1)/m(3,1)-y2]]

   (8)
   [
     [
                     2          2       2
                   x2 sin(alpha)  + 2 x2 y2 cos(alpha)sin(alpha)
                 + 
                      2  2     2  2           2
                   (x2 y2  + x2 y1 )cos(alpha)
              *
                          2
                 sin(beta)
             + 
                                        2
                   2 x1 x2 y2 sin(alpha)
                 + 
                              2             2
                   (2 x1 x2 y2  + 2 x1 x2 y1  - 2 x1 x2)cos(alpha)sin(alpha)
                 + 
                       2                                              2
                   2 x2 y1 cos(alpha)cos(beta) - 2 x1 x2 y2 cos(alpha)
              *
                 sin(beta)
             + 
                  2  2     2  2           2
               (x1 y2  + x1 y1 )sin(alpha)
             + 
                                           2
               (2 x1 x2 y1 cos(beta) - 2 x1 y2 cos(alpha))sin(alpha)
             + 
                 2         2     2          2
               x2 cos(beta)  + x1 cos(alpha)
          *
                       2
             sin(gamma)
         + 
                                                 2
                   - 2 x1 x2 cos(gamma)sin(alpha)
                 + 
                   - 4 x1 x2 y2 cos(alpha)cos(gamma)sin(alpha)
                 + 
                                2             2           2
                   (- 2 x1 x2 y2  - 2 x1 x2 y1 )cos(alpha) cos(gamma)
                 + 
                   - 2 x2 x3 y1 cos(alpha)
              *
                          2
                 sin(beta)
             + 
                         2       2                                             2
                   ((2 x2  - 2 x1 )y2 cos(gamma) + 2 x2 x3 cos(beta))sin(alpha)
                 + 
                                  2       2   2        2       2   2       2
                             (2 x2  - 2 x1 )y2  + (2 x2  - 2 x1 )y1  - 2 x2
                           + 
                                 2
                             2 x1
                      *
                         cos(alpha)cos(gamma)
                     + 
                       4 x2 x3 y2 cos(alpha)cos(beta) + 2 t3 x2 y2 - 2 x1 x3 y1
                     + 
                       - 2 t2 x2
                  *
                     sin(alpha)
                 + 
                       - 4 x1 x2 y1 cos(alpha)cos(beta)
                     + 
                              2       2              2
                       (- 2 x2  + 2 x1 )y2 cos(alpha)
                  *
                     cos(gamma)
                 + 
                               2             2           2
                   ((2 x2 x3 y2  + 2 x2 x3 y1 )cos(alpha)  - 2 x2 x3)cos(beta)
                 + 
                            2                          2
                 (2 t3 x2 y2  - 2 t2 x2 y2 + 2 t3 x2 y1  - 2 t1 x2 y1)cos(alpha)
              *
                 sin(beta)
             + 
                             2             2
                 ((2 x1 x2 y2  + 2 x1 x2 y1 )cos(gamma) + 2 x1 x3 y2 cos(beta))
              *
                           2
                 sin(alpha)
             + 
                           2       2
                     ((2 x2  - 2 x1 )y1 cos(beta) - 4 x1 x2 y2 cos(alpha))
                  *
                     cos(gamma)
                 + 
                              2             2
                   (2 x1 x3 y2  + 2 x1 x3 y1  - 2 x1 x3)cos(alpha)cos(beta)
                 + 
                             2                          2
                   2 t3 x1 y2  - 2 t2 x1 y2 + 2 t3 x1 y1  - 2 t1 x1 y1
              *
                 sin(alpha)
             + 
                                   2                     2
               (- 2 x1 x2 cos(beta)  + 2 x1 x2 cos(alpha) )cos(gamma)
             + 
                                             2
               2 x2 x3 y1 cos(alpha)cos(beta)
             + 
                                       2
               (- 2 x1 x3 y2 cos(alpha)  + 2 t3 x2 y1 - 2 t1 x2)cos(beta)
             + 
               (- 2 t3 x1 y2 + 2 t2 x1)cos(alpha)
          *
             sin(gamma)
         + 
                 2          2          2
               x1 cos(gamma) sin(alpha)
             + 
                   2                       2
               2 x1 y2 cos(alpha)cos(gamma) sin(alpha)
             + 
                  2  2     2  2           2          2
               (x1 y2  + x1 y1 )cos(alpha) cos(gamma)
             + 
                                                   2
               2 x1 x3 y1 cos(alpha)cos(gamma) + x3
          *
                      2
             sin(beta)
         + 
                                         2
                 (- 2 x1 x2 y2 cos(gamma)  - 2 x1 x3 cos(beta)cos(gamma))
              *
                           2
                 sin(alpha)
             + 
                                2             2                               2
                   (- 2 x1 x2 y2  - 2 x1 x2 y1  + 2 x1 x2)cos(alpha)cos(gamma)
                 + 
                       - 4 x1 x3 y2 cos(alpha)cos(beta) - 2 t3 x1 y2
                     + 
                       - 2 x2 x3 y1 + 2 t2 x1
                  *
                     cos(gamma)
              *
                 sin(alpha)
             + 
                    2                                              2           2
               (2 x1 y1 cos(alpha)cos(beta) + 2 x1 x2 y2 cos(alpha) )cos(gamma)
             + 
                                 2             2           2
                   ((- 2 x1 x3 y2  - 2 x1 x3 y1 )cos(alpha)  + 2 x1 x3)cos(beta)
                 + 
                                  2                          2
                     (- 2 t3 x1 y2  + 2 t2 x1 y2 - 2 t3 x1 y1  + 2 t1 x1 y1)
                  *
                     cos(alpha)
              *
                 cos(gamma)
             + 
                     2
               - 2 x3 y1 cos(alpha)cos(beta) - 2 t3 x3 y1 + 2 t1 x3
          *
             sin(beta)
         + 
                  2  2     2  2           2
               (x2 y2  + x2 y1 )cos(gamma)  + 2 x2 x3 y2 cos(beta)cos(gamma)
             + 
                 2         2
               x3 cos(beta)
          *
                       2
             sin(alpha)
         + 
                                             2                        2
               (- 2 x1 x2 y1 cos(beta) - 2 x2 y2 cos(alpha))cos(gamma)
             + 
                              2             2
                   (2 x2 x3 y2  + 2 x2 x3 y1  - 2 x2 x3)cos(alpha)cos(beta)
                 + 
                             2                          2
                   2 t3 x2 y2  - 2 t2 x2 y2 + 2 t3 x2 y1  - 2 t1 x2 y1
              *
                 cos(gamma)
             + 
                   2                      2
               2 x3 y2 cos(alpha)cos(beta)  + (2 t3 x3 y2 - 2 t2 x3)cos(beta)
          *
             sin(alpha)
         + 
              2         2     2          2           2
           (x1 cos(beta)  + x2 cos(alpha) )cos(gamma)
         + 
                                               2
               - 2 x1 x3 y1 cos(alpha)cos(beta)
             + 
                                       2
               (- 2 x2 x3 y2 cos(alpha)  - 2 t3 x1 y1 + 2 t1 x1)cos(beta)
             + 
               (- 2 t3 x2 y2 + 2 t2 x2)cos(alpha)
          *
             cos(gamma)
         + 
              2  2     2  2           2         2
           (x3 y2  + x3 y1 )cos(alpha) cos(beta)
         + 
                        2                          2
             (2 t3 x3 y2  - 2 t2 x3 y2 + 2 t3 x3 y1  - 2 t1 x3 y1)cos(alpha)
          *
             cos(beta)
         + 
             2  2                  2  2                  2     2
           t3 y2  - 2 t2 t3 y2 + t3 y1  - 2 t1 t3 y1 + t2  + t1
      /
                 2          2         2
               x2 cos(alpha) sin(beta)  + 2 x1 x2 cos(alpha)sin(alpha)sin(beta)
             + 
                 2          2
               x1 sin(alpha)
          *
                       2
             sin(gamma)
         + 
                                   2                   2
               - 2 x1 x2 cos(alpha) cos(gamma)sin(beta)
             + 
                        2       2
                   (2 x2  - 2 x1 )cos(alpha)cos(gamma)sin(alpha)
                 + 
                                     2
                   2 x2 x3 cos(alpha) cos(beta) + 2 t3 x2 cos(alpha)
              *
                 sin(beta)
             + 
                                           2
               2 x1 x2 cos(gamma)sin(alpha)
             + 
               (2 x1 x3 cos(alpha)cos(beta) + 2 t3 x1)sin(alpha)
          *
             sin(gamma)
         + 
             2          2          2         2
           x1 cos(alpha) cos(gamma) sin(beta)
         + 
                                             2
               - 2 x1 x2 cos(alpha)cos(gamma) sin(alpha)
             + 
                                    2
               (- 2 x1 x3 cos(alpha) cos(beta) - 2 t3 x1 cos(alpha))cos(gamma)
          *
             sin(beta)
         + 
             2          2          2
           x2 cos(gamma) sin(alpha)
         + 
           (2 x2 x3 cos(alpha)cos(beta) + 2 t3 x2)cos(gamma)sin(alpha)
         + 
             2          2         2                                   2
           x3 cos(alpha) cos(beta)  + 2 t3 x3 cos(alpha)cos(beta) + t3
       ]
     ]