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Edit detail for SandBoxRotationMatrix revision 1 of 1

1
Editor:
Time: 2009/10/16 09:02:06 GMT-7
Note: Test

changed:
-
This says don't use LaTeX output.
\begin{axiom}
)set output tex off
)set output algebra on
\end{axiom}

Rotation matrix
\begin{axiom}
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)

\end{axiom}
Transformation for a 3d point (x1,x2,x3) by rotation and translation (t1,t2,t3)
\begin{axiom}
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);
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]]

\end{axiom}

This says don't use LaTeX output.

fricas
)set output tex off
 
fricas
)set output algebra on

Rotation matrix

fricas
rotationX(alpha) == [[1,0,0],[0,cos(alpha),-sin(alpha)],[0,sin(alpha),cos(alpha)]]
Type: Void
fricas
rotationY(beta) == [[cos(beta),0,sin(beta)],[0,1,0],[-sin(beta),0,cos(beta)]]
Type: Void
fricas
rotationZ(gamma) == [[cos(gamma),-sin(gamma),0],[sin(gamma),cos(gamma),0],[0,0,1]]
Type: Void
fricas
rotation(alpha,beta,gamma) == rotationX(alpha)*rotationY(beta)*rotationZ(gamma)
Type: Void
fricas
rotation(alpha,beta,gamma)
fricas
Compiling function rotationX with type Variable(alpha) -> List(List(
      Expression(Integer)))
fricas
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.
fricas
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)] ]
Type: Matrix(Expression(Integer))

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

fricas
transform(x1,x2,x3,alpha,beta,gamma,t1,t2,t3) == rotation(alpha,beta,gamma) * [[x1],[x2],[x3]]+[[t1],[t2],[t3]]
Type: Void
fricas
m := transform(x1,x2,x3,alpha,beta,gamma,t1,t2,t3);
Cannot compile map: rotation We will attempt to interpret the code.
Type: Matrix(Expression(Integer))
fricas
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 ] ]
Type: Matrix(Expression(Integer))