Finding the matrix of an isometry

The mkisom command finds the matrix of an isometry given the characteristic elements.

mkisom takes two arguments:
- char, the characteristic element (for isometries of ℝ² or a list of characteristic elements (for isometries of ℝ³.
  For isometries of ℝ², char will be the angle of rotation for direct isometries or a vector determining the line (through the origin) of reflection for an indirect symmetry.
  For isometries of ℝ³, char will be the list consisting of the axis direction and angle of rotation for a direct isometry or a vector normal to the plane of reflection for an indirect isometry.
- n, either +1 for a direct isometry or -1 an indirect isometry.
mkisom(char,n) returns a matrix of the corresponding isometry.

To obtain the matrix of the rotation about axis [−1,2,−1] of angle π, input:

mkisom([[-1,2,-1],pi],1)

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

−

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

To obtain the matrix of the symmetry with respect to O, input:

mkisom([pi],-1)

⎡
⎢
⎢
⎣

⎤
⎥
⎥
⎦

To obtain the matrix of the symmetry with respect to the plane x+y+z=0, input:

mkisom([1,1,1],-1)

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

−

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

To obtain the matrix of the product of a rotation of axis [1,1,1] and angle π/3 and of a symmetry with respect to the plane x+y+z=0, input:

mkisom([[1,1,1],pi/3],-1)

⎡
⎢
⎢
⎣

⎤
⎥
⎥
⎦

To obtain the matrix of the plane rotation of angle π/2, input:

mkisom(pi/2,1)

⎡
⎢
⎣

0	−1
1	0

⎤
⎥
⎦

To obtain matrix of the plane symmetry with respect to the line of equation x+2y=0, input:

mkisom([1,2],-1)

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

−

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦