15.5.2 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.
Examples
To obtain the matrix of the rotation about axis [−1,2,−1] of angle π, input:
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| ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
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To obtain the matrix of the symmetry with respect to O, input:
To obtain the matrix of the symmetry with respect to the plane x+y+z=0, input:
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| ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
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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:
To obtain matrix of the plane symmetry with respect to the line
of equation x+2y=0, input:
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| ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
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