Here are some examples of 3 dimensional transforms and their corresponding 3x3 matrices.
I have noted whether the matrix is symmetric or anti-symmetric across the leading diagonal. As you can see, rotation contains the sum of both a symmetric and an anti-symmetric across the leading diagonal, reflection is symmetric across the leading diagonal.
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Identity (symmetric) |
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Scale (symmetric) |
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Swap x and y axes, which is the same as reflecting in a 45° line (symmetric) |
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Swap x and y axes and invert y, which is the same as rotating by 90° around z axis (anti-symmetric) |
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Rotate by a around z axis (anti-symmetric) |
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Reflection in z axis | |||||||||
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Rotation around an axis x,y,z For examples of 3D rotations see this page. |
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Skew symmetric matrix, Matrix equivalent of vector cross multiplication, this transform generates a vector which is mutually perpendicular to both x,y,z and the input vector. | |||||||||
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Reflection in line x,y,z (see this page) (symmetric) |
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