Inverse transformation
From Robotics
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A general homogeneous transformation matrix for three-dimensional space consists of a 3-by-3 rotation matrix and a 3-by-1 translation vector combined with the last row of the identity matrix:
As stated in the article about homogeneous coordinates, multiplication with is equivalent to applying the rotation matrix first and then translating the coordinates by in cartesian coordinates:
The inverse transformation corresponds to the transformation that reverts the rotation and translation effected by . If a vector is pre-multiplied by and subsequently pre-multiplied by , this results in the original coordinates because and multiplication with the identity matrix does not change anything (see transformations).