Inverse transformation
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Let be a general homogeneous transformation matrix. 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).
The 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 in cartesian coordinates to applying the rotation matrix first and then translating the coordinates by :
The equation above is now solved for . Here the fact is used, that the inverse of 3-by-3 rotation matrices equals the transpose of the matrix:
So the inverse of a homogeneous transformation matrix is defined as: