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
: