Difference between revisions of "Inverse transformation"
From Robotics
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\mathbf{R}\cdot \vec{\mathbf{q}}_0 + \vec{\mathbf{p}} | \mathbf{R}\cdot \vec{\mathbf{q}}_0 + \vec{\mathbf{p}} | ||
</math> | </math> | ||
− | The inverse transformation <math>\mathbf{T}^{-1}</math> corresponds to the transformation that reverts the rotation and translation effected by <math>\mathbf{T}</math>. If a vector is pre-multiplied by <math>\mathbf{T}</math> and subsequently pre-multiplied by <math>\mathbf{T}^{-1}</math>, this results in the original coordinates because <math>\mathbf{T}^{-1}\mathbf{T}=\mathbf{I}</math> and multiplication with the identity matrix does not change anything (see | + | The inverse transformation <math>\mathbf{T}^{-1}</math> corresponds to the transformation that reverts the rotation and translation effected by <math>\mathbf{T}</math>. If a vector is pre-multiplied by <math>\mathbf{T}</math> and subsequently pre-multiplied by <math>\mathbf{T}^{-1}</math>, this results in the original coordinates because <math>\mathbf{T}^{-1}\mathbf{T}=\mathbf{I}</math> and multiplication with the identity matrix does not change anything (see [[Transformations|transformations]]). |
[[Category:Article]] | [[Category:Article]] | ||
[[Category:Transformations]] | [[Category:Transformations]] |
Revision as of 14:02, 17 June 2014
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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:
Multiplication with corresponds to applying the rotation matrix first and then translating the coordinates by :
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).