Difference between revisions of "Inverse transformation"
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{{Navigation|before=[[Combinations of transformations]]|overview=[[Transformations]]|next=[[???]]}} | {{Navigation|before=[[Combinations of transformations]]|overview=[[Transformations]]|next=[[???]]}} | ||
| − | + | Let <math>\mathbf{T}</math> be a general homogeneous transformation matrix. 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]]). | |
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| + | The general homogeneous transformation matrix <math>\mathbf{T}</math> for three-dimensional space consists of a 3-by-3 rotation matrix <math>\mathbf{R}</math> and a 3-by-1 translation vector <math>\vec{\mathbf{p}}</math> combined with the last row of the identity matrix:<br/> | ||
:<math> | :<math> | ||
\mathbf{T}= | \mathbf{T}= | ||
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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> | ||
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[[Category:Article]] | [[Category:Article]] | ||
[[Category:Transformations]] | [[Category:Transformations]] | ||
Revision as of 14:20, 17 June 2014
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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:
![\mathbf{T}=
\left[\begin{array}{ccc|c}
& & & \\
& \mathbf{R} & & \vec{\mathbf{p}}\\
& & & \\ \hline
0 & 0 & 0 & 1
\end{array}\right]](/wiki/robotics/images/math/1/3/2/1327593f05795df92e5c12f1a1a27f84.png)
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 :
