Difference between revisions of "Inverse transformation"
From Robotics
| Line 11: | Line 11: | ||
\end{array}\right] | \end{array}\right] | ||
</math> | </math> | ||
| − | + | As stated in the article about [[Homogeneous coordinates|homogeneous coordinates]], multiplication with <math>\mathbf{T}</math> is equivalent to applying the rotation matrix <math>\mathbf{R}</math> first and then translating the coordinates by <math>\vec{\mathbf{p}}</math> in cartesian coordinates:<br/> | |
:<math> | :<math> | ||
\vec{\mathbf{q}}_1= | \vec{\mathbf{q}}_1= | ||
| − | \mathbf{T} \cdot \vec{\mathbf{q}}_0 | + | \mathbf{T} \cdot \vec{\mathbf{q}}_0 \equiv |
\mathbf{R}\cdot \vec{\mathbf{q}}_0 + \vec{\mathbf{p}} | \mathbf{R}\cdot \vec{\mathbf{q}}_0 + \vec{\mathbf{p}} | ||
</math> | </math> | ||
Revision as of 14:10, 17 June 2014
| ← Back: Combinations of transformations | Overview: Transformations | Next: ??? → |
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:
![\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 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).
