Difference between revisions of "Transformations"
From Robotics
Line 12: | Line 12: | ||
x_0\\ | x_0\\ | ||
y_0 | y_0 | ||
+ | \end{array}\right]= | ||
+ | \left[\begin{array}{c} | ||
+ | 1\cdot x_0+0\cdot y_0\\ | ||
+ | 0\cdot y_0+1\cdot y_0 | ||
+ | \end{array}\right]= | ||
+ | \left[\begin{array}{c} | ||
+ | x_0\\ | ||
+ | y_0 | ||
+ | \end{array}\right]= | ||
+ | \vec{\mathbf{p}}_0 | ||
+ | </math> | ||
+ | |||
+ | The same holds for three dimensions:<br/> | ||
+ | :<math> | ||
+ | \vec{\mathbf{p}}_1=\mathbf{I}_3\cdot\vec{\mathbf{p}}_0= | ||
+ | \left[\begin{array}{ccc} | ||
+ | 1&0&0\\ | ||
+ | 0&1&0\\ | ||
+ | 0&0&1 | ||
\end{array}\right] | \end{array}\right] | ||
+ | \cdot | ||
+ | \left[\begin{array}{c} | ||
+ | x_0\\ | ||
+ | y_0\\ | ||
+ | z_0 | ||
+ | \end{array}\right]= | ||
+ | \left[\begin{array}{c} | ||
+ | 1\cdot x_0+0\cdot y_0+0\cdot z_0\\ | ||
+ | 0\cdot y_0+1\cdot y_0+0\cdot z_0\\ | ||
+ | 0\cdot y_0+0\cdot y_0+1\cdot z_0 | ||
+ | \end{array}\right]= | ||
+ | \left[\begin{array}{c} | ||
+ | x_0\\ | ||
+ | y_0\\ | ||
+ | z_0 | ||
+ | \end{array}\right]= | ||
+ | \vec{\mathbf{p}}_0 | ||
</math> | </math> | ||
Revision as of 14:05, 26 May 2014
← Back: Adjugate Formula | Overview: Transformations | Next: Translation → |
In this article general transformations used in the context of robotics and the underlying mathematics are described. All types of transformations can be achieved by matrix multiplication. Basis is at first the multiplication of a vector with the respective identity matrix. The identity matrix corresponds to an empty transformation. So a multiplication with an identity matrix results in the original vector, for example in two-dimensional space:
The same holds for three dimensions:
The different types of transformations are presented in the following subarticles: