Transformations

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Overview: Transformations

Next: Homogeneous coordinates →

In this article general transformations used in the context of robotics and the underlying mathematics are described.

Transformations are applied to vectors or coordinates in vector notation, respectively. All types of transformations can be achieved by matrix multiplication. To transform a vector $\vec{\mathbf{p}}_0$ to $\vec{\mathbf{p}}_1$, $\vec{\mathbf{p}}_0$ has to be multilplied by the transformation matrix $\mathbf{T}$:

$\vec{\mathbf{p}}_1=\mathbf{T}\cdot\vec{\mathbf{p}}_0$

The basic transformation matrix is an identity matrix, which corresponds to an empty transformation. So an identity matrix as transformation matrix results in the original vector no matter how many dimensions the vector has. The following equation shows this for three dimensions:

$\vec{\mathbf{p}}_1=\mathbf{I}_3\cdot\vec{\mathbf{p}}_0= \left[\begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right] \cdot \left[\begin{array}{c} x_0\\ y_0\\ z_0 \end{array}\right]= \left[\begin{array}{c} 1\cdot x_0+0\cdot y_0+0\cdot z_0\\ 0\cdot y_0+1\cdot y_0+0\cdot z_0\\ 0\cdot y_0+0\cdot y_0+1\cdot z_0 \end{array}\right]= \left[\begin{array}{c} x_0\\ y_0\\ z_0 \end{array}\right]= \vec{\mathbf{p}}_0$

This also holds for two, four and more dimensions.

By modifying the components of the transformation matrix different types of transformations can be achieved. In n-dimensional space rotation, scaling and shearing can basically be described by an n-by-n transformation matrix. Translation however is actually a vector addition of an n-by-1 vector. So first the different types of transformations are explained individually. Then homogeneous coordinates are introduced to be able to apply all kinds of transformations by matrix multiplication. Conclusively the last subarticle explains how different transformations are combined in a single transformation matrix.

1. Translation
2. Rotation
3. Scaling
4. Shearing
5. Homogeneous coordinates
6. Combinations of the three transformation types