Translation

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Translation is the easiest kind of transformation. Translating a point $q_0$ means that it is shifted by a translation vector. So the translation vector $\vec{\mathbf{p}}$ is added to the position vector $\vec{\mathbf{q}}_0$ of $q_0$. The position vector $\vec{\mathbf{q}}_1$ of the resulting transformed point $q_1$ is calculated as follows:

$\vec{\mathbf{q}}_1=\vec{\mathbf{q}}_0+\vec{\mathbf{p}}$

The figure on the right shows an example in two-dimensional space. In robotics usually three dimensions are regarded. Considering the particular components of the vectors, a translation looks as follows:

$\left[\begin{array}{c} x_1\\ y_1\\ z_1 \end{array}\right]= \left[\begin{array}{c} x_0\\ y_0\\ z_0 \end{array}\right]+ \left[\begin{array}{c} p_x\\ p_y\\ p_z \end{array}\right]= \left[\begin{array}{c} x_0+p_x\\ y_0+p_y\\ z_0+p_z \end{array}\right]$

For further information about vector addition and examples, please have a look at the article about simple arithmetic vector operations.