Editing 2D computer graphics (section)

===Translation===<!-- This section is linked from [[affine transformation]]. -->

Since a translation is an [[affine transformation]] but not a [[linear transformation]], [[homogeneous coordinates]] are normally used to represent the translation operator by a [[matrix (mathematics)|matrix]] and thus to make it linear. Thus we write the 3-dimensional vector '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>) using 4 homogeneous coordinates as '''w''' = (''w''<sub>''x''</sub>, ''w''<sub>''y''</sub>, ''w''<sub>''z''</sub>, 1).<ref>Richard Paul, 1981, [https://books.google.com/books?id=UzZ3LAYqvRkC Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators], MIT Press, Cambridge, MA</ref>

To translate an object by a [[vector (geometry)|vector]] '''v''', each homogeneous vector '''p''' (written in homogeneous coordinates) would need to be multiplied by this '''translation matrix''':

: <math> T_{\mathbf{v}} = 
\begin{bmatrix}
1 & 0 & 0 & v_x \\
0 & 1 & 0 & v_y \\
0 & 0 & 1 & v_z \\
0 & 0 & 0 & 1 
\end{bmatrix}
</math>
As shown below, the multiplication will give the expected result:
: <math> T_{\mathbf{v}} \mathbf{p} =
\begin{bmatrix}
1 & 0 & 0 & v_x \\
0 & 1 & 0 & v_y\\
0 & 0 & 1 & v_z\\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
p_x \\ p_y \\ p_z \\ 1
\end{bmatrix}
=
\begin{bmatrix}
p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1
\end{bmatrix}
= \mathbf{p} + \mathbf{v} </math>

The inverse of a translation matrix can be obtained by reversing the direction of the vector:
: <math> T^{-1}_{\mathbf{v}} = T_{-\mathbf{v}} . \! </math>

Similarly, the product of translation matrices is given by adding the vectors:
: <math> T_{\mathbf{u}}T_{\mathbf{v}} = T_{\mathbf{u}+\mathbf{v}} . \! </math>
Because addition of vectors is [[commutative]], multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).