Editing Cartesian coordinate system (section)

===Affine transformation===
[[File:2D_affine_transformation_matrix.svg|thumb|Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)]]

[[Affine transformation]]s of the [[Euclidean plane]] are transformations that map lines to lines, but may change distances and angles. As said in the preceding section, they can be represented with augmented matrices:
<math display=block>\begin{pmatrix} A_{1,1} & A_{2,1} & b_{1} \\ A_{1,2} & A_{2,2} & b_{2} \\ 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} x \\ y \\ 1 \end{pmatrix}
=
\begin{pmatrix} x' \\ y' \\ 1 \end{pmatrix}.</math>

The Euclidean transformations are the affine transformations such that the 2×2 matrix of the <math>A_{i,j}</math> is [[orthogonal matrix|orthogonal]].

The augmented matrix that represents the [[function composition|composition]] of two affine transformations is obtained by multiplying their augmented matrices.

Some affine transformations that are not Euclidean transformations have received specific names.

====Scaling====
An example of an affine transformation which is not Euclidean is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number ''m''. If {{nowrap|(''x'', ''y'')}} are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

<math display=block>(x',y') = (m x, m y).</math>

If ''m'' is greater than 1, the figure becomes larger; if ''m'' is between 0 and 1, it becomes smaller.

====Shearing====
A [[shear mapping|shearing transformation]] will push the top of a square sideways to form a [[parallelogram]]. Horizontal shearing is defined by:

<math display=block>(x',y') = (x+y s, y)</math>

Shearing can also be applied vertically:

<math display=block>(x',y') = (x, x s+y)</math>