Editing Affine transformation (section)

==Properties==
=== Properties preserved ===
An affine transformation preserves:
# [[collinearity]] between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation.
# [[Parallel (geometry)|parallelism]]: two or more lines which are parallel, continue to be parallel after the transformation.
# [[Convex set|convexity]] of sets: a convex set continues to be convex after the transformation. Moreover, the [[extreme point]]s of the original set are mapped to the extreme points of the transformed set.<ref name=res>{{cite web|last1=Reinhard Schultz|title=Affine transformations and convexity|url=http://math.ucr.edu/~res/math145A-2014/affine+convex.pdf|access-date=27 February 2017}}</ref>
# ratios of lengths of parallel line segments: for distinct parallel segments defined by points <math>p_1</math> and <math>p_2</math>, <math>p_3</math> and <math>p_4</math>, the ratio of <math>\overrightarrow{p_1p_2}</math> and <math>\overrightarrow{p_3p_4}</math> is the same as that of <math>\overrightarrow{f(p_1)f(p_2)}</math> and <math>\overrightarrow{f(p_3)f(p_4)}</math>.
# [[Barycentric_coordinate_system|barycenters]] of weighted collections of points.

=== Groups ===
As an affine transformation is [[invertible function|invertible]], the [[square matrix]]  <math>A</math> appearing in its [[#Representation|matrix representation]] is [[invertible matrix|invertible]]. The matrix representation of the inverse transformation is thus
:<math>
\left[ \begin{array}{ccc|c} & A^{-1} & & -A^{-1}\vec{b} \ \\ 0 & \ldots & 0 & 1 \end{array} \right].
</math>

The invertible affine transformations (of an affine space onto itself) form the [[affine group]], which has the [[general linear group]] of degree <math>n</math> as subgroup and is itself a subgroup of the general linear group of degree <math>n + 1</math>.

The [[Similarity transformation (geometry)|similarity transformations]] form the subgroup where <math>A</math> is a scalar times an [[orthogonal matrix]]. For example, if the affine transformation acts on the plane and if the [[determinant]] of <math>A</math> is 1 or −1 then the transformation is an [[equiareal map]]ping. Such transformations form a subgroup called the ''equi-affine group''.<ref>[[Oswald Veblen]] (1918) ''Projective Geometry'', volume 2, pp. 105–7.</ref> A transformation that is both equi-affine and a similarity is an [[isometry]] of the plane taken with [[Euclidean distance]].

Each of these groups has a subgroup of ''[[orientability|orientation]]-preserving'' or ''positive'' affine transformations: those where the determinant of <math>A</math> is positive. In the last case this is in 3D the group of [[rigid transformation]]s ([[Improper rotation|proper rotations]] and pure translations).

If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a transformation as a rotation by a certain angle with respect to a certain axis may give a clearer idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context.