An affine transformation is a transformation that preserves affine combinations of points.
It preserves collinearity and ratios
of directed distances along any line
not collapsed to a point. In particular, the midpoint
of a line segment maps to the midpoint
of its image. The nonsingular affine transformations of 
are the projective transformations that preserve the plane
at infinity. An affine transformation is also called an affinity.
Geometric contraction, expansion, dilation, reflection, rotation, shear, similarity transformations, spiral similarities, and translation are all affine transformations, as are their combinations. In general, a nonsingular affine transformation is a composition of rotations, reflections, translations, dilations, and shears.
While an affine transformation preserves proportions on uncollapsed lines, it does not necessarily preserve angles or lengths. Any triangle can be transformed into any other by an affine transformation, so all triangles are affine and, in this sense, affine is a generalization of congruent and similar.
A particular example combining rotation and expansion is the rotation-enlargement transformation, with 
,
![[x^'; y^']](/images/equations/AffineTransformation/Inline3.svg) |  | ![s[cosalpha sinalpha; -sinalpha cosalpha][x-x_0; y-y_0]](/images/equations/AffineTransformation/Inline5.svg) |
(1)
|
 |  | ![s[cosalpha(x-x_0)+sinalpha(y-y_0); -sinalpha(x-x_0)+cosalpha(y-y_0)].](/images/equations/AffineTransformation/Inline8.svg) |
(2)
|
Separating the equations,
 |  |  |
(3)
|
 |  |  |
(4)
|
This can be also written as
 |  |  |
(5)
|
 |  |  |
(6)
|
where
 |  |  |
(7)
|
 |  |  |
(8)
|
The scale factor 
is then defined by
 |
(9)
|
and the rotation angle, modulo 
, by
 |
(10)
|
where 
denotes the complex argument.
An affine transformation of 
is a map 
of the form
 |
(11)
|
for all 
,
where 
is a linear transformation of 
. If 
, the transformation is orientation-preserving;
if 
,
it is orientation-reversing.
