View source for Affine transformation

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''of a Euclidean space''

A one-to-one point mapping of the plane or space into itself, where to
three points lying on a straight line correspond three points also
lying on a straight line. Thus, by an affine transformation straight
lines are transformed into straight lines. An affine transformation of
the plane transforms intersecting lines into intersecting lines, and
parallel lines into parallel lines. An affine transformation of space
results in an affine mapping of each plane onto some plane;
intersecting planes are mapped into intersecting planes, while
parallel planes are mapped into parallel planes. Moreover, the mutual
location of two straight lines is preserved: intersecting straight
lines are mapped into intersecting straight lines, parallel straight
lines are mapped into parallel straight lines, while skew lines are
mapped into skew lines.

Under an affine transformation the ratio between directed segments
lying on the same line or on parallel straight lines is equal to the
ratio of their images. The ratio between the surface areas of two
measurable figures (on the Euclidean plane) and the ratio between the
volumes of two measurable bodies (in the Euclidean space) are
preserved as well. Under an affine transformation a set of vectors in
the plane (in space) is one-to-one mapped on a set of vectors in the
plane (in space), and this mapping is linear. An affine transformation
is defined in an affine coordinate system by a non-degenerate
(non-homogeneous)
[[Linear transformation|linear transformation]]; thus, in the case of
a plane, an affine transformation is analytically expressed by the
formulas 
$$x'=c_{11}x+c_{12}y+c_{13}$$

$$y'=c_{21}x+c_{22}y+c_{23}$$
with the supplementary condition 
$$\begin{vmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{vmatrix} \ne 0$$
Affine transformations in
space are defined in a similar manner.

Under an affine transformation an algebraic curve is converted into an
algebraic curve, the order of the curves being preserved. In
particular, a curve of the second order is converted into a curve of
the second order, moreover, ellipses are converted into ellipses,
hyperbolas into hyperbolas, parabolas into parabolas, etc.

Examples of affine transformations: isometric transformations,
similarity transformations, and uniform contractions of a plane to a
straight line. Each affine transformation in a plane is a product of
an isometric transformation and two uniform contractions to two
mutually-perpendicular straight lines. Each affine transformation in a
space is a product of an isometric transformation and three uniform
contractions to three pairwise-perpendicular planes.

The affine transformations form a group; the similarity
transformations constitute a subgroup of this group; the set of
isometric transformations is a subgroup of the group of similarity
transformations.

Affine transformations are the most general one-to-one mappings of a
plane (a space) into itself under which straight lines are preserved.

====Comments====
In terms of coordinates like above similarity
transformations take the form 
$$x'=c_{11}x+c_{12}y+c_{13},\quad y'=-ec_{21}x+ec_{22}y+c_{23}$$
Similitudes, or homothetic
transformations, which in terms of coordinates take the form $x'=c_{11}x+c_{13}$, $ y'=c_{22}y+c_{23}$,
constitute another interesting subgroup of the group of affine
transformations.


====References====
{|
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|valign="top"|{{Ref|Al}}||valign="top"| P.S. Aleksandrov, "Lectures on analytical geometry", Moscow (1968) (In Russian)  {{MR|0244836}}  
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|valign="top"|{{Ref|Co}}||valign="top"| H.S.M. Coxeter, "Introduction to geometry", Wiley (1961)  {{MR|1531486}} {{MR|0123930}}  {{ZBL|0095.34502}} 
|-
|valign="top"|{{Ref|Me}}||valign="top"| B.E. Meserve, "Fundamental concepts of geometry", Addison-Wesley (1955)  {{MR|1529461}}  {{ZBL|0070.37802}} {{ZBL|0053.28502}} 
|-
|valign="top"|{{Ref|Po}}||valign="top"| M.M. Postnikov, "Analytic geometry", Moscow (1973) (In Russian)  {{MR|0987205}} {{MR|0890748}}  {{ZBL|0662.51001}} {{ZBL|0482.51001}} 
|-
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