Editing Affine transformation (section)

==In the plane==
[[File:Central dilation.svg|thumb|right|300px|A [[homothety]]. The triangles A{{sub|1}}B{{sub|1}}Z, B{{sub|1}}C{{sub|1}}Z, and A{{sub|1}}C{{sub|1}}Z get mapped to A{{sub|2}}B{{sub|2}}Z, B{{sub|2}}C{{sub|2}}Z, and A{{sub|2}}C{{sub|2}}Z, respectively.]] 
Every affine transformations in a [[Euclidean plane]] is the composition of a [[translation]] and an affine transformation that fixes a point; the latter may be 
* a [[homothety]],
* [[rotation]]s around the fixed point,
* a [[scaling (geometry)|scaling]], with possibly negative scaling factors, in two directions (not necessarily perpendicular); this includes [[Reflection (mathematics)|reflections]],
* a [[shear mapping]]
* a [[squeeze mapping]].

Given two non-degenerate [[triangle]]s ''ABC'' and ''A′B′C′'' in a Euclidean plane, there is a unique affine transformation ''T'' that maps ''A'' to ''A′'', ''B'' to ''B′'' and ''C'' to ''C′''. Each of ''ABC'' and ''A′B′C′'' defines an [[affine coordinate system]] and a [[barycentric coordinate system]]. Given a point ''P'', the point ''T''(P) is the point that has the same coordinates on the second system as the coordinates of ''P'' on the first system. 

Affine transformations do not respect lengths or angles; they multiply areas by the constant factor
:area of ''A′B′C′'' / area of ''ABC''.

A given ''T'' may either be ''direct'' (respect orientation), or ''indirect'' (reverse orientation), and this may be determined by comparing the orientations of the triangles.