Editing Affine transformation (section)

{{Short description|Geometric transformation that preserves lines but not angles nor the origin}}
[[File:Fractal fern explained.png|thumb|right|200px|An image of a fern-like [[fractal]] ([[Barnsley fern|Barnsley's fern]]) that exhibits affine [[self-similarity]]. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the dark blue leaf and any of the light blue leaves by a combination of reflection, rotation, scaling, and translation.]]

In [[Euclidean geometry]], an '''affine transformation''' or '''affinity''' (from the Latin, ''[[wikt:affine|affinis]]'', "connected with") is a [[geometric transformation]] that preserves [[line (geometry)|lines]] and [[parallel (geometry)|parallelism]], but not necessarily [[Euclidean distance]]s and [[angle]]s.

More generally, an affine transformation is an [[automorphism]] of an [[affine space]] (Euclidean spaces are specific affine spaces), that is, a [[Function (mathematics)|function]] which [[Map (mathematics)|maps]] an affine space onto itself while preserving both the [[dimension]] of any [[affine subspace]]s (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of [[Parallel (geometry)|parallel]] line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

If {{mvar|X}} is the point set of an affine space, then every affine transformation on {{mvar|X}} can be represented as the [[function composition|composition]] of a [[linear transformation]] on {{mvar|X}} and a [[Translation (geometry)|translation]] of {{mvar|X}}. Unlike a purely linear transformation, an affine transformation need not preserve the origin of the affine space. Thus, every linear transformation is affine, but not every affine transformation is linear.

Examples of affine transformations include translation, [[Scaling (geometry)|scaling]], [[homothety]], [[Similarity (geometry)|similarity]], [[Reflection (mathematics)|reflection]], [[Rotation (mathematics)|rotation]], [[hyperbolic rotation]], [[shear mapping]], and compositions of them in any combination and sequence.

Viewing an affine space as the complement of a [[hyperplane at infinity]] of a [[projective space]], the affine transformations are the [[projective transformation]]s of that projective space that leave the hyperplane at infinity [[Invariant (mathematics)|invariant]], restricted to the complement of that hyperplane.

A [[generalization]] of an affine transformation is an '''affine map'''{{sfn|Berger|1987|p=38}} (or '''affine homomorphism''' or '''affine mapping''') between two (potentially different) affine spaces over the same [[Field (mathematics)|field]] {{mvar|k}}. Let {{math|(''X'', ''V'', ''k'')}} and {{math|(''Z'', ''W'', ''k'')}} be two affine spaces with {{mvar|X}} and {{mvar|Z}} the point sets and {{mvar|V}} and {{mvar|W}} the respective associated [[vector space]]s over the field {{mvar|k}}. A map {{math|''f'' : ''X'' → ''Z''}} is an affine map if there exists a [[linear map]] {{math|''m''<sub>''f''</sub> : ''V'' → ''W''}} such that {{math|1=''m''<sub>''f''</sub> (''x'' − ''y'') = ''f'' (''x'') − ''f'' (''y'')}} for all {{mvar|x, y}} in {{mvar|X}}.{{sfn|Samuel|1988|p=11}}