# Transformation (function)

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{{Short description|Function that applies a set to itself}}
{{Redirect|Transformation (mathematics)||Transformation (disambiguation)}}
{{broader|Function (mathematics)}}
[[File:A code snippet for a rhombic repetitive pattern.svg|thumb|upright=1.5|A [composition](/source/Function_composition) of four [mappings](/source/Map_(mathematics)) coded [in&nbsp;SVG](/source/Scalable_Vector_Graphics),<br/>which '''transforms''' a [rectangular](/source/Rectangle) repetitive [pattern](/source/pattern)<br/>into a [rhombic](/source/Rhombus) pattern. The four transformations are [linear](/source/Linear_map).]]

In [mathematics](/source/mathematics), a '''transformation''', '''transform''', or '''self-map'''<ref>{{Cite web|title=Self-Map -- from Wolfram MathWorld|url=https://mathworld.wolfram.com/Self-Map.html|access-date=March 4, 2024}}</ref> is a [function](/source/Function_(mathematics)) ''f'', usually with some [geometrical](/source/Geometry) underpinning, that maps a [set](/source/set_(mathematics)) ''X'' to itself, i.e. {{nowrap|''f'': ''X'' → ''X''}}.<ref>{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n73 1]}}</ref><ref name="Grillet1995">{{cite book|author=Pierre A. Grillet|title=Semigroups: An Introduction to the Structure Theory|url=https://books.google.com/books?id=yM544W1N2UUC&pg=PA2|year=1995|publisher=CRC Press|isbn=978-0-8247-9662-4|page=2}}</ref><ref>{{cite book|author=Wilkinson, Leland |title=The Grammar of Graphics|publisher=Springer|year=2005|isbn=978-0-387-24544-7|page=29|url=https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA29|edition=2nd}}</ref>
Examples include [linear transformation](/source/linear_transformation)s of [vector spaces](/source/vector_spaces) and [geometric transformation](/source/geometric_transformation)s, which include [projective transformation](/source/projective_transformation)s, [affine transformation](/source/affine_transformation)s, and specific affine transformations, such as [rotation](/source/rotation)s, [reflections](/source/reflection_(mathematics)) and [translations](/source/translation_(geometry)).<ref>{{Cite web|url=https://www.mathsisfun.com/geometry/transformations.html|title=Transformations|website=www.mathsisfun.com|access-date=2019-12-13}}</ref><ref name=":0">{{Cite web|url=https://www.basic-mathematics.com/transformations-in-math.html|title=Types of Transformations in Math|website=Basic-mathematics.com|access-date=2019-12-13}}</ref>

== Partial transformations ==
While it is common to use the term '''transformation''' for any function of a set into itself (especially in terms like "[transformation semigroup](/source/transformation_semigroup)" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to [partial functions](/source/partial_functions), then a '''partial transformation''' is a function ''f'': ''A'' → ''B'', where both ''A'' and ''B'' are [subset](/source/subset)s of some set ''X''.<ref name="Hollings2014">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=251}}</ref>

==Algebraic structures==
The set of all transformations on a given base set, together with [function composition](/source/function_composition), forms a [regular semigroup](/source/regular_semigroup).

==Combinatorics==
For a finite set of [cardinality](/source/cardinality) ''n'', there are ''n''<sup>''n''</sup> transformations and (''n''+1)<sup>''n''</sup> partial transformations.<ref>{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n74 2]}}</ref>

==See also==
*[Endofunction](/source/Endofunction)
*[Coordinate transformation](/source/Coordinate_transformation)
*[Data transformation (statistics)](/source/Data_transformation_(statistics))
*[Geometric transformation](/source/Geometric_transformation)
*[Infinitesimal transformation](/source/Infinitesimal_transformation)
*[Linear transformation](/source/Linear_transformation)
*[List of transforms](/source/List_of_transforms)
*[Rigid transformation](/source/Rigid_transformation)
*[Transformation geometry](/source/Transformation_geometry)
*[Transformation semigroup](/source/Transformation_semigroup)
*[Transformation group](/source/Transformation_group)
*[Transformation matrix](/source/Transformation_matrix)

==References==
{{Reflist}}

==External links==
*{{Commonscatinline}}

{{Authority control}}

{{DEFAULTSORT:Transformation (Geometry)}}
Category:Transformation (function)
Category:Functions and mappings

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Adapted from the Wikipedia article [Transformation (function)](https://en.wikipedia.org/wiki/Transformation_(function)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Transformation_(function)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.