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In mathematics, a transformation is a function ''f'', usually with some geometrical underpinning, that maps a set ''X'' to itself, i.e. .
Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations,
In mathematics, a transformation is a function ''f'', usually with some geometrical underpinning, that maps a set ''X'' to itself, i.e. .
Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
s, and specific affine transformations, such as rotations, reflections and translations.
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" 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, then a partial transformation is a function ''f'': ''A'' → ''B'', where both ''A'' and ''B'' aresubset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of some set ''X''.
Algebraic structures
The set of all transformations on a given base set, together withfunction composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, forms a regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
.
Combinatorics
For a finite set ofcardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
''n'', there are ''n''''n'' transformations and (''n''+1)''n'' partial transformations.
See also
* Coordinate transformation * Data transformation (statistics) * Geometric transformation * Infinitesimal transformation * Linear transformation * Rigid transformation * Transformation geometry * Transformation semigroup * Transformation group * Transformation matrixReferences
External links
* {{DEFAULTSORT:Transformation (Geometry) Functions and mappings