Transformation (geometry)
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In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both \mathbb^2 or both \mathbb^3 — such that the function is bijective so that its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
exists. The study of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
may be approached by the study of these transformations.


Classifications

Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve: * Displacements preserve distances and oriented angles (e.g., translations); * Isometries preserve angles and distances (e.g., Euclidean transformations); * Similarities preserve angles and ratios between distances (e.g., resizing); *
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 preserve parallelism (e.g., scaling, shear); * Projective transformations preserve collinearity;Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – ' Each of these classes contains the previous one. * Möbius transformations using complex coordinates on the plane (as well as
circle inversion A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
) preserve the set of all lines and circles, but may interchange lines and circles. France identique.gif , Original image (based on the map of France) France par rotation 180deg.gif , Isometry France par similitude.gif , Similarity France affine (1).gif ,
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 ...
France homographie.gif , Projective transformation France circ.gif ,
Inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
* Conformal transformations preserve angles, and are, in the first order, similarities. * Equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case. and are, in the first order, affine transformations of determinant 1. * Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points. *
Diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
s (bidifferentiable transformations) are the transformations that are affine in the first order; they contain the preceding ones as special cases, and can be further refined. Fconf.gif , Conformal transformation France aire.gif , Equiareal transformation France homothetie.gif , Homeomorphism France diff.gif ,
Diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
Transformations of the same type form groups that may be sub-groups of other transformation groups.


Opposite group actions

Many geometric transformations are expressed with linear algebra. The bijective linear transformations are elements of a general linear group. The linear transformation ''A'' is non-singular. For a row vector ''v'', the matrix product ''vA'' gives another row vector ''w'' = ''vA''. The transpose of a row vector ''v'' is a column vector ''v''T, and the transpose of the above equality is w^T = (vA)^T = A^T v^T . Here ''A''T provides a left action on column vectors. In transformation geometry there are compositions ''AB''. Starting with a row vector ''v'', the right action of the composed transformation is ''w'' = ''vAB''. After transposition, :w^T = (vAB)^T = (AB)^Tv^T = B^T A^T v^T . Thus for ''AB'' the associated left group action is B^T A^T . In the study of opposite groups, the distinction is made between opposite group actions for the only groups for which these opposites are equal are commutative groups.


See also

* Coordinate transformation * Erlangen program * Symmetry (geometry) * Reflection * Rigid transformation * Rotation *
Topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
* Transformation matrix


References


Further reading

* * Dienes, Z. P.; Golding, E. W. (1967) . ''Geometry Through Transformations'' (3 vols.): ''Geometry of Distortion'', ''Geometry of Congruence'', and ''Groups and Coordinates''. New York: Herder and Herder. * David Gans – ''Transformations and geometries''. *{{cite book , first1=David, last1=Hilbert, author1-link=David Hilbert , first2=Stephan, last2=Cohn-Vossen, author2-link=Stephan Cohn-Vossen , title = Geometry and the Imagination , edition = 2nd , year = 1952 , publisher = Chelsea , isbn = 0-8284-1087-9 * John McCleary – ''Geometry from a Differentiable Viewpoint''. * Modenov, P. S.; Parkhomenko, A. S. (1965) . ''Geometric Transformations'' (2 vols.): ''Euclidean and Affine Transformations'', and ''Projective Transformations''. New York: Academic Press. * A. N. Pressley – ''Elementary Differential Geometry''. * Yaglom, I. M. (1962, 1968, 1973, 2009) . ''Geometric Transformations'' (4 vols.).
Random House Random House is an American book publisher and the largest general-interest paperback publisher in the world. The company has several independently managed subsidiaries around the world. It is part of Penguin Random House, which is owned by Ger ...
(I, II & III), MAA (I, II, III & IV). Geometry Functions and mappings Symmetry Transformation (function)