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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a geometric transformation is any bijection of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to itself (or to another such set) with some salient
geometrical 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 ...
underpinning. More specifically, it is a function whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
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 a ...
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 transformations 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) 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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
France par similitude.gif , Similarity France affine (1).gif , Affine transformation 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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1. * Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points. * Diffeomorphisms (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 Transformations of the same type form
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
''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) In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symme ...
* 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 ...
* 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 (I, II & III), MAA (I, II, III & IV). Geometry Functions and mappings Symmetry Transformation (function)