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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
of a
set to itself (or to another such set) with some salient
geometrical 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
* Do ...
and
range are sets of points — most often both
or both
— 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 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 generally, ...
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 transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s 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
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 generally, ...
France homographie.gif , Projective transformation
France circ.gif , Inversion
*
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.
*
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.
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
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 ...
''AB''. Starting with a row vector ''v'', the right action of the composed transformation is ''w'' = ''vAB''. After transposition,
:
Thus for ''AB'' the associated left
group action is
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 symmetr ...
*
Reflection
*
Rigid transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations ...
*
Rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
*
Topology
*
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)