mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a geometric transformation is any

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

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 underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function whose domain and range are sets of points – most often a

real coordinate space In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as '' coordinate vectors''. ...

\mathbb^2

\mathbb^3

– such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations, such as in transformation geometry.

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: *

Displacement Displacement may refer to: Physical sciences Mathematics and physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...

s preserve

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

s and oriented angles (e.g., translations); *

Isometries 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'' mea ...

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 general ...

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 number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...

s using complex coordinates on the plane (as well as

circle inversion In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry ...

) 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'' me ...

France par similitude.gif , Similarity France affine (1).gif ,

France homographie.gif , Projective transformation France circ.gif , Inversion *

Conformal transformation In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...

s preserve angles, and are, in the first order, similarities. * Equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case. Bruce E. Meserve – Fundamental Concepts of Geometry, page 191.] and are, in the first order, affine transformations of

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

1. *

Homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

s (bicontinuous transformations) preserve the neighborhoods of points. *

Diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

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 ,

France aire.gif , Equiareal transformation France homothetie.gif ,

France diff.gif ,

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 In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

. The

linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

''A'' is non-singular. For a

row vector In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some , co ...

''v'', the

matrix product In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

''vA'' gives another row vector ''w'' = ''vA''. The

transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...

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 In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...

B^T A^T .

In the study of opposite groups, the distinction is made between opposite group actions because commutative groups are the only groups for which these opposites are equal.

Active and passive transformations

Classifications

Opposite group actions

Active and passive transformations

See also

References

Further reading