Isometry

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Isometry

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an isometry, isometric isomorphism or congruence mapping is a distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
-preserving isomorphism

Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
between metric spaces. Geometric figures which can be related by an isometry are called congruent

Congruence (geometry)

In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
.

Isometries are often used in constructions where one space is embedded

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in another space. For instance, the completion

Complete space

In mathematical analysis, a metric space M is said to be complete if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M....
of a metric space M involves an isometry from M into M, a quotient set of the space of Cauchy sequence
Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses....
s on M.

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Encyclopedia

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an isometry, isometric isomorphism or congruence mapping is a distance
Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
-preserving isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
between metric spaces. Geometric figures which can be related by an isometry are called congruent
Congruence (geometry)

In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
.

Isometries are often used in constructions where one space is embedded
Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in another space. For instance, the completion
Complete space

In mathematical analysis, a metric space M is said to be complete if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M....
of a metric space M involves an isometry from M into M, a quotient set of the space of Cauchy sequence
Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses....
s on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
of some normed vector space
Normed vector space

In mathematics, with 2- or 3-dimensional Vector s with real number-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any Vector space Rn....
and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space
Banach space

In mathematics, Banach spaces are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them....
.

Definitions

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and one should determine from context which one is intended.

Let X and Y be metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
s with metrics dX and dY. A map
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
ƒ : X → Y is called distance preserving if for any x,y ∈ X one has

A distance preserving map is automatically injective. Clearly, every isometry between metric spaces is necessarily a topological imbedding.

A global isometry is a bijective distance preserving map. A path isometry or arcwise isometry is a map which preserves the lengths of curves
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
(not necessarily bijective).

Two metric spaces X and Y are called isometric if there is an isometry from X to Y. The set of isometries from a metric space to itself forms a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
with respect to function composition
Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
, called the isometry group
Isometry group

In mathematics, the isometry group of a metric space is the Set of all isometry from the metric space onto itself, with the function composition as group operation....
.

Examples

Any reflection
Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
, translation
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
and rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
is a global isometry on Euclidean spaces. See also Euclidean group
Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
.

The map R'R defined by is a path isometry but not a global isometry.

The isometric linear maps from Cn to itself are the unitary matrices
Unitary matrix

In mathematics, a unitary matrix is an n by n complex number matrix U satisfying the condition where is the identity matrix and is the conjugate transpose of U....
.

Linear isometries

Given two normed vector space
Normed vector space

In mathematics, with 2- or 3-dimensional Vector s with real number-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any Vector space Rn....
s V and W, a linear isometry is a linear map f : V → W that preserves the norms: for all v in V. Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective.

By the Mazur-Ulam theorem
Mazur-Ulam theorem

In mathematics, the Mazur?Ulam theorem states that if and are normed spaces over R and the map pingis a isometry, then is affine transformation....
, any isometry of normed vector spaces over R is affine
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
.

Generalizations

Given a positive real number e, an e-isometry or almost isometry (also called a Hausdorff
Felix Hausdorff

Felix Hausdorff was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis....
approximation) is a map between metric spaces such that for x,x′ ∈ X one has |dY(ƒ(x),ƒ(x′))−dX(x,x′)| < ε, and
for any point y ∈ Y there exists a point x ∈ X with dY(y,ƒ(x)) < ε

That is, an e-isometry preserves distances to within e and leaves no element of the codomain further than e away from the image of an element of the domain. Note that e-isometries are not assumed to be continuous
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
.

Quasi-isometry
Glossary of Riemannian and metric geometry

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful....
is yet another useful generalization.

Beckman-Quarles theorem
The Beckman-Quarles theorem states that for a Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
E of dimension d at least 2, any mapping f from E to itself that preserves the property of being at a unit distance apart must be an isometry.

See also

Isometric projection
Isometric projection

File:Isometric projection.jpgIsometric projection is a form of graphical projection, more specifically, a form of axonometric projection. It is a method of visually representing three-dimensional objects in two dimensions, in which the three Cartesian coordinate system appear equally foreshortened and the angles between any two of them are 1...
Congruence (geometry)
Congruence (geometry)

In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
Euclidean plane isometry
Euclidean plane isometry

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length....
3D isometries which leave the origin fixed
Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
space group
Space group

The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. The word 'group' in the name comes from the group , which is used to build the set of space groups....
involution
Involution

In mathematics, an involution, or an involutary function, is a function that is its own inverse function, so that...
Isometries in physics
Isometry group
Isometry group

In mathematics, the isometry group of a metric space is the Set of all isometry from the metric space onto itself, with the function composition as group operation....
Homeomorphism group
Homeomorphism group

In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group binary operation....