In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Distance is a numerical description of how far apart things lie. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... An example of congruence. ...

Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences 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 of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

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 guess from context which one is intended.

Let X and Y be metric spaces with metrics d_X and d_Y. A map is called distance preserving if for any one has d_Y(f(x),f(y)) = d_X(x,y). A distance preserving map is automatically injective. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Partial plot of a function f. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

A global isometry is a bijective distance preserving map. A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective). In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

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 with respect to function composition, called the isometry group. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...

Examples

• Any reflection, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group.

• The map RR defined by is a path isometry but not a global isometry.

• The isometric linear maps from Cⁿ to itself are the unitary matrices.

In mathematics, a reflection (also spelt reflexion) is a map that transforms an object into its mirror image. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... A sphere rotating around its axis. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...

Linear isometries

Given two normed vector spaces V and W, a linear isometry is a linear map f : V → W that preserves the norms: In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

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. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

Generalizations

• Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map between metric spaces such that
  1. for one has | d_Y(f(x),f(x')) − d_X(x,x') | < ε, and
  2. for any point there exists a point with d_Y(y,f(x)) < ε.

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

• Quasi-isometry is yet another useful generalization.

Felix Hausdorff Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... This is a glossary of some terms used in Riemannian geometry and metric geometry &#8212; it doesnt cover the terminology of differential topology. ...

See also

• Euclidean group
• Congruence (geometry)
• Euclidean plane isometry
• 3D isometries which leave the origin fixed
• space group
• involution