In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

"We shall find it convenient to use the word ''transformation'' in the special sense of a one-to-one correspondence $P\; \backslash to\; P\text{'}$ among all points in the plane (or in space), that is, a rule for associating pairs of points, with the understanding that each pair has a first member and a second member and that every point occurs as the first member of just one pair and also as the second member of just one pair...

In particular, an ''isometry'' (or "congruent transformation," or "congruence") is a transformation which preserves length..."
** Introduction **
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection.
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.
An isometric surjective linear operator on a Hilbert space is called a unitary operator.

Properties

A collection of isometries typically form a group, the isometry group. When the group is a continuous group, the infinitesimal generators of the group are the Killing vector fields. The Myers–Steenrod theorem states that every isometry between two connected Riemannian manifolds is smooth (differentiable). A second form of this theorem states that the isometry group of a Riemannian manifold is a Lie group. Riemannian manifolds that have isometries defined at every point are called symmetric spaces.

Generalizations

* Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map $f:X\backslash to\; Y$ between metric spaces such that *# for ''x'',''x''′ ∈ ''X'' one has |''d''

Bibliography

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