# Isometry

*For the mechanical engineering and architecture usage, see*isometric projection.

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.

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.

## 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 and be metric spaces with metrics and . A map is called **distance preserving** if for any one has A distance preserving map is automatically injective.

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).

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**.

## Examples

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

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

- The isometric linear maps from
**C**^{n}to itself are the unitary matrices.

## 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- for one has , and
- for any point there exists a point with

- 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 an other useful generalization.