# Dictionary Definition

isometry

### Noun

1 the growth rates in different parts of a growing organism are the same
2 a one-to-one mapping of one metric space into another metric space that preserves the distances between each pair of points; "the isometries of the cube"
3 equality of elevation above sea level
4 equality of measure (e.g., equality of height above sea level or equality of loudness etc.)

# User Contributed Dictionary

## English

### Etymology

From iso- and -metry.

### Noun

1. A function between metric spaces (or on a single metric space) having the property that the distance between two images is equal to the distance between their pre-images.

#### Translations

A function
• Finnish: isometria
• French: isométrie
• German: Isometrie
• Hebrew: איזומטריה
• Swedish: isometri

# Extensive Definition

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 determine from context which one is intended.
Let X and Y be metric spaces with metrics d_X and d_Y. A map f\colon X\to Y is called distance preserving if for any x,y\in X one has d_Y\left(f(x),f(y)\right)=d_X(x,y). 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

• The map R\toR defined by x\mapsto |x| is a path isometry but not a global isometry.

## Linear isometries

Given two normed vector spaces V and W, a linear isometry is a linear map f : V → W that preserves the norms:
\|f(v)\| = \|v\|
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.

## Generalizations

• Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map f:X\to Y between metric spaces such that
1. for x,x'\in X one has |d_Y(f(x),f(x'))-d_X(x,x')|, and
2. for any point y\in Y there exists a point x\in X with d_Y(y,f(x))
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.
isometry in Czech: Izometrické zobrazení
isometry in German: Isometrie
isometry in Spanish: Isometría
isometry in Esperanto: Izometrio
isometry in French: Isométrie
isometry in Italian: Isometria
isometry in Hebrew: איזומטריה
isometry in Dutch: Isometrie
isometry in Japanese: 等長写像
isometry in Polish: Izometria
isometry in Portuguese: Isometria (transformação geométrica)
isometry in Russian: Изометрия (математика)
isometry in Slovenian: Togi premik
isometry in Finnish: Isometria
isometry in Chinese: 等距同构