# 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

- 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

- For the mechanical engineering and architecture usage, see isometric projection. For isometry in differential geometry, see isometry (Riemannian geometry).

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

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

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

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

## 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\|

## 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
- for x,x'\in X one has |d_Y(f(x),f(x'))-d_X(x,x')|, and
- 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.

- Quasi-isometry is yet another useful generalization.

## See also

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