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Isometry

This article is about isometry in mathematics. For the usage of the term in mechanical engineering and architecture see isometric projection.

In the mathematical discipline of geometry and mathematical analysis, 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 of the space 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.

Contents

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 you should guess from context which one is used.

Let $X$ and $Y$ be metric spaces with metrics $d X$ and $d Y$ , a map $f:X\to Y$ is called distance preserving if for any $x,y\in X$ we have $d Y (f (x), f (y)) = d X (x, y).$ 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).

Metric spaces X and Y are called isometric if there is an isometry $X\to Y$ . The set of isometries from a metric space to itself form a group with respect to composition (called isometry group).

Examples

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

In Euclidean space with the usual Euclidean metric, the (global) isometries are the mappings composed of rotations, reflections and translations. In algebraic terms the isometries form a group called Euclidean group which is the semidirect product of the orthogonal group and the group of translations.

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

Generalizations

ε-isometry or almost isometry also called Hausdorff approximation, it is a map $f:X\to Y$ between metric spaces such that for any point in the target space there is a point in the image on distance $\le\epsilon$ and for any $x,y\in X$ we have

$|d_Y(f(x),f(y))-d_X(x,y)|\le\epsilon.$

Note that ε-isometry is not assumed to be continuous.

Quasi-isometry is yet an other useful generalization.

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