Isometry

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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.

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[edit] Definitions

Isometry is defined for two cases: global isometry and the more constrained path (or arcwise) isometry. Both cases are often [ambiguously] called isometry and based on context, one must determine which sense is being used.

Let X and Y be metric spaces with metrics dX and dY. A map f:X\to Y is called distance preserving if for any x,y\in X one has dY(f(x),f(y)) = dX(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 onto Y, that is if there exists an isometric isomorphism between X and Y. The set of isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.

[edit] Examples

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

[edit] 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 | dY(f(x),f(x')) − dX(x,x') | < ε, and
    2. for any point y\in Y there exists a point x\in X with dY(y,f(x)) < ε.
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.

[edit] See also

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