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isometry

 
Dictionary: i·som·e·try   (ī-sŏm'ĭ-trē) pronunciation
n.
  1. Equality of measure.
  2. Equality of elevation above sea level.
  3. Mathematics. A function between metric spaces which preserves distances, such as a rotation or translation in a plane.

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WordNet: isometry
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Note: click on a word meaning below to see its connections and related words.

The noun has one meaning:

Meaning #1: equality of elevation above sea level


Wikipedia: 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 map 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.

An isometric surjective linear operator on a Hilbert space is called a unitary operator.

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 one should determine from context which one is intended.

Let X and Y be metric spaces with metrics dX and dY. A map ƒ : X → Y is called distance preserving if for any a,b ∈ X one has

d_Y\left(f(a),f(b)\right)=d_X(a,b).

A distance preserving map is automatically injective. Clearly, every isometry between metric spaces is necessarily a topological embedding.

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 isometry

Given two normed vector spaces V and W, a linear isometry is a linear map f : VW 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.

By the Mazur-Ulam theorem, any isometry of normed vector spaces over R is affine.

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′ ∈ X one has |dY(ƒ(x),ƒ(x′))−dX(x,x′)| < ε, and
    2. for any point y ∈ Y there exists a point x ∈ X with dY(y,ƒ(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.

See also

References

  • F. S. Beckman and D. A. Quarles, Jr., On isometries of Euclidean space, Proc. Amer. Math. Soc., 4 (1953) 810-815.

Translations: Isometry
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Dansk (Danish)
n. - isometri

Nederlands (Dutch)
isometrie

Français (French)
n. - isométrie

Deutsch (German)
n. - Isometrie

Ελληνική (Greek)
n. - ισομετρία

Italiano (Italian)
isometria

Português (Portuguese)
n. - isometria (f)

Русский (Russian)
изометрия

Español (Spanish)
n. - isometría

Svenska (Swedish)
n. - isometria

中文(简体)(Chinese (Simplified))
等距, 等容

中文(繁體)(Chinese (Traditional))
n. - 等距, 等容

한국어 (Korean)
n. - 균등 , 등고

日本語 (Japanese)
n. - 大きさの等しいこと, 等長変換, 等高

العربيه (Arabic)
‏(الاسم) التقايس : تساوي القياس, , التشاهق : التساوي في الإرتفاع عن سطح البحر‏

עברית (Hebrew)
n. - ‮המרת מרחבים תוך שמירת מידות (מתמטיקה)‬


Best of the Web: isometry
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Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.  Read more
WordNet. WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved.  Read more
Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Isometry" Read more
Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved.  Read more