- Equality of measure.
- Equality of elevation above sea level.
- Mathematics. A function between metric spaces which preserves distances, such as a rotation or translation in a plane.
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Results for isometry
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The noun has one meaning:
Meaning #1:
equality of elevation above sea level
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.
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 guess from context which one is intended.
Let X and Y be metric
spaces with metrics dX and dY. A map f:X→Y is called distance preserving if for any
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 to Y. The set of isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.
is a path isometry but not a global
isometry.Given two normed vector spaces V and W, a linear isometry is a linear map f : V → W that preserves the norms:

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.
one has
| dY(f(x),f(x')) -
dX(x,x') | < ε, and
there
exists a point
with
dY(y,f(x)) < ε.This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
Français (French)
n. - isométrie
Deutsch (German)
n. - Isometrie
Ελληνική (Greek)
n. - ισομετρία
Português (Portuguese)
n. - isometria (f)
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
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)
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Some good "isometry" pages on the web:
Math mathworld.wolfram.com |
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![]() | Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. 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 GNU Free Documentation License. It uses material from the Wikipedia article "Isometry". Read more | |
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