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单词 Isometry
释义

isometry


Let (X1,d1) and (X2,d2) be metric spaces.A function f:X1X2 is said to be an isometric mapping(or isometric embedding) if

d1(x,y)=d2(f(x),f(y))

for all x,yX1.

Every isometric mapping is injectivePlanetmathPlanetmath,for if x,yX1 with xy then d1(x,y)>0,and so d2(f(x),f(y))>0, and then f(x)f(y).One can also easily show that every isometric mapping is continuousPlanetmathPlanetmath.

An isometric mapping that is surjectivePlanetmathPlanetmath (and therefore bijectiveMathworldPlanetmath)is called an isometry.(Readers are warned, however,that some authors do not require isometries to be surjective;that is, they use the term isometryfor what we have called an isometric mapping.)Every isometry is a homeomorphismPlanetmathPlanetmath.

If there is an isometry between the metric spaces (X1,d1) and (X2,d2),then they are said to be isometric.Isometric spaces are essentially identical as metric spaces,and in particular they are homeomorphic.

Given any metric space (X,d),the set of all isometries XX forms a group under compositionMathworldPlanetmath.This group is called the isometry group(or group of isometries) of X,and may be denoted by Iso(X) or Isom(X).In general, an (as opposed to the) isometry group(or group of isometries) of X is any subgroup of Iso(X).

Titleisometry
Canonical nameIsometry
Date of creation2013-03-22 12:19:08
Last modified on2013-03-22 12:19:08
Owneryark (2760)
Last modified byyark (2760)
Numerical id13
Authoryark (2760)
Entry typeDefinition
Classificationmsc 54E35
Related topicRealTree
Related topicIsometricIsomorphism
Definesisometric
Definesisometric mapping
Definesisometric embedding
Definesisometry group
Definesgroup of isometries
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