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单词 MobiusTransformation
释义

Möbius transformation


A Möbius transformationMathworldPlanetmath is a bijectionMathworldPlanetmath on the extended complex plane {} given by

f(z)={acif z=if z=-dcaz+bcz+dotherwise

where a,b,c,d and ad-bc0

It can be shown that the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and compositionMathworldPlanetmathPlanetmath of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition.

The geometric interpretationMathworldPlanetmathPlanetmath of the Möbius group is that it is the group of automorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the Riemann sphere.

Any Möbius map can be composed from the elementary transformations - dilationsMathworldPlanetmath, translations and inversionsMathworldPlanetmathPlanetmath. If we define a line to be a circle passing through then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.

TitleMöbius transformation
Canonical nameMobiusTransformation
Date of creation2013-03-22 12:23:19
Last modified on2013-03-22 12:23:19
OwnerKoro (127)
Last modified byKoro (127)
Numerical id21
AuthorKoro (127)
Entry typeDefinition
Classificationmsc 30D99
Synonymfractional linear transformation
Synonymlinear fractional transformation
Related topicProofOfConformalMobiusCircleMapTheorem
Related topicAutomorphismsOfUnitDisk
Related topicUnitDiskUpperHalfPlaneConformalEquivalenceTheorem
Related topicInversionOfPlane
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更新时间:2025/5/4 19:57:19