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单词 ProofOfConverseOfMobiusTransformationCrossratioPreservationTheorem
释义

proof of converse of Möbius transformation cross-ratio preservation theorem


Suppose that a,b,c,d are distinct. Consider the transform μ defined as

μ(z)=(b-d)(c-d)(c-b)(z-d)-b-dc-b.

Simple calculation reveals that μ(b)=1, μ(c)=0, and μ(d)=.Furthermore, μ(a) equals the cross-ratioMathworldPlanetmath of a,b,c,d.

Suppose we have two tetrads with a common cross-ratio λ. Then, as above, we mayconstruct a transform μ1 which maps the first tetrad to (λ,1,0,) and atransform μ2 which maps the first tetrad to (λ,1,0,). Then μ2-1μ1 maps the former tetrad to the latter and, by the group property, it is also aMöbius transformationMathworldPlanetmathPlanetmath.

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更新时间:2025/5/4 19:06:27