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Möbius transformation cross-ratio preservation theorem
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(Theorem)
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A Möbius transformation $f: z \mapsto w$ preserves the cross-ratios, i.e.
Conversely, given two quadruplets which have the same cross-ratio, there exists a Möbius transformation which maps one quadruplet to the other.
A consequence of this result is that the cross-ratio of $(a,b,c,d)$ is the value at $a$ of the Möbius transformation that takes $b$ $c$ $d$ to $1$ $0$ $\infty$ respectively.
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"Möbius transformation cross-ratio preservation theorem" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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Cross-references: consequence, maps, cross-ratio, quadruplets, conversely, preserves, Möbius transformation
This is version 6 of Möbius transformation cross-ratio preservation theorem, born on 2003-04-28, modified 2007-05-02.
Object id is 4222, canonical name is MobiusTransformationCrossRatioPreservationTheorem.
Accessed 4661 times total.
Classification:
| AMS MSC: | 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions) |
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Pending Errata and Addenda
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