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Möbius transformation cross-ratio preservation theorem
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(Theorem)
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A M\"obius transformation $f: z \mapsto w$ preserves the cross-ratios, i.e. \\
\begin{displaymath}
\frac{(w_1-w_2)(w_3-w_4)}{(w_1-w_4)(w_3-w_2)} = \frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_4)(z_3-z_2)}
\end{displaymath}
Conversely, given two quadruplets which have the same cross-ratio, there
exists a M\"obius 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 M\"obius transformation that takes $b$, $c$, $d$, to
$1$, $0$, $\infty$ respectively.
"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 5135 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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