M\"obius transformation cross-ratio preservation theoremMobiusTransformationCrossRatioPreservationTheorem2003-04-28 01:07:092007-05-02 00:48:39Theoremcross ratio% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA M\"obius transformation $f: z \mapsto w$ preserves the cross-ratios, i.e. \\
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\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)}
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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.