Möbius transformation cross-ratio preservation theorem

A Möbius transformation $f:z\mapsto w$ preserves the cross-ratios, i.e.

\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})}

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.

Title	Möbius transformation cross-ratio preservation theorem
Canonical name	MobiusTransformationCrossratioPreservationTheorem
Date of creation	2013-03-22 13:35:50
Last modified on	2013-03-22 13:35:50
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	9
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 30E20
Related topic	CrossRatio