# 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 MobiusTransformationCrossratioPreservationTheorem 2013-03-22 13:35:50 2013-03-22 13:35:50 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Theorem msc 30E20 CrossRatio