Editing Möbius transformation (section)

=== Cross-ratio preservation ===
[[Cross-ratio]]s are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points <math>z_1, z_2, z_3, z_4</math> to four distinct points <math>w_1, w_2, w_3, w_4</math> respectively, then
<math display="block">\frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} =\frac{(w_1-w_3)(w_2-w_4)}{(w_2-w_3)(w_1-w_4)}. </math>

If one of the points <math>z_1, z_2, z_3, z_4</math> is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of <math>z_1, z_2, z_3, \infin</math> is
<math display="block">\frac{(z_1-z_3)}{(z_2-z_3)}.</math>

The cross ratio of four different points is real if and only if there is a line or a circle passing through them. This is another way to show that Möbius transformations preserve generalized circles.