Möbius transformation

Möbius transformations are conformal, bilinear functions of the form

$t=f(z)={\frac {az+b}{cz+d}},$

where $t,z\in {\widehat {\mathbb {C} }}$, $a,b,c,d\in \mathbb {C}$, and $ad-bc\neq 0$. It is an automorphism of the Riemann sphere, and the set of all Möbius transformations on the Riemann sphere forms the automorphism group.