Definition:Möbius Transformation

Definition
A Möbius transformation is a mapping $f: \overline \C \to \overline \C$ of the form:


 * $f \left({z}\right) = \dfrac {a z + b} {c z + d}$

where $a, b, c, d \in \C$ and $a d - b c \ne 0$.

We define:


 * $f \left({- \dfrac d c }\right) = \infty$

if $c \ne 0$, and:


 * $f \left({\infty}\right) =

\left\{ \begin{array}{ll} \dfrac a c & \mbox{if } c \ne 0 \\ \infty & \mbox{if } c = 0 \end{array} \right.$

Also see

 * Möbius Transformations are Bijections
 * Möbius Transformations form Group under Composition