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) = \begin{cases}

\dfrac a c & : c \ne 0 \\ \infty & : c = 0 \end{cases}$

Also see

 * Möbius Transformation is Bijection
 * Möbius Transformations form Group under Composition

Also known as
Möbius transformations are also known as complex bilinear transformations or fractional linear transformations.