Category:Möbius Transformation is Bijection

This category contains pages concerning Möbius Transformation is Bijection:

Let $a, b, c, d \in \C$ be complex numbers.

Let $f: \overline \C \to \overline \C$ be the Möbius transformation:

$\map f z = \begin {cases} \dfrac {a z + b} {c z + d} & : z \ne -\dfrac d c \\ \infty & : z = -\dfrac d c \\ \dfrac a c & : z = \infty \\ \infty & : z = \infty \text { and } c = 0 \end {cases}$

where $\overline \C$ denotes the extended complex plane.

Then:

$f: \overline \C \to \overline \C$ is a bijection

if and only if:

$a c - b d \ne 0$