Definition:Möbius Transformation

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Definition

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

$\map f z = \dfrac {a z + b} {c z + d}$

where:

$\overline \C$ denotes the extended complex plane
$a, b, c, d \in \C$ such that $a d - b c \ne 0$


We define:

$\map f {-\dfrac d c} = \infty$

if $c \ne 0$, and:

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


Real Numbers

The Möbius transformation is often seen restricted to the Alexandroff extension $\R^*$ of the real number line:


A Möbius transformation is a mapping $f: \R^* \to \R^*$ of the form:

$\map f x = \dfrac {a x + b} {c x + d}$

where:

$\R^*$ denotes the Alexandroff extension of the real number line
$a, b, c, d \in \R$ such that $a d - b c \ne 0$


We define:

$\map f {-\dfrac d c} = \infty$

if $c \ne 0$, and:

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


Also defined as

Some sources, when defining a Möbius transformation, do not insist that $a d - b c \ne 0$.

However, it needs to be pointed out that when $a d - b c = 0$, the resulting mapping is not a bijection.


Also known as

Möbius transformations are also known as:

bilinear functions
bilinear transformations
complex bilinear transformations (when on the complex plane)
fractional linear transformations.

The term bilinear arises from the fact that both the numerator and denominator are linear functions.


Also see

  • Results about Möbius transformations can be found here.


Do not confuse this with the Definition:Möbius Function.


Source of Name

This entry was named for August Ferdinand Möbius.


Sources

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