# User Contributed Dictionary

# Extensive Definition

- Möbius transformations should not be confused with the Möbius transform or the Möbius function.

In geometry, a Möbius
transformation is a function:

- f(z) = \frac

where z, a, b, c, d are complex
numbers satisfying ad − bc ≠ 0. Möbius
transformations are named in honor of
August Ferdinand Möbius, although they are also called
homographic transformations or fractional linear
transformations.

## Overview

A Möbius transformation is a bijective conformal
map of the extended
complex plane (i.e. the complex
plane augmented by the point at
infinity):

- \widehat = \mathbb\cup\.

The set of all Möbius transformations forms a
group
under composition
called the Möbius group.

The Möbius group is the automorphism
group of the Riemann
sphere, sometimes denoted

- \mbox(\widehat\mathbb C).

A particularly important subgroup of the Möbius
group is the modular
group; it is central to the theory of many fractals, modular
forms, elliptic
curves and Pellian
equations.

In physics, the identity
component of the Lorentz
group acts on the celestial
sphere the same way that the Möbius group acts on the Riemann
sphere. In fact, these two groups are isomorphic. An observer
who accelerates to relativistic velocities will see the pattern of
constellations as seen near the Earth continuously transform
according to infinitesimal Möbius transformations. This observation
is often taken as the starting point of twistor
theory.

## Definition

The general form of a Möbius transformation is
given by

- z \mapsto \frac

The set of all Möbius transformations forms a
group
under composition.
This group can be given the structure of a complex
manifold in such a way that composition and inversion are
holomorphic
maps. The Möbius group is then a complex
Lie group. The Möbius group is usually denoted
\mbox(\widehat\mathbb C) as it is the automorphism
group of the Riemann sphere.

## Decomposition and elementary properties

A Möbius transformation is equivalent to a
sequence of simpler transformations. Let:

- f_1(z)= z+d/c \! (translation)
- f_2(z)= 1/z \! (inversion and reflection)
- f_3(z)= (- (ad-bc)/c^2) \cdot z \! (dilation and rotation)
- f_4(z)= z+a/c \! (translation)

then these functions can be composed
on each other, giving

- f_4\circ f_3\circ f_2\circ f_1 (z)= \frac.\!

This decomposition makes many properties of the
Möbius transform obvious.

For example, the preservation of angles is
reduced to proving the angle preservation property of circle
inversion, since all other transformation are dilatations or
isometries, which
trivially preserve angles.

The existence of an inverse Möbius transformation
function and its explicit formula is easily derived by a
composition of the inverse function of the simpler transformations.
That is, define functions g_1, g_2, g_3, g_4 such that g_i is the
inverse of f_i, then composition g_1\circ g_2\circ g_3\circ g_4 (z)
would be the explicit expression for the inverse Möbius
transformation:

- \frac

From this decomposition, we also see that Möbius
transformation carries over all non-trivial properties of circle
inversion. Namely, that circles are mapped to circles, and
angles are preserved. Also, because of the circle inversion, is
carried over the convenience of defining Möbius transformation over
a plane with a point at infinity, which makes statements and
concepts of Möbius transformation's properties simpler.

For another example, look at f_3. If ad-bc= 0,
then the transformation collapses to the point 0, then f_4 moves to
a/c. Collapsing to a point is not an interesting transformation,
thus we require in the definition of Möbius transformation that
ad-bc \ne 0.

### Preservation of angles and circles

As seen from the above decomposition, Möbius
transformation contains this transformation 1/z, called complex
inversion. Geometrically, a complex inversion is a circle inversion
followed by a reflection around the x-axis.

In circle inversion, circles are mapped to
circles (here, lines are considered as circles with infinite
radius), and angles are preserved. See circle
inversion for various properties and proofs.

### Cross-ratio preservation

The cross-ratio preservation theorem states that
the cross-ratio
\frac = \frac

is invariant under a Möbius transformation that
maps from z to w.

The action of the Möbius group on the Riemann
sphere is sharply 3-transitive in the sense that there is a unique
Möbius transformation which takes any three distinct points on the
Riemann sphere to any other set of three distinct points. See the
section below on
specifying a transformation by three points.

## Projective matrix representations

The transformation

- f(z) = \frac

- \mathfrak = \begin a & b \\ c & d \end.

The usefulness of this representation is that the
composition of two Möbius transformations corresponds precisely to
matrix
multiplication of the corresponding matrices. That is, if we
define a map

- \pi\colon \mbox(2,\mathbb C) \to \mbox(\widehat\mathbb C)

The map \pi is not an isomorphism, since it maps
any scalar multiple of \mathfrak to the same transformation. The
kernel
of this homomorphism is then the set of all scalar matrices kI,
which is the center of GL(2,C). The quotient
group GL(2,C)/Z(GL(2,C)) is called the projective
linear group and is usually denoted PGL(2,C). By the first
isomorphism theorem of group theory we conclude that the Möbius
group is isomorphic to PGL(2,C). Since Z(GL(2,C)) is the kernel of
the group action given by GL(2,C) acting on itself by conjugation,
PGL(2, C) is isomorphic to the inner
automorphism group of GL(2,C). Moreover, the natural action of
PGL(2,C) on the complex
projective line CP1 is exactly the natural action of the Möbius
group on the Riemann sphere when the sphere and the projective line
are identified as follows:

- [z_1 : z_2]\leftrightarrow z_1/z_2.

If one normalizes \mathfrak so that the
determinant is equal to one, the map \pi restricts to a surjective
map from the special
linear group SL(2,C) to the Möbius group. The Möbius group is
therefore also isomorphic to PSL(2,C). We then have the following
isomorphisms:

- \mbox(\widehat\mathbb C) \cong \mbox(2,\mathbb C) \cong \mbox(2,\mathbb C).

Note that there are precisely two matrices with
unit determinant which can be used to represent any given Möbius
transformation. That is, SL(2,C) is a double cover
of PSL(2,C). Since SL(2,C) is simply-connected
it is the universal
cover of the Möbius group. The fundamental
group of the Möbius group is then Z2.

## Classification

Möbius transformations are commonly classified
into four types, parabolic, elliptic, hyperbolic and loxodromic
(actually hyperbolic is a special case of loxodromic). The
classification has both algebraic and geometric significance.
Geometrically, the different types result in different
transformations of the complex plane, as the figures below
illustrate. These types can be distinguished by looking at the
trace
\mbox\,\mathfrak=a+d. Note that the trace is invariant under
conjugation,
that is,

- \mbox\,\mathfrak^ = \mbox\,\mathfrak,

and so every member of a conjugacy class will
have the same trace. Every Möbius transformation can be written
such that its representing matrix \mathfrak has determinant one (by
multiplying the entries with a suitable scalar). Two Möbius
transformations \mathfrak, \mathfrak' (both not equal to the
identity transform) with \det \mathfrak=\det\mathfrak'=1 are
conjugate if and only if \mbox^2\,\mathfrak= \mbox^2\,\mathfrak'
.

In the following discussion we will always assume
that the representing matrix \mathfrak is normalized such that
\det=ad-bc=1 .

### Parabolic transforms

The transform is said to be parabolic if- \mbox^2\mathfrak = (a+d)^2 = 4.

A transform is parabolic if and only if it has
one fixed point in the compactified
complex plane \widehat=\mathbb\cup\. It is parabolic if and
only if it is conjugate to

- \begin 1 & 1 \\ 0 & 1 \end.

The subgroup
consisting of all parabolic transforms of this form:

- \begin 1 & b \\ 0 & 1 \end

is an example of a Borel
subgroup, which generalizes the idea to higher
dimensions.

All other non-identity transformations have two
fixed points. All non-parabolic (non-identity) transforms are
conjugate to

- \begin \lambda & 0 \\ 0 & \lambda^ \end

with \lambda not equal to 0, 1 or −1.
The square k=\lambda^2 is called the characteristic constant or
multiplier of the transformation.

### Elliptic transforms

The transform is said to be elliptic if- 0 \le \mbox^2\mathfrak

A transform is elliptic if and only if
|\lambda|=1. Writing \lambda=e^, an elliptic transform is conjugate
to

- \begin \cos\alpha & \sin\alpha \\

with \alpha real. Note that for any \mathfrak,
the characteristic constant of \mathfrak^n is k^n. Thus, the only
Möbius transformations of finite order
are the elliptic transformations, and these only when λ
is a root of
unity; equivalently, when α is a rational multiple of pi.

### Hyperbolic transforms

The transform is said to be hyperbolic if

- \mbox^2\mathfrak > 4.\,

A transform is hyperbolic if and only if
λ is real and
positive.

### Loxodromic transforms

The transform is said to be loxodromic if \mbox^2\mathfrak is not in the closed interval of [0, 4]. Hyperbolic transforms are thus a special case of loxodromic transformations. A transformation is loxodromic if and only if |\lambda|\ne 1. Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below.## Fixed points

Every non-identity Möbius transformation has two
fixed
points \gamma_1, \gamma_2 on the Riemann sphere. Note that the
fixed points are counted here with multiplicity; for parabolic
transformations, the fixed points coincide. Either or both of these
fixed points may be the point at infinity.

The fixed points of the transformation

- f(z) = \frac

- \gamma_ = \frac = \frac.

When c = 0, one of the fixed points is at
infinity; the other is given by

- \gamma=-\frac.

The transformation will be a simple
transformation composed of translations, rotations, and dilations:

- z \mapsto \alpha z + \beta.\,

If c = 0 and a = d, then both fixed points are at
infinity, and the Möbius transformation corresponds to a pure
translation: z \mapsto z + \beta.

### Normal form

Möbius transformations are also sometimes written
in terms of their fixed points in so-called normal form. We first
treat the non-parabolic case, for which there are two distinct
fixed points.

Non-parabolic case:

Every non-parabolic transformation is conjugate
to a dilation,
i.e. a transformation of the form

- z \mapsto k z

- g(z) = \frac

If f has distinct fixed points (\gamma_1,
\gamma_2) then the transformation gfg^ has fixed points at 0 and
∞ and is therefore a dilation: gfg^(z) = kz. The fixed
point equation for the transformation f can then be written

- \frac = k \frac.

Solving for f gives (in matrix form):

- \mathfrak(k; \gamma_1, \gamma_2) =

or, if one of the fixed points is at
infinity:

- \mathfrak(k; \gamma, \infty) =

From the above expressions one can calculate the
derivatives of f at the fixed points:

- f'(\gamma_1)= k\, and f'(\gamma_2)= 1/k.\,

Observe that, given an ordering of the fixed
points, we can distinguish one of the multipliers (k) of f as the
characteristic constant of f. Reversing the order of the fixed
points is equivalent to taking the inverse multiplier for the
characteristic constant:

- \mathfrak(k; \gamma_1, \gamma_2) = \mathfrak(1/k; \gamma_2, \gamma_1).

For loxodromic transformations, whenever
|k|>1, one says that \gamma_1 is the repulsive fixed point, and
\gamma_2 is the attractive fixed point. For |k|, the roles are
reversed.

Parabolic case:

In the parabolic case there is only one fixed
point \gamma. The transformation sending that point to ∞
is

- g(z) = \frac

or the identity if \gamma is already at infinity.
The transformation gfg^ fixes infinity and is therefore a
translation:

- gfg^(z) = z + \beta\,.

Here, β is called the translation
length. The fixed point formula for a parabolic transformation is
then

- \frac = \frac + \beta.

Solving for f (in matrix form) gives

- \mathfrak(\beta; \gamma) =

or, if \gamma = \infty:

- \mathfrak(\beta; \infty) =

Note that \beta is not the characteristic
constant of f, which is always 1 for a parabolic transformation.
From the above expressions one can calculate:

- f'(\gamma) = 1.\,

## Geometric interpretation of the characteristic constant

The following picture depicts (after
stereographic transformation from the sphere to the plane) the two
fixed points of a Möbius transformation in the non-parabolic
case:

The characteristic constant can be expressed in
terms of its logarithm:

- e^ = k \;

### Elliptic transformations

If \rho = 0, then the fixed points are neither
attractive nor repulsive but indifferent, and the transformation is
said to be elliptical. These transformations tend to move all
points in circles around the two fixed points. If one of the fixed
points is at infinity, this is equivalent to doing an affine
rotation around a point.

If we take the one-parameter
subgroup generated by any elliptic Möbius transformation, we
obtain a continuous transformation, such that every transformation
in the subgroup fixes the same two points. All other points flow
along a family of circles which is nested between the two fixed
points on the Riemann sphere. In general, the two fixed points can
be any two distinct points.

This has an important physical interpretation.
Imagine that some observer rotates with constant angular velocity
about some axis. Then we can take the two fixed points to be the
North and South poles of the celestial sphere. The appearance of
the night sky is now transformed continuously in exactly the manner
described by the one-parameter subgroup of elliptic transformations
sharing the fixed points 0, \infty, and with the number \alpha
corresponding to the constant angular velocity of our
observer.

Here are some figures illustrating the effect of
an elliptic Möbius transformation on the Riemann sphere (after
stereographic projection to the plane):

## Iterating a transformation

If a transformation \mathfrak has fixed points
\gamma_1, \gamma_2, and characteristic constant k, then \mathfrak'
= \mathfrak^n will have \gamma_1' = \gamma_1, \gamma_2' = \gamma_2,
k' = k^n.

This can be used to iterate a
transformation, or to animate one by breaking it up into
steps.

These images show three points (red, blue and
black) continuously iterated under transformations with various
characteristic constants.