Möbius transformations should not be confused with the Möbius transform or the Möbius function. The Möbius transform should not be confused with Möbius transformations In Mathematics, the Möbius transform Tf of For the rational functions defined on the complex numbers see Möbius transformation.

In geometry, a Möbius transformation is a rational function of the form:

$f(z) = \frac{a z + b}{c z + d}$

where z, a, b, c, d are complex numbers satisfying adbc ≠ 0. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Möbius transformations are named in honor of August Ferdinand Möbius, although they are also called homographic transformations or fractional linear transformations. August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and

## Overview

A Möbius transformation is a bijective conformal map of the extended complex plane (i. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that e. the complex plane augmented by the point at infinity):

$\widehat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}.$

The set of all Möbius transformations forms a group under composition called the Möbius group. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis The point at infinity, also called ideal point, is a point which when added to the real Number line yields a Closed curve called the Real In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a composite function represents the application of one function to the results of another

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

$\mbox{Aut}(\widehat\mathbb C).$

Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In As such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Fuchsian group is a particular type of group of isometries of the Hyperbolic plane. In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformal Möbius transformations are also closely related to isometries of hyperbolic 3-manifolds. For the Mechanical engineering and Architecture usage see Isometric projection. A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant Sectional curvature -1

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 Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x

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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Astronomy and Navigation, the celestial sphere is an imaginary rotating Sphere of "gigantic Radius " In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that 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. The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the Geometric objects of the four dimensional space-time

## Definition

The general form of a Möbius transformation is given by

$z \mapsto \frac{a z + b}{c z + d}$

where a, b, c, d are any complex numbers satisfying adbc ≠ 0. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted (If ad = bc the rational function defined above is a constant. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In ) This definition can be extended to the whole Riemann sphere (the complex plane plus the point at infinity). In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis The point at infinity, also called ideal point, is a point which when added to the real Number line yields a Closed curve called the Real

The set of all Möbius transformations forms a group under composition. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a composite function represents the application of one function to the results of another This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane The Möbius group is then a complex Lie group. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The Möbius group is usually denoted $\mbox{Aut}(\widehat\mathbb C)$ as it is the automorphism group of the Riemann sphere. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

## 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

$f_4\circ f_3\circ f_2\circ f_1 (z)= \frac{az+b}{cz+d}.\!$

This decomposition makes many properties of the Möbius transform obvious. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions. In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. In Mathematics, a dilation is a function &fnof from a Metric space into itself that satisfies the identity d(f(xf(y=rd(xy \ In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed In Mathematics, a composite function represents the application of one function to the results of another

For example, the preservation of angles is reduced to proving the angle preservation property of circle inversion, since all other transformation are dilations or isometries, which trivially preserve angles. In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions. For the Mechanical engineering and Architecture usage see Isometric projection.

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 g1,g2,g3,g4 such that gi is the inverse of fi, then composition $g_1\circ g_2\circ g_3\circ g_4 (z)$ would be the explicit expression for the inverse Möbius transformation:

$\frac{dz-b}{-cz+a}$

From this decomposition, we also see that Möbius transformation carries over all non-trivial properties of circle inversion. In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions. 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 f3. If adbc = 0, then the transformation collapses to the point 0, then f4 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. In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions.

### Cross-ratio preservation

The cross-ratio preservation theorem states that the cross-ratio

$\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} =\frac{(w_1-w_3)(w_2-w_4)}{(w_1-w_4)(w_2-w_3)}$

is invariant under a Möbius transformation that maps from z to w. In Mathematics, the cross-ratio of a set of four distinct points on the Complex plane is given by (z_1z_2z_3z_4 = \frac{(z_1-z_3(z_2-z_4}{(z_1-z_4(z_2-z_3}

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

$f(z) = \frac{a z + b}{c z + d}$

is determined by the matrix of complex numbers

$\mathfrak{H} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.$

The condition adbc ≠ 0 is equivalent to the condition that the determinant of above matrix be nonzero (i. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n e. the matrix should be non-singular). In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- Note that any matrix obtained by multiplying $\mathfrak H$ by a complex scalar λ determines the same transformation, so the transformation does not determine the matrix. The freedom of choosing the matrix for a given transformation can be restricted by requiring that the determinant of $\mathfrak H$ be equal to 1; then $\mathfrak H$ will be unique up to sign. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose

The usefulness of using matrices to describe Möbius transformations stems from the fact that matrix multiplication gives rise to composition of the corresponding Möbius transformations. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In other words, the map

$\pi\colon \mbox{GL}(2,\mathbb C) \to \mbox{Aut}(\widehat\mathbb C)$

from the general linear group GL(2,C) to the Möbius group, which sends the matrix $\mathfrak{H}$ to the transformation f is a group homomorphism. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function This map is called a projective representation of GL(2,C) for reasons explained below. In the mathematical field of Representation theory, a projective representation of a group G on a Vector space V over a

The map π is not an isomorphism, since it maps any scalar multiple of $\mathfrak{H}$ to the same transformation. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The kernel of this homomorphism is then the set of all scalar multiples of the identity matrix, which is the center Z(GL(2,C) of GL(2,C). In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism The quotient group GL(2,C)/Z(GL(2,C)) is called the projective linear group and is usually denoted PGL(2,C). In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group By the first isomorphism theorem of group theory we conclude that the Möbius group is isomorphic to PGL(2,C). In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural 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). In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G 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, where the projective line CP1 and the Riemann sphere are identified as follows:

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

Here [z1:z2] are homogeneous coordinates on CP1; the point [1:0] corresposnds to the point ∞ of the Riemann sphere. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations

If one restricts $\mathfrak{H}$ to matrices of determinant one, the map π restricts to a surjective map from the special linear group SL(2,C) to the Möbius group; in the restricted setting the kernel (formed by plus and minus the identity) is still the center of the group. In Mathematics, the special linear group of degree n over a field F is the set of n × n matrices with The Möbius group is therefore also isomorphic to PSL(2,C). We then have the following isomorphisms:

$\mbox{Aut}(\widehat\mathbb C) \cong \mbox{PGL}(2,\mathbb C) \cong \mbox{PSL}(2,\mathbb C).$

From the last identification we see that the Möbius group is a 3-dimensional complex Lie group (or a 6-dimensional real Lie group).

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). In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism Since SL(2,C) is simply-connected it is the universal cover of the Möbius group. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism Therefore the fundamental group of the Möbius group is Z2. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

## 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 $\mbox{tr}\,\mathfrak{H}=a+d$. In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal Note that the trace is invariant under conjugation, that is,

$\mbox{tr}\,\mathfrak{GHG}^{-1} = \mbox{tr}\,\mathfrak{H},$

and so every member of a conjugacy class will have the same trace. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class Every Möbius transformation can be written such that its representing matrix $\mathfrak{H}$ has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations $\mathfrak{H}, \mathfrak{H}'$ (both not equal to the identity transform) with $\det \mathfrak{H}=\det\mathfrak{H}'=1$ are conjugate if and only if $\mbox{tr}^2\,\mathfrak{H}= \mbox{tr}^2\,\mathfrak{H}'$.

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

### Parabolic transforms

A non-identity Möbius transformation defined by a matrix $\mathfrak{H}$ of determinant one is said to be parabolic if

$\mbox{tr}^2\mathfrak{H} = (a+d)^2 = 4$

(so the trace is plus or minus 2; either can occur for a given transformation since $\mathfrak{H}$ is determined only up to sign). In fact one of the choices for $\mathfrak{H}$ has the same characteristic polynomial X2−2X+1 as the identity matrix, and is therefore unipotent. In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. In Mathematics, a unipotent element r of a ring R is one such that r  &minus 1 is a Nilpotent element, in A Möbius transform is parabolic if and only if it has exactly one fixed point in the compactified complex plane $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$, which happens if and only if it can be defined by a matrix conjugate to

$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}.$

The set of all parabolic Möbius transformations with a given fixed point in $\widehat{\mathbb{C}}$, together with the identity, forms a subgroup isomorphic to the group of matrices

$\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix}$

for $b\in\mathbb C$; this is an example of a Borel subgroup (of the Möbius group, or of SL(2,C) for the matrix group; the notion is defined for any reductive Lie group). In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In the theory of Algebraic groups, a Borel subgroup of an Algebraic group G is a maximal Zariski closed and connected solvable In Mathematics, a reductive group is an Algebraic group G such that the Unipotent radical of the Identity component of G

All other non-identity transformations have two fixed points. All non-parabolic (non-identity) transformations are defined by a matrix conjugate to

$\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}$

with λ not equal to 0, 1 or −1. The square k = λ2 is called the characteristic constant or multiplier of the transformation.

### Elliptic transforms

The transformation is said to be elliptic if it can be represented by a matrix $\mathfrak H$ whose trace is real with

$0 \le \mbox{tr}^2\mathfrak{H} < 4.\,$

A transform is elliptic if and only if | λ | = 1. In Mathematics, the real numbers may be described informally in several different ways Writing λ = eiα, an elliptic transform is conjugate to

$\begin{pmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{pmatrix}$

with α real. Note that for any $\mathfrak{H}$ with characteristic constant k, the characteristic constant of $\mathfrak{H}^n$ is kn. 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 π. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems

### Hyperbolic transforms

The transform is said to be hyperbolic if it can be represented by a matrix $\mathfrak H$ whose trace is real with

$\mbox{tr}^2\mathfrak{H} > 4.\,$

A transform is hyperbolic if and only if λ is real and positive. In Mathematics, the real numbers may be described informally in several different ways

### Loxodromic transforms

The transform is said to be loxodromic if $\mbox{tr}^2\mathfrak{H}$ 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. Navigation is the process of reading and controlling the movement of a craft or vehicle from one place to another See also Great circle Small circle See also Great circle Small circle In Navigation, a bearing is the direction one object is from another object See the geometric figures below.

TransformationTrace squaredMultipliersClass representative
Elliptic$0 \leq \sigma < 4$| k | = 1
$k = e^{\pm i\theta} \neq 1$
$\begin{pmatrix}e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2}\end{pmatrix}$$z\mapsto e^{i\theta}z$
Parabolicσ = 4k = 1$\begin{pmatrix}1 & a \\ 0 & 1\end{pmatrix}$$z\mapsto z + a$
Hyperbolic$4 < \sigma < \infty$$k \in \mathbb R^{+}$
$k = e^{\pm \theta} \neq 1$
$\begin{pmatrix}e^{\theta/2} & 0 \\ 0 & e^{-\theta/2}\end{pmatrix}$$z \mapsto e^\theta z$
Loxodromic$\sigma\in\mathbb C, \sigma \not\in [0,4]$$|k| \neq 1$
k = λ2− 2
$\begin{pmatrix}\lambda & 0 \\ 0 & \lambda^{-1}\end{pmatrix}$$z \mapsto k z$

## Fixed points

Every non-identity Möbius transformation has two fixed points γ12 on the Riemann sphere. In Mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function 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

$f(z) = \frac{az + b}{cz + d}$

are obtained by solving the fixed point equation f(γ) = γ. For $c\ne 0$, this has two roots (proof):

$\gamma_{1,2} = \frac{(a - d) \pm \sqrt{(a-d)^2 + 4bc}}{2c} = \frac{(a - d) \pm \sqrt{(a+d)^2 - 4(ad-bc)}}{2c}.$

Note that for parabolic transformations, which satisfy (a + d)2 = 4(adbc), the fixed points coincide. Fixed Points The article claims that for c\ne 0 the two roots are \gamma = \frac{(a - d \pm \sqrt{(a - d^2 + 4 c b}}{2 c} of the

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

$\gamma=-\frac{b}{a-d}.$

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: $z \mapsto z + \beta$. Translation is the interpreting of the meaning of a text and the subsequent production of an equivalent text likewise called a " translation A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a dilation is a function &fnof from a Metric space into itself that satisfies the identity d(f(xf(y=rd(xy \

### 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. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class In Mathematics, a dilation is a function &fnof from a Metric space into itself that satisfies the identity d(f(xf(y=rd(xy \ e. a transformation of the form

$z \mapsto k z$

with fixed points at 0 and ∞. To see this define a map

$g(z) = \frac{z - \gamma_1}{z - \gamma_2}$

which sends the points 12) to $(0,\infty)$. Here we assume that both γ1 and γ2 are finite. If one of them is already at infinity then g can be modified so as to fix infinity and send the other point to 0.

If f has distinct fixed points 12) then the transformation gfg − 1 has fixed points at 0 and ∞ and is therefore a dilation: gfg − 1(z) = kz. The fixed point equation for the transformation f can then be written

$\frac{f(z)-\gamma_1}{f(z)-\gamma_2} = k \frac{z-\gamma_1}{z-\gamma_2}.$

Solving for f gives (in matrix form):

$\mathfrak{H}(k; \gamma_1, \gamma_2) =\begin{pmatrix} \gamma_1 - k\gamma_2 & (k - 1) \gamma_1\gamma_2 \\ 1 - k & k\gamma_1 - \gamma_2\end{pmatrix}$

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

$\mathfrak{H}(k; \gamma, \infty) =\begin{pmatrix} k & (1 - k) \gamma \\ 0 & 1\end{pmatrix}.$

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:

$\mathfrak{H}(k; \gamma_1, \gamma_2) = \mathfrak{H}(1/k; \gamma_2, \gamma_1).$

For loxodromic transformations, whenever | k | > 1, one says that γ1 is the repulsive fixed point, and γ2 is the attractive fixed point. For | k | < 1, the roles are reversed.

Parabolic case:

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

$g(z) = \frac{1}{z - \gamma}$

or the identity if γ is already at infinity. The transformation gfg − 1 fixes infinity and is therefore a translation:

$gfg^{-1}(z) = z + \beta\,.$

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

$\frac{1}{f(z)-\gamma} = \frac{1}{z-\gamma} + \beta.$

Solving for f (in matrix form) gives

$\mathfrak{H}(\beta; \gamma) =\begin{pmatrix} 1+\gamma\beta & - \beta \gamma^2 \\ \beta & 1-\gamma \beta \end{pmatrix}$

or, if $\gamma = \infty$:

$\mathfrak{H}(\beta; \infty) =\begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix}$

Note that β 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:

$e^{\rho + \alpha i} = k \;$

When expressed in this way, the real number ρ becomes an expansion factor. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational It indicates how repulsive the fixed point γ1 is, and how attractive γ2 is. The real number α is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about γ1 and clockwise about γ2.

### Elliptic transformations

If ρ = 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. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R 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 α 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):

These pictures illustrate the effect of a single Möbius transformation. The one-parameter subgroup which it generates continuously moves points along the family of circular arcs suggested by the pictures.

### Hyperbolic transformations

If α is zero (or a multiple of ), then the transformation is said to be hyperbolic. These transformations tend to move points along circular paths from one fixed point toward the other.

If we take the one-parameter subgroup generated by any hyperbolic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the same two points. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R All other points flow along a certain family of circular arcs away from the first fixed point and toward the second fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.

This too has an important physical interpretation. Imagine that an observer accelerates (with constant magnitude of acceleration) in the direction of the North pole on his celestial sphere. Then the appearance of the night sky is transformed in exactly the manner described by the one-parameter subgroup of hyperbolic transformations sharing the fixed points $0,\infty$, with the real number ρ corresponding to the magnitude of his acceleration vector. The stars seem to move along longitudes, away from the South pole toward the North pole. (The longitudes appear as circular arcs under stereographic projection from the sphere to the plane).

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

It is not surprising that these pictures look very much like the field lines of bar magnets, since the circular flow lines subtend a constant angle between the two fixed points.

### Loxodromic transformations

If both ρ and α are nonzero, then the transformation is said to be loxodromic. These transformations tend to move all points in S-shaped paths from one fixed point to the other.

The word "loxodrome" is from the Greek: "λοξος (loxos), slanting + δρόμος (dromos), course". See also Great circle Small circle When sailing on a constant bearing - if you maintain a heading of (say) north-east, you will eventually wind up sailing around the north pole in a logarithmic spiral. Sailing is the art of controlling a Sailing vessel. By changing the Rigging, Rudder and dagger or centre board a Sailor manages the force The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is subject to the caveats explained below defined as the point in the northern Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta On the mercator projection such a course is a straight line, as the north and south poles project to infinity. The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569 The angle that the loxodrome subtends relative to the lines of longitude (i. e. its slope, the "tightness" of the spiral) is the argument of k. Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles. But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes.

If we take the one-parameter subgroup generated by any loxodromic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the same two points. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R All other points flow along a certain family of curves, away from the first fixed point and toward the second fixed point. Unlike the hyperbolic case, these curves are not circular arcs, but certain curves which under stereographic projection from the sphere to the plane appear as spiral curves which twist counterclockwise infinitely often around one fixed point and twist clockwise infinitely often around the other fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.

You can probably guess the physical interpretation in the case when the two fixed points are $0, \infty$: an observer who is both rotating (with constant angular velocity) about some axis and boosting (with constant magnitude acceleration) along the same axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points $0, \infty$, and with ρ,α determined respectively by the magnitude of acceleration and angular velocity.

### Stereographic projection

These images show Möbius transformations stereographically projected onto the Riemann sphere. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location.

 Elliptic Hyperbolic Loxodromic One fixed point at infinity Full size Full size Full size Fixed points diametrically opposite Full size Full size Full size Fixed points in an arbitrary location Full size Full size Full size

## Iterating a transformation

If a transformation $\mathfrak{H}$ has fixed points γ12, and characteristic constant k, then $\mathfrak{H}' = \mathfrak{H}^n$ will have γ1' = γ1, γ2' = γ2, k' = kn.

This can be used to iterate a transformation, or to animate one by breaking it up into steps. Iteration means the act of repeating Mathematics Iteration in mathematics may refer to the process of iterating a function, or to the techniques used

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

And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms. Note that in the elliptical and loxodromic images, the α value is 1/10 .

## Poles of the transformation

The point

$z_\infty = - \frac{d}{c}$

is called the pole of $\mathfrak{H}$; it is that point which is transformed to the point at infinity under $\mathfrak{H}$.

The inverse pole

$Z_\infty = \frac{a}{c}$

is that point to which the point at infinity is transformed. The point midway between the two poles is always the same as the point midway between the two fixed points:

$\gamma_1 + \gamma_2 = z_\infty + Z_\infty.$

These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation.

A transform $\mathfrak{H}$ can be specified with two fixed points γ12 and the pole $z_\infty$.

$\mathfrak{H} =\begin{pmatrix} Z_\infty & - \gamma_1 \gamma_2 \\ 1 & - z_\infty\end{pmatrix}, \;\; Z_\infty = \gamma_1 + \gamma_2 - z_\infty.$

This allows us to derive a formula for conversion between k and $z_\infty$ given γ12:

$z_\infty = \frac{k \gamma_1 - \gamma_2}{1 - k}$
$k = \frac{\gamma_2 - z_\infty}{\gamma_1 - z_\infty}= \frac{Z_\infty - \gamma_1}{Z_\infty - \gamma_2}= \frac {a - c \gamma_1}{a - c \gamma_2},$

which reduces down to

$k = \frac{(a + d) + \sqrt {(a - d)^2 + 4 b c}}{(a + d) - \sqrt {(a - d)^2 + 4 b c}}.$

The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios $\lambda_1\over \lambda_2$ of the matrix

$\mathfrak{H} =\begin{pmatrix} a & b \\ c & d\end{pmatrix}$

representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Its characteristic polynomial is equal to

$\mbox{det} (\lambda I_2- \mathfrak{H})=\lambda^2-\mbox{tr} \mathfrak{H}\,\lambda+\mbox{det} \mathfrak{H}=\lambda^2-(a+d)\lambda+(ad-bc)$

which has roots

$\lambda_{i}=\frac{(a + d) \pm \sqrt {(a - d)^2 + 4 b c}}{2}=\frac{(a + d) \pm \sqrt {(a + d)^2 - 4(ad-b c)}}{2} \ .$

## Specifying a transformation by three points

### Direct approach

Any set of three points

$Z_1 = \mathfrak{H}(z_1), \;\; Z_2 = \mathfrak{H}(z_2), \;\; Z_3 = \mathfrak{H}(z_3)$

uniquely defines a transformation $\mathfrak{H}$. To calculate this out, it is handy to make use of a transformation that is able to map three points onto (0,0), (1, 0) and the point at infinity.

$\mathfrak{H}_1 = \begin{pmatrix} \frac{z_2 - z_3}{z_2 - z_1} & -z_1 \frac{z_2 - z_3}{z_2 - z_1} \\ 1 & -z_3 \end{pmatrix}, \;\;\mathfrak{H}_2 = \begin{pmatrix} \frac{Z_2 - Z_3}{Z_2 - Z_1} & -Z_1 \frac{Z_2 - Z_3}{Z_2 - Z_1} \\ 1 & -Z_3 \end{pmatrix}$

One can get rid of the infinities by multiplying out by z2z1 and Z2Z1 as previously noted.

$\mathfrak{H}_1 = \begin{pmatrix} z_2 - z_3 & z_1 z_3 - z_1 z_2 \\ z_2 - z_1 & z_1 z_3 - z_3 z_2 \end{pmatrix}, \;\;\mathfrak{H}_2 = \begin{pmatrix} Z_2 - Z_3 & Z_1 Z_3 - Z_1 Z_2 \\ Z_2 - Z_1 & Z_1 Z_3 - Z_3 Z_2 \end{pmatrix}$

The matrix $\mathfrak{H}$ to map z1,2,3 onto Z1,2,3 then becomes

$\mathfrak{H} = \mathfrak{H}_2^{-1} \mathfrak{H}_1$

You can multiply this out, if you want, but if you are writing code then it's easier to use temporary variables for the middle terms.

### Explicit determinant formula

The problem of constructing a Möbius transformation $\mathfrak{H}(z)$ mapping a triple (z1,z2,z3) to another triple (w1,w2,w3) is equivalent to finding the equation of a standard hyperbola

$\, c wz -az+dw -b=0$

in the (z,w)-plane passing through the points (zi,wi). In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions An explicit equation can be found by evaluating the determinant

$\det \begin{pmatrix} zw & z & w & 1 \\ z_1w_1 & z_1 & w_1 & 1 \\ z_2w_2 & z_2 & w_2 & 1 \\ z_3w_3 & z_3 & w_3 & 1 \end{pmatrix} \,$

by means of a Laplace expansion along the first row. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Linear algebra, the Laplace expansion of the Determinant ofan n  ×  n square matrix B expresses the determinant This results in the determinant formulae

$a=\det \begin{pmatrix} z_1w_1 & w_1 & 1 \\ z_2w_2 & w_2 & 1 \\ z_3w_3 & w_3 & 1 \end{pmatrix} \,$
$b=\det \begin{pmatrix} z_1w_1 & z_1 & w_1 \\ z_2w_2 & z_2 & w_2 \\ z_3w_3 & z_3 & w_3 \end{pmatrix} \,$
$c=\det \begin{pmatrix} z_1 & w_1 & 1 \\ z_2 & w_2 & 1 \\ z_3 & w_3 & 1 \end{pmatrix} \,$
$d=\det \begin{pmatrix} z_1w_1 & z_1 & 1 \\ z_2w_2 & z_2 & 1 \\ z_3w_3 & z_3 & 1 \end{pmatrix}$

for the coefficients a,b,c,d of the representing matrix $\, \mathfrak{H} =\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. The constructed matrix $\mathfrak{H}$ has determinant equal to (z1z2)(z1z3)(z2z3)(w1w2)(w1w3)(w2w3) which does not vanish if the zi resp. wi are pairwise different thus the Möbius transformation is well-defined.

Remark: A similar determinant (with wz replaced by w2 + z2) leads to the equation of a circle through three different (non-collinear) points in the plane. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

### Alternate method using cross-ratios of point quadruples

This construction exploits the fact (mentioned in the first section) that the cross-ratio

$\mbox{cr}(z_1,z_2,z_3,z_4)={{(z_1-z_3)(z_2-z_4)}\over{(z_1-z_4)(z_2-z_3)}}$

is invariant under a Möbius transformation mapping a quadruple (z1,z2,z3,z4) to (w1,w2,w3,w4) via $w_i=\mathfrak{H}z_i$. In Mathematics, the cross-ratio of a set of four distinct points on the Complex plane is given by (z_1z_2z_3z_4 = \frac{(z_1-z_3(z_2-z_4}{(z_1-z_4(z_2-z_3} If $\mathfrak{H}$ maps a triple (z1,z2,z3) of pairwise different zi to another triple (w1,w2,w3), then the

Möbius transformation $\mathfrak{H}$ is determined by the equation

$\mbox{cr}(\mathfrak{H}(z),w_1,w_2,w_3)=\mbox{cr}(z,z_1,z_2,z_3),$

or written out in concrete terms:

${{(\mathfrak{H}(z)-w_2)(w_1-w_3)}\over{(\mathfrak{H}(z)-w_3)(w_1-w_2)}}={{(z-z_2)(z_1-z_3)}\over{(z-z_3)(z_1-z_2)}}\ .$

The last equation can be transformed into

${{\mathfrak{H}(z)-w_2}\over{\mathfrak{H}(z)-w_3}}={{(z-z_2)(w_1-w_2)(z_1-z_3)}\over{(z-z_3)(w_1-w_3)(z_1-z_2)}} \ .$

Solving this equation for $\mathfrak{H}(z)$ one obtains the sought transformation.

Relation to the fixed point normal form

Assume that the points $z_2,\, z_3$ are the two (different) fixed points of the Möbius transform $\mathfrak{H}$ i. e. $w_2=z_2, \, w_3=z_3$. Write $z_2 =\gamma_1,\, z_3 =\gamma_2$. The last equation

${{\mathfrak{H}(z)-w_2}\over{\mathfrak{H}(z)-w_3}}={{(z-z_2)(w_1-w_2)(z_1-z_3)}\over{(z-z_3)(w_1-w_3)(z_1-z_2)}}$

then reads

${{\mathfrak{H}(z)-\gamma_1}\over{\mathfrak{H}(z)-\gamma_2}}={{(w_1-\gamma_1)(z_1-\gamma_2)}\over {(w_1-\gamma_2)(z_1-\gamma_1)}}\cdot {{z-\gamma_1}\over {z-\gamma_2}}\ .$

In the previous section on normal form a Möbius transform with two fixed points γ12 was expressed using the characteristic constant k of the transform as

${{\mathfrak{H}(z)-\gamma_1}\over{\mathfrak{H}(z)-\gamma_2}}=k\,{{z-\gamma_1}\over {z-\gamma_2}}\ .$

Comparing both expressions one derives the equality

$k={{(w_1-\gamma_1)(z_1-\gamma_2)}\over {(w_1-\gamma_2)(z_1-\gamma_1)}}=\mbox{cr}(w_1,z_1,\gamma_1,\gamma_2) \ ,$

where z1 is different from the fixed points $\gamma_1 ,\, \gamma_2$ and $w_1=\mathfrak{H}(z_1)$ is the image of z1 under $\mathfrak{H}$. In particular the cross-ratio $\mbox{cr}(\mathfrak{H}(z),z,\gamma_1,\gamma_2)$ does not depend on the choice of the point z (different from the two fixed points) and is equal to the characteristic constant.

## References

• David Mumford, Caroline Series and David Wright (2002). In Mathematics, a Fuchsian group is a particular type of group of isometries of the Hyperbolic plane. In In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformal In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced In Non-Euclidean geometry, the Poincaré half-plane model is the Upper half-plane, together with a metric the Poincaré metric, that makes it a model Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. The bilinear transform (also known as Tustin's method) is used in Digital signal processing and discrete-time Control theory to transform continuous-time Inversion transformations are a natural extension of Poincaré transformations to include all conformal One-to-one transformations on coordinate Space-time A Circle &Gamma is the set of points p that lie at Radius r from a center point &gamma. Indra's Pearls: The Vision of Felix Klein. Indra's Pearls The Vision of Felix Klein is a Geometry book written by David Mumford, Caroline Series and David Wright and published by Cambridge Cambridge University Press. ISBN 0-521-35253-3.   (Aimed at non-mathematicians, provides an excellent exposition of theory and results, richly illustrated with diagrams. )
• Konrad Knopp (1952). Elements of the Theory of Functions. New York: Dover. ISBN 0-486-60154-4.   (See Chapters 3-5 of this classic book for a beautiful introduction to the Riemann sphere, stereographic projection, and Möbius transformations. )
• Tristan Needham (1997). Visual Complex Analysis. Oxford: Clarendon Press. ISBN 0-19-853446-9.   (See Chapter 3 for a beautifully illustrated introduction to Möbius transformations, including their classification up to conjugacy. )
• Alan F. Beardon (1995). The Geometry of Discrete Groups. New York: Springer-Verlag. ISBN 0-3879-0788-2.
• Svetlana Katok (1992). Fuchsian Groups. Chicago:University of Chicago Press. ISBN 0-226-42583-5.   See Chapter 2.
• Gareth A. Jones and David Singerman (1987). Complex Functions: an Algebraic and Geometric Viewpoint. Cambridge: Cambridge Univerisy Press. ISBN 0-521-313666-X.   (See Chapter 2 for various isomorphisms, and for the Lorentz group viewed as a Galois group. )
• G. S. Hall (2004). Symmetries and Curvature Structure in General Relativity. Singapore: World Scientific. ISBN 981-02-1051-5.   (See Chapter 6 for the classification, up to conjugacy, of the Lie subalgebras of the Lie algebra of the Lorentz group. )