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

In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i.e. the complex plane augmented by the point at infinity):

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

The set of all Möbius transformations forms a group under composition called the Möbius group. Möbius transformations are named in honor of August Ferdinand Möbius, although they are also called homographic transformations or fractional linear transformations.

Overview

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

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

Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The covering group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). Möbius transformations are also closely related to isometries of hyperbolic 3-manifolds.

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

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The general form of a Möbius transformation is given by

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

where a, b, c, d are any complex numbers satisfying ad − bc ≠ 0. This definition can be extended to the whole Riemann sphere (the complex plane plus the point at infinity) with the following two special cases:

• the point $z\; =\; -d/c$ is mapped to $\backslash infty$
• the point $z=\backslash infty$ is mapped to $a/c$

We can have Möbius transformations over the real numbers, as well as for the complex numbers. In both cases, we need to augment the domain with a point at infinity.

The condition ad − bc ≠ 0 ensures that the transformation is invertible. The inverse transformation is given by

$z\; \backslash mapsto\; \backslash frac\{\backslash left(\backslash frac\{d\}\{ad-bc\}\backslash right)\; z\; -\; \backslash left(\backslash frac\{b\}\{ad-bc\}\backslash right)\}\{-c\; z\; +\; a\}$

with the usual special cases understood.

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 $\backslash mbox\{Aut\}(\backslash widehat\backslash mathbb\; C)$ as it is the automorphism group of the Riemann sphere.

Projective matrix representations

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The transformation

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

can be usefully expressed as a matrix

The condition ad − bc ≠ 0 is equivalent to the condition that the determinant of above matrix be nonzero (i.e. the matrix should be non-singular). Note that multiplying $\backslash mathfrak\; H$ by any complex number λ gives rise to the same transformation. Such matrix representations are called projective representations for reasons explained below. It is often convenient to normalize $\backslash mathfrak\; H$ so that its determinant is equal to 1. The matrix $\backslash mathfrak\; H$ is then unique up to sign.

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

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

from the general linear group GL(2,C) to the Möbius group which sends the matrix $\backslash mathfrak\{H\}$ to the transformation f, then this map is a group homomorphism.

The map $\backslash pi$ is not an isomorphism, since it maps any scalar multiple of $\backslash mathfrak\{H\}$ 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:

$:\; z\_2\backslash leftrightarrow\; z\_1/z\_2.$

Here ^* are homogeneous coordinates on CP¹.

If one normalizes $\backslash mathfrak\{H\}$ so that the determinant is equal to one, the map $\backslash pi$ 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:

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

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). 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.

Properties

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Any Möbius transformation can be composed from the elementary transformations: dilations, translations and inversions. If we define a line to be a circle passing through infinity, then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.

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.

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.

Classification

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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 $\backslash mbox\{tr\}\backslash ,\backslash mathfrak\{H\}=a+d$. Note that the trace is invariant under conjugation, that is,

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

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

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

Parabolic transforms The transform is said to be parabolic if

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

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

.

The subgroup consisting of all parabolic transforms of this form:

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

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

Elliptic transforms The transform is said to be elliptic if

.

A transform is elliptic if and only if $|\backslash lambda|=1$. Writing $\backslash lambda=e^\{i\backslash alpha\}$, an elliptic transform is conjugate to

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

with $\backslash alpha$ real. Note that for any $\backslash mathfrak\{H\}$, the characteristic constant of $\backslash mathfrak\{H\}^n$ 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

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

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

Loxodromic transforms The transform is said to be loxodromic if $\backslash mbox\{tr\}^2\backslash mathfrak\{H\}$ is not in the closed interval of ^*. Hyperbolic transforms are thus a special case of loxodromic transformations. A transformation is loxodromic if and only if $|\backslash lambda|\backslash ne\; 1$. 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.

Transformation	Trace squared	Multipliers	Class representative
Elliptic		>k		$z\backslash mapsto\; e^\{i\backslash theta\}z$
Parabolic	$\backslash sigma\; =\; 4$	$k\; =\; 1$		$z\backslash mapsto\; z\; +\; a$
Hyperbolic		$k\; \backslash in\; \backslash mathbb\; R^\{+\}$ $k\; =\; e^\{\backslash pm\; \backslash theta\}\; \backslash neq\; 1$		$z\; \backslash mapsto\; e^\backslash theta\; z$
Loxodromic	$\backslash sigma\backslash in\backslash mathbb\; C,\; \backslash sigma\; \backslash not\backslash in*$	>k		$z\; \backslash mapsto\; k\; z$

Fixed points

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Every non-identity Möbius transformation has two fixed points $\backslash gamma\_1,\; \backslash gamma\_2$ 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

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

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

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

Note that for parabolic transformations, which satisfy $(a+d)^2\; =\; 4(ad-bc)$, the fixed points coincide.

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

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

The transformation will be a simple transformation composed of translations, rotations, and dilations: $z\; \backslash mapsto\; \backslash alpha\; z\; +\; \backslash beta$.

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

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

$z\; \backslash mapsto\; k\; z$

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

$g(z)\; =\; \backslash frac\{z\; -\; \backslash gamma\_1\}\{z\; -\; \backslash gamma\_2\}$

which sends the points $(\backslash gamma\_1,\; \backslash gamma\_2)$ to $(0,\backslash infty)$. Here we assume that both $\backslash gamma\_1$ and $\backslash gamma\_2$ 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 $(\backslash gamma\_1,\; \backslash gamma\_2)$ 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

$\backslash frac\{f(z)-\backslash gamma\_1\}\{f(z)-\backslash gamma\_2\}\; =\; k\; \backslash frac\{z-\backslash gamma\_1\}\{z-\backslash gamma\_2\}.$

Solving for f gives (in matrix form):

$\backslash mathfrak\{H\}(k;\; \backslash gamma\_1,\; \backslash gamma\_2)\; =$

\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:

$\backslash mathfrak\{H\}(k;\; \backslash gamma,\; \backslash infty)\; =$

\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:

$f\text{'}(\backslash gamma\_1)=\; k\backslash ,$ and $f\text{'}(\backslash gamma\_2)=\; 1/k.\backslash ,$

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:

$\backslash mathfrak\{H\}(k;\; \backslash gamma\_1,\; \backslash gamma\_2)\; =\; \backslash mathfrak\{H\}(1/k;\; \backslash gamma\_2,\; \backslash gamma\_1).$

For loxodromic transformations, whenever $|k|>1$, one says that $\backslash gamma\_1$ is the repulsive fixed point, and $\backslash gamma\_2$ is the attractive fixed point. For , the roles are reversed.

Parabolic case:

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

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

or the identity if $\backslash gamma$ is already at infinity. The transformation $gfg^\{-1\}$ fixes infinity and is therefore a translation:

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

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

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

Solving for f (in matrix form) gives

$\backslash mathfrak\{H\}(\backslash beta;\; \backslash gamma)\; =$

\begin{pmatrix} 1+\gamma\beta & - \beta \gamma^2 \\ \beta & 1-\gamma \beta \end{pmatrix}

or, if $\backslash gamma\; =\; \backslash infty$:

$\backslash mathfrak\{H\}(\backslash beta;\; \backslash infty)\; =$

\begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix}

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

$f\text{'}(\backslash gamma)\; =\; 1.\backslash ,$

Geometric interpretation of the characteristic constant

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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:

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

When expressed in this way, the real number $\backslash rho$ becomes an expansion factor. It indicates how repulsive the fixed point $\backslash gamma\_1$ is, and how attractive $\backslash gamma\_2$ is. The real number $\backslash alpha$ is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about $\backslash gamma\_1$ and clockwise about $\backslash gamma\_2$.

Elliptic transformations

If $\backslash rho\; =\; 0$, 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 $0,\; \backslash infty$, and with the number $\backslash alpha$ 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):

||

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 $\backslash alpha$ is zero (or a multiple of $2\; \backslash pi$), 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. 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 $0,\backslash infty$, with the real number $\backslash rho$ 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: "loxos, slanting + dromos, course". 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. On the mercator projection such a course is a straight line, as the north and south poles project to infinity. The angle that the loxodrome subtends relative to the lines of longitude (ie, 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. 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 $0,\; \backslash infty$: 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 $0,\; \backslash infty$, and with $\backslash rho,\; \backslash alpha$ determined respectively by the magnitude of acceleration and angular velocity.

Here are some pictures illustrating the effect of a loxodromic transformation:

Stereographic projection

These images show Möbius transformations stereographically projected onto the Riemann sphere. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different to having the fixed points in an arbitrary location.

|| Mob3d-hyp-inf-200.png || Mob3d-lox-inf-200.png || Mob3d-hyp-opp-200.png || Mob3d-lox-opp-200.png || Mob3d-hyp-arb-200.png || Mob3d-lox-arb-200.png

	Elliptic	Hyperbolic	Loxodromic
One fixed point at Infinity
	Mob3d-elip-inf-480.png	Mob3d-hyp-inf-480.png	Mob3d-lox-inf-480.png
Fixed points diametrically opposite
	Mob3d-elip-opp-480.png	Mob3d-hyp-opp-480.png	Mob3d-lox-opp-480.png
Fixed points in an arbitrary location
	Mob3d-elip-arb-480.png	Mob3d-hyp-arb-480.png	Mob3d-lox-arb-480.png

Iterating a transformation

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If a transformation $\backslash mathfrak\{H\}$ has fixed points $\backslash gamma\_1,\; \backslash gamma\_2$, and characteristic constant 'k'', then $\backslash mathfrak\{H\}\text{'}\; =\; \backslash mathfrak\{H\}^n$ will have $\backslash gamma\_1\text{'}\; =\; \backslash gamma\_1,\; \backslash gamma\_2\text{'}\; =\; \backslash gamma\_2,\; k\text{'}\; =\; k^n$.

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.

|| Mobius23622.jpeg ||

Poles of the transformation

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The point

$z\_\backslash infty\; =\; -\; \backslash frac\{d\}\{c\}$

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

The inverse pole

$Z\_\backslash infty\; =\; \backslash frac\{a\}\{c\}$

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:

$\backslash gamma\_1\; +\; \backslash gamma\_2\; =\; z\_\backslash infty\; +\; Z\_\backslash infty$

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

A transform $\backslash mathfrak\{H\}$ can be specified with two fixed points $\backslash gamma\_1,\; \backslash gamma\_2$ and the pole $z\_\backslash infty$.

$\backslash mathfrak\{H\}\; =$

\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 $z\_\backslash infty$ given $\backslash gamma\_1,\; \backslash gamma\_2$:

$z\_\backslash infty\; =\; \backslash frac\{k\; \backslash gamma\_1\; -\; \backslash gamma\_2\}\{1\; -\; k\}$

$k$

= \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

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

The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios $\backslash lambda\_1\backslash over\; \backslash lambda\_2$ of the matrix

$\backslash mathfrak\{H\}\; =$

\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). 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

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

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 $\backslash mathfrak\{H\}$. 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.

$\backslash mathfrak\{H\}\_1\; =\; \backslash begin\{pmatrix\}$

\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 $z\_2\; -\; z\_1$ and $Z\_2\; -\; Z\_1$ as previously noted.

$\backslash mathfrak\{H\}\_1\; =\; \backslash begin\{pmatrix\}$

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 $\backslash mathfrak\{H\}$ to map $z\_\{1,2,3\}$ onto $Z\_\{1,2,3\}$ then becomes

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

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 $\backslash mathfrak\{H\}(z)$ mapping a triple $(z\_1,\; z\_2,\; z\_3\; )$ to another triple $(\; w\_1,\; w\_2,\; w\_3\; )$ is equivalent to finding the equation of a standard hyperbola

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

in the (z,w)-plane passing through the points $(z\_i,\; w\_i\; )$. An explicit equation can be found by evaluating the determinant

\end{pmatrix} \, by means of a Laplace expansion along the first row. This results in the determinant formulae

\end{pmatrix} \,

\end{pmatrix} \,

\end{pmatrix} \,

\end{pmatrix} for the coefficients $a,b,c,d$ of the representing matrix . The constructed matrix $\backslash mathfrak\{H\}$ has determinant equal to $(z\_1-z\_2)\; (z\_1-z\_3)(z\_2-z\_3)(w\_1-w\_2)\; (w\_1-w\_3)(w\_2-w\_3)$ which does not vanish if the z_i resp. w_i are pairwise different thus the Möbius transformation is well-defined.

Remark: A similar determinant (with $wz$ replaced by $w^2+z^2$) leads to the equation of a circle through three different (non collinear) points in the plane.

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)=

is invariant under a Möbius transformation mapping a quadruple $(z\_1,z\_2,z\_3,z\_4)$ to $(w\_1,w\_2,w\_3,w\_4)$ via $w\_i=\backslash mathfrak\{H\}z\_i$. If $\backslash mathfrak\{H\}$ maps a triple $(z\_1,z\_2,z\_3)$ of pairwise different z_i to another triple $(w\_1,w\_2,w\_3)$, then the Möbius transformation $\backslash mathfrak\{H\}$ 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:

$$

=\ .

The last equation can be transformed into

$$

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

Relation to the fixed point normal form

Assume that the points $z\_2,\backslash ,\; z\_3$ are the two (different) fixed points of the Möbius transform $\backslash mathfrak\{H\}$ i.e. $w\_2=z\_2,\; \backslash ,\; w\_3=z\_3$. Write $z\_2\; =\backslash gamma\_1,\backslash ,\; z\_3\; =\backslash gamma\_2$. The last equation

$$

= then reads

$$

=\cdot \ . In the previous section on normal form a Möbius transform with two fixed points $\backslash gamma\_1,\; \backslash gamma\_2$ was expressed using the characteristic constant k of the transform as

$$

=k\,\ . Comparing both expressions one derives the equality

$k==\backslash mbox\{cr\}(w\_1,z\_1,\backslash gamma\_1,\backslash gamma\_2)\; \backslash \; ,$

where $z\_1$ is different from the fixed points $\backslash gamma\_1\; ,\backslash ,\; \backslash gamma\_2$ and $w\_1=\backslash mathfrak\{H\}(z\_1)$ is the image of z₁ under $\backslash mathfrak\{H\}$. In particular the cross-ratio $\backslash mbox\{cr\}(\backslash mathfrak\{H\}(z),z,\backslash gamma\_1,\backslash gamma\_2)$ does not depend on the choice of the point z (different from the two fixed points) and is equal to the characteristic constant.

See also

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• Fuchsian group
• Hyperbolic geometry
• Inversive ring geometry
• Kleinian group
• Lorentz group
• Modular group
• Poincaré half-plane model
• Projective geometry
• Bilinear transform

References

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• (Aimed at non-mathematicians, provides an excellent exposition of theory and results, richly illustrated with diagrams.)

• (See Chapters 3-5 of this classic book for a beautiful introduction to the Riemann sphere, stereographic projection, and Möbius transformations.)

• (See Chapter 3 for a beautifully illustrated introduction to Möbius transformations, including their classification up to conjugacy.)

• See Chapter 2.

• (See Chapter 2 for various isomorphisms, and for the Lorentz group viewed as a Galois group.)

• (See Chapter 6 for the classification, up to conjugacy, of the Lie subalgebras of the Lie algebra of the Lorentz group.)

External links

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• A java applet allowing you to specify a transformation via its fixed points and so on can be found at ^*.

Projective geometry | Conformal geometry | Lie groups | Riemann surfaces

Möbiustransformation | Transformée de Möbius | Дробно-линейное преобразование