Möbius transformation

The fixed points of the transformation

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

are obtained by solving the fixed point equation f(γ) = γ. For c ≠ 0, this has two roots obtained by expanding this equation to

$${\displaystyle c\gamma ^{2}-(a-d)\gamma -b=0\ ,}$$

and applying the quadratic formula. The roots are

$${\displaystyle \gamma _{1,2}={\frac {(a-d)\pm {\sqrt {\left(a-d\right)^{2}+4bc}}}{2c}}={\frac {(a-d)\pm {\sqrt {\Delta }}}{2c}}}$$

with discriminant

$${\displaystyle \Delta =(\operatorname {tr} {\mathfrak {H}})^{2}-4\det {\mathfrak {H}}=(a+d)^{2}-4(ad-bc).}$$

Parabolic transforms have coincidental fixed points due to zero discriminant. For c nonzero and nonzero discriminant the transform is elliptic or hyperbolic.

When c = 0, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of the fixed points is the point at infinity. When a ≠ d the second fixed point is finite and is given by

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

In this case the transformation will be a simple transformation composed of translations, rotations, and dilations:

$${\displaystyle z\mapsto \alpha z+\beta .}$$

If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:

$${\displaystyle z\mapsto z+\beta .}$$

Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2:

$${\displaystyle \chi ({\hat {\mathbb {C} }})=2.}$$

Firstly, the projective linear group PGL(2,K) is sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially dimension counting, as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity.

Next, one can see by identifying the Möbius group with ${\displaystyle \mathrm {PGL} (2,\mathbb {C} )}$ that any Möbius function is homotopic to the identity. Indeed, any member of the general linear group can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the Lefschetz number of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic.

By contrast, the projective linear group of the real projective line, PGL(2,R) need not fix any points – for example ${\displaystyle (1+x)/(1-x)}$ has no (real) fixed points: as a complex transformation it fixes ±i[note 1] – while the map 2x fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more.

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/rotation, i.e., a transformation of the form

$${\displaystyle z\mapsto kz}$$

(k ∈ C) with fixed points at 0 and ∞. To see this define a map

$${\displaystyle g(z)={\frac {z-\gamma _{1}}{z-\gamma _{2}}}}$$

which sends the points (γ₁, γ₂) to (0, ∞). Here we assume that γ₁ and γ₂ are distinct and 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 (γ₁, γ₂) then the transformation ${\displaystyle gfg^{-1}}$ has fixed points at 0 and ∞ and is therefore a dilation: ${\displaystyle gfg^{-1}(z)=kz}$. The fixed point equation for the transformation f can then be written

$${\displaystyle {\frac {f(z)-\gamma _{1}}{f(z)-\gamma _{2}}}=k{\frac {z-\gamma _{1}}{z-\gamma _{2}}}.}$$

Solving for f gives (in matrix form):

$${\displaystyle {\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\begin{pmatrix}\gamma _{1}-k\gamma _{2}&(k-1)\gamma _{1}\gamma _{2}\\1-k&k\gamma _{1}-\gamma _{2}\end{pmatrix}}}$$

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

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

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

$${\displaystyle f'(\gamma _{1})=k}$$

and

$${\displaystyle f'(\gamma _{2})=1/k.}$$

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:

$${\displaystyle {\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\mathfrak {H}}(1/k;\gamma _{2},\gamma _{1}).}$$

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

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

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

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

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

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

Solving for f (in matrix form) gives

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

or, if γ = ∞:

$${\displaystyle {\mathfrak {H}}(\beta$$ ;\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:

$${\displaystyle f'(\gamma )=1.}$$

If ${\displaystyle c\neq 0}$, let:

• ${\displaystyle f_{1}(z)=z+d/c\quad }$ (translation by d/c)
• ${\displaystyle f_{2}(z)=1/z\quad }$ (inversion and reflection with respect to the real axis)
• ${\displaystyle f_{3}(z)={\frac {bc-ad}{c^{2}}}z\quad }$ (homothety and rotation)
• ${\displaystyle f_{4}(z)=z+a/c\quad }$ (translation by a/c)

Then these functions can be composed, giving

$${\displaystyle f_{4}\circ f_{3}\circ f_{2}\circ f_{1}(z)=f(z)={\frac {az+b}{cz+d}}.}$$

That is,

$${\displaystyle {\frac {az+b}{cz+d}}={\frac {a}{c}}+{\frac {e}{z+{\frac {d}{c}}}},}$$

with

$${\displaystyle e={\frac {bc-ad}{c^{2}}}.}$$

This decomposition makes many properties of the Möbius transformation obvious.

The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions g₁, g₂, g₃, g₄ such that each g_i is the inverse of f_i. Then the composition

$${\displaystyle g_{1}\circ g_{2}\circ g_{3}\circ g_{4}(z)=f^{-1}(z)={\frac {dz-b}{-cz+a}}}$$

gives a formula for the inverse.

From this decomposition, we see that Möbius transformations carry over all non-trivial properties of circle inversion. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations and isometries (translation, reflection, rotation), which trivially preserve angles.

Furthermore, Möbius transformations map generalized circles to generalized circles since circle inversion has this property. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center.

Cross-ratios are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points ${\displaystyle z_{1},z_{2},z_{3},z_{4}}$ to four distinct points ${\displaystyle w_{1},w_{2},w_{3},w_{4}}$ respectively, then

$${\displaystyle {\frac {(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1}-z_{4})}}={\frac {(w_{1}-w_{3})(w_{2}-w_{4})}{(w_{2}-w_{3})(w_{1}-w_{4})}}.}$$

If one of the points ${\displaystyle z_{1},z_{2},z_{3},z_{4}}$ is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of ${\displaystyle z_{1},z_{2},z_{3},\infty }$ is

$${\displaystyle {\frac {(z_{1}-z_{3})}{(z_{2}-z_{3})}}.}$$

The cross ratio of four different points is real if and only if there is a line or a circle passing through them. This is another way to show that Möbius transformations preserve generalized circles.

Two points z₁ and z₂ are conjugate with respect to a generalized circle C, if, given a generalized circle D passing through z₁ and z₂ and cutting C in two points a and b, (z₁, z₂; a, b) are in harmonic cross-ratio (i.e. their cross ratio is −1). This property does not depend on the choice of the circle D. This property is also sometimes referred to as being symmetric with respect to a line or circle.[3][4]

Two points z, z^∗ are conjugate with respect to a line, if they are symmetric with respect to the line. Two points are conjugate with respect to a circle if they are exchanged by the inversion with respect to this circle.

The point z^∗ conjugate to z when L is the line determined by the vector based e^iθ at the point z₀ can be explicitly given as

$${\displaystyle z^{*}=e^{2i\theta }{\overline {z-z_{0}}}+z_{0}.}$$

The point z^∗ conjugate to z when C is the circle of radius r centered z₀ can be explicitly given as

$${\displaystyle z^{*}={\frac {r^{2}}{\overline {z-z_{0}}}}+z_{0}}$$

Since Möbius transformations preserve generalized circles and cross-ratios, they preserve also the conjugation.

Given a set of three distinct points ${\displaystyle z_{1},z_{2},z_{3}}$ on the Riemann sphere and a second set of distinct points ${\displaystyle w_{1},w_{2},w_{3}}$, there exists precisely one Möbius transformation ${\displaystyle f(z)}$ with ${\displaystyle f(z_{j})=w_{j}}$ for ${\displaystyle j=1,2,3}$. (In other words: the action of the Möbius group on the Riemann sphere is sharply 3-transitive.) There are several ways to determine ${\displaystyle f(z)}$ from the given sets of points.

It is easy to check that the Möbius transformation

$${\displaystyle f_{1}(z)={\frac {(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}}}$$

with matrix

$${\displaystyle {\mathfrak {H}}_{1}={\begin{pmatrix}z_{2}-z_{3}&-z_{1}(z_{2}-z_{3})\\z_{2}-z_{1}&-z_{3}(z_{2}-z_{1})\end{pmatrix}}}$$

maps ${\displaystyle z_{1},z_{2}{\text{ and }}z_{3}}$ to ${\displaystyle 0,1,\ {\text{and}}\ \infty ,}$ respectively. If one of the ${\displaystyle z_{j}}$ is ${\displaystyle \infty }$, then the proper formula for ${\displaystyle {\mathfrak {H}}_{1}}$ is obtained from the above one by first dividing all entries by ${\displaystyle z_{j}}$ and then taking the limit ${\displaystyle z_{j}\to \infty }$.

If ${\displaystyle {\mathfrak {H}}_{2}}$ is similarly defined to map ${\displaystyle w_{1},w_{2},w_{3}}$ to ${\displaystyle 0,1,\ {\text{and}}\ \infty ,}$ then the matrix ${\displaystyle {\mathfrak {H}}}$ which maps ${\displaystyle z_{1,2,3}}$ to ${\displaystyle w_{1,2,3}}$ becomes

$${\displaystyle {\mathfrak {H}}={\mathfrak {H}}_{2}^{-1}{\mathfrak {H}}_{1}}$$

.

The stabilizer of ${\displaystyle \{0,1,\infty \}}$ (as an unordered set) is a subgroup known as the anharmonic group.

The equation

$${\displaystyle w={\frac {az+b}{cz+d}}}$$

is equivalent to the equation of a standard hyperbola

$${\displaystyle cwz-az+dw-b=0}$$

in the ${\displaystyle (z,w)}$-plane. The problem of constructing a Möbius transformation ${\displaystyle {\mathfrak {H}}(z)}$ mapping a triple ${\displaystyle (z_{1},z_{2},z_{3})}$ to another triple ${\displaystyle (w_{1},w_{2},w_{3})}$ is thus equivalent to finding the coefficients ${\displaystyle a,b,c,d}$ of the hyperbola passing through the points ${\displaystyle (z_{i},w_{i})}$. An explicit equation can be found by evaluating the determinant

$${\displaystyle \det {\begin{pmatrix}zw&z&w&1\\z_{1}w_{1}&z_{1}&w_{1}&1\\z_{2}w_{2}&z_{2}&w_{2}&1\\z_{3}w_{3}&z_{3}&w_{3}&1\end{pmatrix}}\,}$$

by means of a Laplace expansion along the first row. This results in the determinant formulae

$${\displaystyle a=\det {\begin{pmatrix}z_{1}w_{1}&w_{1}&1\\z_{2}w_{2}&w_{2}&1\\z_{3}w_{3}&w_{3}&1\end{pmatrix}}}$$

$${\displaystyle b=\det {\begin{pmatrix}z_{1}w_{1}&z_{1}&w_{1}\\z_{2}w_{2}&z_{2}&w_{2}\\z_{3}w_{3}&z_{3}&w_{3}\end{pmatrix}}}$$

$${\displaystyle c=\det {\begin{pmatrix}z_{1}&w_{1}&1\\z_{2}&w_{2}&1\\z_{3}&w_{3}&1\end{pmatrix}}}$$

$${\displaystyle d=\det {\begin{pmatrix}z_{1}w_{1}&z_{1}&1\\z_{2}w_{2}&z_{2}&1\\z_{3}w_{3}&z_{3}&1\end{pmatrix}}}$$

for the coefficients ${\displaystyle a,b,c,d}$ of the representing matrix ${\displaystyle {\mathfrak {H}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$. The constructed matrix ${\displaystyle {\mathfrak {H}}}$ has determinant equal to ${\displaystyle (z_{1}-z_{2})(z_{1}-z_{3})(z_{2}-z_{3})(w_{1}-w_{2})(w_{1}-w_{3})(w_{2}-w_{3})}$ which does not vanish if the ${\displaystyle z_{j}}$ resp. ${\displaystyle w_{j}}$ are pairwise different thus the Möbius transformation is well-defined. If one of the points ${\displaystyle z_{j}}$ or ${\displaystyle w_{j}}$ is ${\displaystyle \infty }$, then we first divide all four determinants by this variable and then take the limit as the variable approaches ${\displaystyle \infty }$.

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

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

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

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

which describes a translation in the complex plane.

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

$${\displaystyle \left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}\mid b\in \mathbb {C} \right\};}$$

this is an example of the unipotent radical 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).

All non-parabolic transformations have two fixed points and are defined by a matrix conjugate to

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

with the complex number λ not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number k = λ², called the characteristic constant or multiplier of the transformation.

The Smith chart, used by electrical engineers for analyzing transmission lines, is a visual depiction of the elliptic Möbius transformation Γ = (z-1)/(z+1). Each point on the Smith chart simultaneously represents both a value of z (bottom left), and the corresponding value of Γ (bottom right), for |Γ |<1.

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

$${\displaystyle 0\leq \operatorname {tr} ^{2}{\mathfrak {H}}<4.}$$

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

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

with α real.

Note that for any ${\displaystyle {\mathfrak {H}}}$ with characteristic constant k, the characteristic constant of ${\displaystyle {\mathfrak {H}}^{n}}$ is kⁿ. Thus, all Möbius transformations of finite order are elliptic transformations, namely exactly those where λ is a root of unity, or, equivalently, where α is a rational multiple of π. The simplest possibility of a fractional multiple means α = π/2, which is also the unique case of ${\displaystyle \operatorname {tr} {\mathfrak {H}}=0}$, is also denoted as a circular transform; this corresponds geometrically to rotation by 180° about two fixed points. This class is represented in matrix form as:

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

There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: ${\displaystyle 1/z,}$ which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), ${\displaystyle 1-z}$, which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and ${\displaystyle z/(z-1)}$ which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2).

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

$${\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}>4.}$$

A transform is hyperbolic if and only if λ is real and λ ≠ ±1.

The transform is said to be loxodromic if ${\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}}$ is not in [0,4]. A transformation is loxodromic if and only if ${\displaystyle |\lambda |\neq 1}$.

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.

Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real conics. The terminology is due to considering half the absolute value of the trace, |tr|/2, as the eccentricity of the transformation – division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/n is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL. Alternatively one may use half the trace squared as a proxy for the eccentricity squared, as was done above; these classifications (but not the exact eccentricity values, since squaring and absolute values are different) agree for real traces but not complex traces. The same terminology is used for the classification of elements of SL(2, R) (the 2-fold cover), and analogous classifications are used elsewhere. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities.

If ρ = 0, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptic. 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, ∞, 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.

If α is zero (or a multiple of 2π), 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, ∞, 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):

These pictures resemble the field lines of a positive and a negative electrical charge located at the fixed points, because the circular flow lines subtend a constant angle between the two fixed points.

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". 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 (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. 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, ∞: an observer who is both rotating (with constant angular velocity) about some axis and moving 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, ∞, and with ρ, α determined respectively by the magnitude of the actual linear and angular velocities.

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 from having the fixed points in an arbitrary location.

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

An isomorphism of the Möbius group with the Lorentz group was noted by several authors: Based on previous work of Felix Klein (1893, 1897)[11] on automorphic functions related to hyperbolic geometry and Möbius geometry, Gustav Herglotz (1909)[12] showed that hyperbolic motions (i.e. isometric automorphisms of a hyperbolic space) transforming the unit sphere into itself correspond to Lorentz transformations, by which Herglotz was able to classify the one-parameter Lorentz transformations into loxodromic, elliptic, hyperbolic, and parabolic groups. Other authors include Emil Artin (1957),[13] H. S. M. Coxeter (1965),[14] and Roger Penrose, Wolfgang Rindler (1984),[15] Tristan Needham (1997)[16] and W. M. Olivia (2002).[17]

Minkowski space consists of the four-dimensional real coordinate space R⁴ consisting of the space of ordered quadruples (x₀,x₁,x₂,x₃) of real numbers, together with a quadratic form

$${\displaystyle Q(x_{0},x_{1},x_{2},x_{3})=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.}$$

Borrowing terminology from special relativity, points with Q > 0 are considered timelike; in addition, if x₀ > 0, then the point is called future-pointing. Points with Q < 0 are called spacelike. The null cone S consists of those points where Q = 0; the future null cone N⁺ are those points on the null cone with x₀ > 0. The celestial sphere is then identified with the collection of rays in N⁺ whose initial point is the origin of R⁴. The collection of linear transformations on R⁴ with positive determinant preserving the quadratic form Q and preserving the time direction form the restricted Lorentz group SO⁺(1,3).

In connection with the geometry of the celestial sphere, the group of transformations SO⁺(1,3) is identified with the group PSL(2,C) of Möbius transformations of the sphere. To each (x₀, x₁, x₂, x₃) ∈ R⁴, associate the hermitian matrix

$${\displaystyle X={\begin{bmatrix}x_{0}+x_{1}&x_{2}+ix_{3}\\x_{2}-ix_{3}&x_{0}-x_{1}\end{bmatrix}}.}$$

The determinant of the matrix X is equal to Q(x₀, x₁, x₂, x₃). The special linear group acts on the space of such matrices via

${\displaystyle X\mapsto AXA^{*}}$

(1)

for each A ∈ SL(2,C), and this action of SL(2,C) preserves the determinant of X because det A = 1. Since the determinant of X is identified with the quadratic form Q, SL(2,C) acts by Lorentz transformations. On dimensional grounds, SL(2,C) covers a neighborhood of the identity of SO(1,3). Since SL(2,C) is connected, it covers the entire restricted Lorentz group SO⁺(1,3). Furthermore, since the kernel of the action (1) is the subgroup {±I}, then passing to the quotient group gives the group isomorphism

${\displaystyle \operatorname {PSL} (2,\mathbb {C} )\cong \operatorname {SO} ^{+}(1,3).}$

(2)

Focusing now attention on the case when (x₀,x₁,x₂,x₃) is null, the matrix X has zero determinant, and therefore splits as the outer product of a complex two-vector ξ with its complex conjugate:

${\displaystyle X=\xi {\bar {\xi }}^{T}=\xi \xi ^{*}.}$

(3)

The two-component vector ξ is acted upon by SL(2,C) in a manner compatible with (1). It is now clear that the kernel of the representation of SL(2,C) on hermitian matrices is {±I}.

The action of PSL(2,C) on the celestial sphere may also be described geometrically using stereographic projection. Consider first the hyperplane in R⁴ given by x₀ = 1. The celestial sphere may be identified with the sphere S⁺ of intersection of the hyperplane with the future null cone N⁺. The stereographic projection from the north pole (1,0,0,1) of this sphere onto the plane x₃ = 0 takes a point with coordinates (1,x₁,x₂,x₃) with

$${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1}$$

to the point

$${\displaystyle \left(1,{\frac {x_{1}}{1-x_{3}}},{\frac {x_{2}}{1-x_{3}}},0\right).}$$

Introducing the complex coordinate

$${\displaystyle \zeta ={\frac {x_{1}+ix_{2}}{1-x_{3}}},}$$

the inverse stereographic projection gives the following formula for a point (x₁, x₂, x₃) on S⁺:

${\displaystyle {\begin{aligned}x_{1}&={\frac {\zeta +{\bar {\zeta }}}{\zeta {\bar {\zeta }}+1}}\\x_{2}&={\frac {\zeta -{\bar {\zeta }}}{i(\zeta {\bar {\zeta }}+1)}}\\x_{3}&={\frac {\zeta {\bar {\zeta }}-1}{\zeta {\bar {\zeta }}+1}}.\end{aligned}}}$

(4)

The action of SO⁺(1,3) on the points of N⁺ does not preserve the hyperplane S⁺, but acting on points in S⁺ and then rescaling so that the result is again in S⁺ gives an action of SO⁺(1,3) on the sphere which goes over to an action on the complex variable ζ. In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere. Conversely, for any fractional linear transformation of ζ variable goes over to a unique Lorentz transformation on N⁺, possibly after a suitable (uniquely determined) rescaling.

A more invariant description of the stereographic projection which allows the action to be more clearly seen is to consider the variable ζ = z:w as a ratio of a pair of homogeneous coordinates for the complex projective line CP¹. The stereographic projection goes over to a transformation from C² − {0} to N⁺ which is homogeneous of degree two with respect to real scalings

${\displaystyle (z,w)\mapsto (x_{0},x_{1},x_{2},x_{3})=(z{\bar {z}}+w{\bar {w}},z{\bar {z}}-w{\bar {w}},z{\bar {w}}+w{\bar {z}},i^{-1}(z{\bar {w}}-w{\bar {z}}))}$

(5)

which agrees with (4) upon restriction to scales in which ${\displaystyle z{\bar {z}}+w{\bar {w}}=1.}$ The components of (5) are precisely those obtained from the outer product

$${\displaystyle {\begin{bmatrix}x_{0}+x_{1}&x_{2}+ix_{3}\\x_{2}-ix_{3}&x_{0}-x_{1}\end{bmatrix}}=2{\begin{bmatrix}z\\w\end{bmatrix}}{\begin{bmatrix}{\bar {z}}&{\bar {w}}\end{bmatrix}}.}$$

In summary, the action of the restricted Lorentz group SO⁺(1,3) agrees with that of the Möbius group PSL(2,C). This motivates the following definition. In dimension n ≥ 2, the Möbius group Möb(n) is the group of all orientation-preserving conformal isometries of the round sphere Sⁿ to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space R^1,n+1, there is an isomorphism of Möb(n) with the restricted Lorentz group SO⁺(1,n+1) of Lorentz transformations with positive determinant, preserving the direction of time.

Coxeter began instead with the equivalent quadratic form ${\displaystyle Q(x_{1},\ x_{2},\ x_{3}\ x_{4})=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}.}$

He identified the Lorentz group with transformations for which {x : Q(x) = -1} is stable. Then he interpreted the x's as homogeneous coordinates and {x : Q(x) = 0}, the null cone, as the Cayley absolute for a hyperbolic space of points {x : Q(x) < 0}. Next, Coxeter introduced the variables

$${\displaystyle \xi ={\frac {x_{1}}{x_{4}}},\ \eta ={\frac {x_{2}}{x_{4}}},\ \zeta ={\frac {x_{3}}{x_{4}}}}$$

so that the Lorentz-invariant quadric corresponds to the sphere ${\displaystyle \xi ^{2}+\eta ^{2}+\zeta ^{2}=1.}$ Coxeter notes that Felix Klein also wrote of this correspondence, applying stereographic projection from (0, 0, 1) to the complex plane ${\textstyle z={\frac {\xi +i\eta }{1-\zeta }}.}$ Coxeter used the fact that circles of the inversive plane represent planes of hyperbolic space, and the general homography is the product of inversions in two or four circles, corresponding to the general hyperbolic displacement which is the product of inversions in two or four planes.

As seen above, the Möbius group PSL(2,C) acts on Minkowski space as the group of those isometries that preserve the origin, the orientation of space and the direction of time. Restricting to the points where Q=1 in the positive light cone, which form a model of hyperbolic 3-space H³, we see that the Möbius group acts on H³ as a group of orientation-preserving isometries. In fact, the Möbius group is equal to the group of orientation-preserving isometries of hyperbolic 3-space.

If we use the Poincaré ball model, identifying the unit ball in R³ with H³, then we can think of the Riemann sphere as the "conformal boundary" of H³. Every orientation-preserving isometry of H³ gives rise to a Möbius transformation on the Riemann sphere and vice versa; this is the very first observation leading to the AdS/CFT correspondence conjectures in physics.