Ingeometry andcomplex analysis, aMöbius transformation of thecomplex plane is arational function of the form$${\displaystyle f(z)=\frac{az+b}{cz+d}}$$of onecomplex variablez; here the coefficientsa,b,c,d are complex numbers satisfyingad −bc ≠ 0.

Geometrically, a Möbius transformation can be obtained by first applying the inversestereographic projection from the plane to the unitsphere, moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane.^[1] These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.

The Möbius transformations are theprojective transformations of thecomplex projective line. They form agroup called theMöbius group, which is theprojective linear groupPGL(2,C). Together with itssubgroups, it has numerous applications in mathematics and physics.

Möbius geometries and their transformations generalize this case to any number of dimensions over other fields.

Möbius transformations are named in honor ofAugust Ferdinand Möbius; they are an example ofhomographies,linear fractional transformations, bilinear transformations, and spin transformations (in relativity theory).^[2]

Möbius transformations are defined on theextended complex plane${\displaystyle \hat{\mathbb{C}}=\mathbb{C}\cup \{\mathrm{\infty }\}}$ (i.e., thecomplex plane augmented by thepoint at infinity).

Stereographic projection identifies${\displaystyle \hat{\mathbb{C}}}$ with a sphere, which is then called theRiemann sphere; alternatively,${\displaystyle \hat{\mathbb{C}}}$ can be thought of as the complexprojective line${\displaystyle \mathbb{C}{\mathbb{P}}^1}$. The Möbius transformations are exactly thebijectiveconformal maps from the Riemann sphere to itself, i.e., theautomorphisms of the Riemann sphere as acomplex manifold; alternatively, they are the automorphisms of${\displaystyle \mathbb{C}{\mathbb{P}}^1}$ as an algebraic variety. Therefore, the set of all Möbius transformations forms agroup undercomposition. This group is called the Möbius group, and is sometimes denoted${\displaystyle \mathrm{Aut}(\hat{\mathbb{C}})}$.

The Möbius group isisomorphic to the group of orientation-preservingisometries ofhyperbolic 3-space and therefore plays an important role when studyinghyperbolic 3-manifolds.

Inphysics, theidentity component of theLorentz group acts on thecelestial sphere in 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 oftwistor theory.

Certainsubgroups of the Möbius group form the automorphism groups of the othersimply-connected Riemann surfaces (thecomplex plane and thehyperbolic plane). As such, Möbius transformations play an important role in the theory ofRiemann surfaces. Thefundamental group of every Riemann surface is adiscrete subgroup of the Möbius group (seeFuchsian group andKleinian group). A particularly important discrete subgroup of the Möbius group is themodular group; it is central to the theory of manyfractals,modular forms,elliptic curves andPellian equations.

Möbius transformations can be more generally defined in spaces of dimensionn > 2 as the bijective conformal orientation-preserving maps from then-sphere to then-sphere. Such a transformation is the most general form of conformal mapping of a domain. According toLiouville's theorem a Möbius transformation can be expressed as a composition of translations,similarities, orthogonal transformations and inversions.

The general form of a Möbius transformation is given by$${\displaystyle f(z)=\frac{az+b}{cz+d},}$$wherea,b,c,d are anycomplex numbers that satisfyad −bc ≠ 0.

In casec ≠ 0, this definition is extended to the wholeRiemann sphere by defining

Ifc = 0, we define$${\displaystyle f(\mathrm{\infty })=\mathrm{\infty }.}$$

Thus a Möbius transformation is always a bijectiveholomorphic function from the Riemann sphere to the Riemann sphere.

The set of all Möbius transformations forms agroup undercomposition. This group can be given the structure of acomplex manifold in such a way that composition and inversion areholomorphic maps. The Möbius group is then acomplex Lie group. The Möbius group is usually denoted${\displaystyle \mathrm{Aut}(\hat{\mathbb{C}})}$ as it is theautomorphism group of the Riemann sphere.

Ifad =bc, the rational function defined above is a constant (unlessc =d = 0, when it is undefined):$${\displaystyle \frac{az+b}{cz+d}=\frac{a}{c}=\frac{b}{d},}$$where a fraction with a zero denominator is ignored. A constant function is not bijective and is thus not considered a Möbius transformation.

An alternative definition is given as the kernel of theSchwarzian derivative.

Every non-identity Möbius transformation has twofixed points${\displaystyle {\gamma }_1,{\gamma }_2}$ on the Riemann sphere. The fixed points are counted here withmultiplicity; theparabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity.

The fixed points of the transformation$${\displaystyle f(z)=\frac{az+b}{cz+d}}$$are obtained by solving the fixed point equationf(γ) =γ. Forc ≠ 0, this has two roots obtained by expanding this equation to$${\displaystyle c{\gamma }^2-(a-d)\gamma -b=0,}$$and applying thequadratic formula. The roots are$${\displaystyle {\gamma }_{1,2}=\frac{(a-d)\pm \sqrt{(a-d{)}^2+4bc}}{2c}=\frac{(a-d)\pm \sqrt{\mathrm{\Delta }}}{2c}}$$with discriminant$${\displaystyle \mathrm{\Delta }=(\mathrm{tr}\mathfrak{H}{)}^2-4det\mathfrak{H}=(a+d{)}^2-4(ad-bc),}$$where the matrixrepresents the transformation.Parabolic transforms have coincidental fixed points due to zero discriminant. Forc nonzero and nonzero discriminant the transform is elliptic or hyperbolic.

Whenc = 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. Whena ≠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 oftranslations,rotations, anddilations:$${\displaystyle z\mapsto \alpha z+\beta .}$$

Ifc = 0 anda =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 theEuler characteristic of the sphere being 2:$${\displaystyle \chi (\hat{\mathbb{C}})=2.}$$

Firstly, theprojective linear groupPGL(2,K) issharply 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 (essentiallydimension 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${\displaystyle \mathrm{P}\mathrm{G}\mathrm{L}(2,\mathbb{C})}$ that any Möbius function is homotopic to the identity. Indeed, any member of thegeneral linear group can be reduced to the identity map byGauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. TheLefschetz–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 theLefschetz 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-callednormal form. We first treat the non-parabolic case, for which there are two distinct fixed points.

Non-parabolic case:

Every non-parabolic transformation isconjugate 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 thatγ₁ andγ₂ are distinct and finite. If one of them is already at infinity theng can be modified so as to fix infinity and send the other point to 0.

Iff has distinct fixed points (γ₁,γ₂) then the transformation${\displaystyle gf{g}^{-1}}$ has fixed points at 0 and ∞ and is therefore a dilation:${\displaystyle gf{g}^{-1}(z)=kz}$. The fixed point equation for the transformationf can then be written$${\displaystyle \frac{f(z)-{\gamma }_1}{f(z)-{\gamma }_2}=k\frac{z-{\gamma }_1}{z-{\gamma }_2}.}$$

Solving forf gives (in matrix form):or, if one of the fixed points is at infinity:

From the above expressions one can calculate the derivatives off at the fixed points:$${\displaystyle f^{\prime }({\gamma }_1)=k}$$ and$${\displaystyle f^{\prime }({\gamma }_2)=1/k.}$$

Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (k) off as thecharacteristic constant off. 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 thatγ₁ is therepulsive fixed point, andγ₂ is theattractive fixed point. For|k| < 1, the roles are reversed.

Parabolic case:

In the parabolic case there is only one fixed pointγ. The transformation sending that point to ∞ is$${\displaystyle g(z)=\frac{1}{z-\gamma }}$$or the identity ifγ is already at infinity. The transformation${\displaystyle gf{g}^{-1}}$ fixes infinity and is therefore a translation:$${\displaystyle gf{g}^{-1}(z)=z+\beta .}$$

Here,β is called thetranslation length. The fixed point formula for a parabolic transformation is then$${\displaystyle \frac{1}{f(z)-\gamma }=\frac{1}{z-\gamma }+\beta .}$$

Solving forf (in matrix form) givesNote that

Ifγ = ∞:

Note thatβ isnot the characteristic constant off, which is always 1 for a parabolic transformation. From the above expressions one can calculate:$${\displaystyle f^{\prime }(\gamma )=1.}$$

The point$z_{\mathrm{\infty }}=-\frac{d}{c}$ is called thepole of${\displaystyle \mathfrak{H}}$; it is that point which is transformed to the point at infinity under⁠${\displaystyle \mathfrak{H}}$⁠.

The inverse pole$Z_{\mathrm{\infty }}=\frac{a}{c}$ 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:$${\displaystyle {\gamma }_1+{\gamma }_2=z_{\mathrm{\infty }}+Z_{\mathrm{\infty }}.}$$

These four points are the vertices of aparallelogram which is sometimes called thecharacteristic parallelogram of the transformation.

A transform${\displaystyle \mathfrak{H}}$ can be specified with two fixed pointsγ₁,γ₂ and the pole${\displaystyle z_{\mathrm{\infty }}}$.

This allows us to derive a formula for conversion betweenk and${\displaystyle z_{\mathrm{\infty }}}$ given${\displaystyle {\gamma }_1,{\gamma }_2}$:$${\displaystyle z_{\mathrm{\infty }}=\frac{k{\gamma }_1-{\gamma }_2}{1-k}}$$$${\displaystyle k=\frac{{\gamma }_2-z_{\mathrm{\infty }}}{{\gamma }_1-z_{\mathrm{\infty }}}=\frac{Z_{\mathrm{\infty }}-{\gamma }_1}{Z_{\mathrm{\infty }}-{\gamma }_2}=\frac{a-c{\gamma }_1}{a-c{\gamma }_2},}$$which reduces down to$${\displaystyle k=\frac{(a+d)+\sqrt{(a-d{)}^2+4bc}}{(a+d)-\sqrt{(a-d{)}^2+4bc}}.}$$

The last expression coincides with one of the (mutually reciprocal)eigenvalue ratios$\frac{{\lambda }_1}{{\lambda }_2}$ of${\displaystyle \mathfrak{H}}$ (compare the discussion in the preceding section about the characteristic constant of a transformation). Itscharacteristic polynomial is equal to$${\displaystyle det(\lambda I_2-\mathfrak{H})={\lambda }^2-\mathrm{tr}\mathfrak{H}\lambda +det\mathfrak{H}={\lambda }^2-(a+d)\lambda +(ad-bc)}$$which has roots$${\displaystyle {\lambda }_i=\frac{(a+d)\pm \sqrt{(a-d{)}^2+4bc}}{2}=\frac{(a+d)\pm \sqrt{(a+d{)}^2-4(ad-bc)}}{2}=c{\gamma }_i+d.}$$

A Möbius transformation can becomposed as a sequence of simple transformations.

The following simple transformations are also Möbius transformations:

• ${\displaystyle f(z)=z+b(a=1,c=0,d=1)}$ is atranslation.
• ${\displaystyle f(z)=az(b=0,c=0,d=1)}$ is a combination of ahomothety (uniformscaling) and arotation. If${\displaystyle |a|=1}$ then it is a rotation, if${\displaystyle a\in \mathbb{R}}$ then it is a homothety.
• ${\displaystyle f(z)=1/z(a=0,b=1,c=1,d=0)}$ (inversion andreflection with respect to the real axis)

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

• ${\displaystyle f_1(z)=z+d/c}$ (translation byd/c)
• ${\displaystyle f_2(z)=1/z}$ (inversion and reflection with respect to the real axis)
• ${\displaystyle f_3(z)=\frac{bc-ad}{c^2}z}$ (homothety and rotation)
• ${\displaystyle f_4(z)=z+a/c}$ (translation bya/c)

Then these functions can becomposed, showing that, if$${\displaystyle f(z)=\frac{az+b}{cz+d},}$$one has$${\displaystyle f=f_4\circ f_3\circ f_2\circ f_1.}$$In other terms, one has$${\displaystyle \frac{az+b}{cz+d}=\frac{a}{c}+\frac{e}{z+\frac{d}{c}},}$$with$${\displaystyle e=\frac{bc-ad}{c^2}.}$$

This decomposition makes many properties of the Möbius transformation obvious.

A Möbius transformation is equivalent to a sequence of simpler transformations. The composition 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 functionsg₁,g₂,g₃,g₄ such that eachg_i is the inverse off_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 ofcircle inversion. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations andisometries (translation, reflection, rotation), which trivially preserve angles.

Furthermore, Möbius transformations mapgeneralized 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${\displaystyle z_1,z_2,z_3,\mathrm{\infty }}$ 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 pointsz₁ andz₂ areconjugate with respect to a generalized circleC, if, given a generalized circleD passing throughz₁ andz₂ and cuttingC in two pointsa andb,(z₁,z₂;a,b) are inharmonic cross-ratio (i.e. their cross ratio is −1). This property does not depend on the choice of the circleD. This property is also sometimes referred to as beingsymmetric with respect to a line or circle.^[3][4]

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

The pointz^∗ is conjugate toz whenL is the line determined by the vector based upone^iθ, at the pointz₀. This can be explicitly given as$${\displaystyle z^{\ast }=e^{2i\theta }\overline{z-z_0}+z_0.}$$

The pointz^∗ is conjugate toz whenC is the circle of a radiusr, centered aboutz₀. This can be explicitly given as$${\displaystyle z^{\ast }=\frac{r^2}{\overline{z-z_0}}+z_0.}$$

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

The naturalaction ofPGL(2,C) on thecomplex projective lineCP¹ is exactly the natural action of the Möbius group on the Riemann sphere

Here, the projective lineCP¹ and the Riemann sphere are identified as follows:$${\displaystyle [z_1:z_2]\sim \frac{z_1}{z_2}.}$$

Here [z₁:z₂] arehomogeneous coordinates onCP¹; the point [1:0] corresponds to the point∞ of the Riemann sphere. By using homogeneous coordinates, many calculations involving Möbius transformations can be simplified, since no case distinctions dealing with∞ are required.

Everyinvertible complex 2×2 matrixacts on the projective line as$${\displaystyle z=[z_1:z_2]\mapsto w=[w_1:w_2],}$$where

The result is therefore$${\displaystyle w=[w_1:w_2]=[a{z}_1+b{z}_2:c{z}_1+d{z}_2]}$$

Which, using the above identification, corresponds to the following point on the Riemann sphere :

$${\displaystyle w=[a{z}_1+b{z}_2:c{z}_1+d{z}_2]\sim \frac{a{z}_1+b{z}_2}{c{z}_1+d{z}_2}=\frac{a\frac{z_1}{z_2}+b}{c\frac{z_1}{z_2}+d}.}$$

Since the above matrix is invertible if and only if itsdeterminantad −bc is not zero, this induces an identification of the action of the group of Möbius transformations with the action ofPGL(2,C) on the complex projective line. In this identification, the above matrix${\displaystyle \mathfrak{H}}$ corresponds to the Möbius transformation${\displaystyle z\mapsto \frac{az+b}{cz+d}.}$

This identification is agroup isomorphism, since the multiplication of${\displaystyle \mathfrak{H}}$ by a non zero scalar${\displaystyle \lambda }$ does not change the element ofPGL(2,C), and, as this multiplication consists of multiplying all matrix entries by${\displaystyle \lambda ,}$ this does not change the corresponding Möbius transformation.

For anyfieldK, one can similarly identify the groupPGL(2,K) of the projective linear automorphisms with the group of fractional linear transformations. This is widely used; for example in the study ofhomographies of thereal line and its applications inoptics.

If one divides${\displaystyle \mathfrak{H}}$ by a square root of its determinant, one gets a matrix of determinant one. This induces a surjective group homomorphism from thespecial linear groupSL(2,C) toPGL(2,C), with${\displaystyle \pm I}$ as its kernel.

This allows showing that the Möbius group is a 3-dimensional complexLie group (or a 6-dimensional real Lie group), which is asemisimple and non-compact, and that SL(2,C) is adouble cover ofPSL(2,C). SinceSL(2,C) issimply-connected, it is theuniversal cover of the Möbius group, and thefundamental group of the Möbius group is Z₂.

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: theaction of the Möbius group on the Riemann sphere issharply 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 matrixmaps${\displaystyle z_1,z_2\text{ and }{z}_3}$ to⁠${\displaystyle 0,1,\text{and}\mathrm{\infty }}$⁠, respectively. If one of the${\displaystyle z_j}$ is${\displaystyle \mathrm{\infty }}$, 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${\displaystyle z_j\to \mathrm{\infty }}$⁠.

If${\displaystyle {\mathfrak{H}}_2}$ is similarly defined to map${\displaystyle w_1,w_2,w_3}$ to${\displaystyle 0,1,\text{and}\mathrm{\infty },}$ 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${\displaystyle \{0,1,\mathrm{\infty }\}}$ (as an unordered set) is a subgroup known as theanharmonic group.

The equation$${\displaystyle w=\frac{az+b}{cz+d}}$$is equivalent to the equation of a standardhyperbola$${\displaystyle cwz-az+dw-b=0}$$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 thedeterminantby means of aLaplace expansion along the first row, resulting in explicit formulae,for the coefficients${\displaystyle a,b,c,d}$ of the representing matrix⁠${\displaystyle \mathfrak{H}}$⁠. 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${\displaystyle \mathrm{\infty }}$⁠, then we first divide all four determinants by this variable and then take the limit as the variable approaches⁠${\displaystyle \mathrm{\infty }}$⁠.

If we require the coefficients${\displaystyle a,b,c,d}$ of a Möbius transformation to be real numbers with⁠${\displaystyle ad-bc=1}$⁠, we obtain a subgroup of the Möbius group denoted asPSL(2,R). This is the group of those Möbius transformations that map theupper half-planeH = {x + iy :y > 0} to itself, and is equal to the group of allbiholomorphic (or equivalently:bijective,conformal and orientation-preserving) mapsH →H. If a propermetric is introduced, the upper half-plane becomes a model of thehyperbolic planeH², thePoincaré half-plane model, andPSL(2,R) is the group of all orientation-preserving isometries ofH² in this model.

The subgroup of all Möbius transformations that map the open diskD = {z : |z| < 1} to itself consists of all transformations of the form$${\displaystyle f(z)=e^{i\varphi }\frac{z+b}{\overline{b}z+1}}$$with${\displaystyle \varphi }$ ∈R,b ∈C and|b| < 1. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) mapsD →D. By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, thePoincaré disk model, and this group is the group of all orientation-preserving isometries ofH² in this model.

Since both of the above subgroups serve as isometry groups ofH², they are isomorphic. A concrete isomorphism is given byconjugation with the transformation$${\displaystyle f(z)=\frac{z+i}{iz+1}}$$which bijectively maps the open unit disk to the upper half plane.

Alternatively, consider an open disk with radiusr, centered atr i. The Poincaré disk model in this disk becomes identical to the upper-half-plane model asr approaches ∞.

Amaximal compact subgroup of the Möbius group${\displaystyle \mathcal{M}}$ is given by (Tóth 2002)^[5]$${\displaystyle {\mathcal{M}}_0:=\left\{z\mapsto \frac{uz-\overline{v}}{vz+\overline{u}}:|u{|}^2+|v{|}^2=1\right\},}$$and corresponds under the isomorphism${\displaystyle \mathcal{M}\cong \mathrm{PSL}(2,\mathbb{C})}$ to theprojective special unitary groupPSU(2,C) which is isomorphic to thespecial orthogonal group SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, thepoint groups in three dimensions.

Icosahedral groups of Möbius transformations were used byFelix Klein to give an analytic solution to thequintic equation in (Klein 1913); a modern exposition is given in (Tóth 2002).^[6]

If we require the coefficientsa,b,c,d of a Möbius transformation to beintegers withad −bc = 1, we obtain themodular groupPSL(2,Z), a discrete subgroup ofPSL(2,R) important in the study oflattices in the complex plane,elliptic functions andelliptic curves. The discrete subgroups ofPSL(2,R) are known asFuchsian groups; they are important in the study ofRiemann surfaces.

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

Non-identity Möbius transformations are commonly classified into four types,parabolic,elliptic,hyperbolic andloxodromic, with the hyperbolic ones being a subclass of the loxodromic ones. 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.

The four types can be distinguished by looking at thetrace${\displaystyle \mathrm{tr}\mathfrak{H}=a+d}$. The trace is invariant underconjugation, that is,$${\displaystyle \mathrm{tr}{\mathfrak{G}\mathfrak{H}\mathfrak{G}}^{-1}=\mathrm{tr}\mathfrak{H},}$$and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing matrix${\displaystyle \mathfrak{H}}$ has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations${\displaystyle \mathfrak{H},{\mathfrak{H}}^{\prime }}$ (both not equal to the identity transform) with${\displaystyle det\mathfrak{H}=det{\mathfrak{H}}^{\prime }=1}$ are conjugate if and only if${\displaystyle {\mathrm{tr}}^2\mathfrak{H}={\mathrm{tr}}^2{\mathfrak{H}}^{\prime }.}$

A non-identity Möbius transformation defined by a matrix${\displaystyle \mathfrak{H}}$ of determinant one is said to beparabolic if$${\displaystyle {\mathrm{tr}}^2\mathfrak{H}=(a+d{)}^2=4}$$(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 samecharacteristic polynomialX² − 2X + 1 as the identity matrix, and is thereforeunipotent. A Möbius transform is parabolic if and only if it has exactly one fixed point in theextended complex plane${\displaystyle \hat{\mathbb{C}}=\mathbb{C}\cup \{\mathrm{\infty }\}}$, which happens if and only if it can be defined by a matrixconjugate towhich describes a translation in the complex plane.

The set of all parabolic Möbius transformations with agiven fixed point in${\displaystyle \hat{\mathbb{C}}}$, together with the identity, forms asubgroup isomorphic to the group of matricesthis is an example of theunipotent radical of aBorel subgroup (of the Möbius group, or ofSL(2,C) for the matrix group; the notion is defined for anyreductive Lie group).

All non-parabolic transformations have two fixed points and are defined by a matrix conjugate towith the complex numberλ not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex numberk =λ², called thecharacteristic constant ormultiplier of the transformation.

The transformation is said to beelliptic if it can be represented by a matrix${\displaystyle \mathfrak{H}}$ of determinant 1 such that

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

Forany${\displaystyle \mathfrak{H}}$ with characteristic constantk, the characteristic constant of${\displaystyle {\mathfrak{H}}^n}$ iskⁿ. Thus, all Möbius transformations of finiteorder are elliptic transformations, namely exactly those whereλ is aroot of unity, or, equivalently, whereα is arational multiple ofπ. The simplest possibility of a fractional multiple meansα =π/2, which is also the unique case of${\displaystyle \mathrm{tr}\mathfrak{H}=0}$, is also denoted as acircular transform; this corresponds geometrically to rotation by 180° about two fixed points. This class is represented in matrix form as: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 behyperbolic if it can be represented by a matrix${\displaystyle \mathfrak{H}}$ whose trace isreal with$${\displaystyle {\mathrm{tr}}^2\mathfrak{H}>4.}$$

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

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

Historically,navigation byloxodrome orrhumb line refers to a path of constantbearing; the resulting path is alogarithmic 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
Circular	σ = 0	k = −1		z ↦ −z
Elliptic	0 ≤σ < 4 ${\displaystyle \sigma =2+2\cos (\theta )}$	|k| = 1 ${\displaystyle k=e^{\pm i\theta }\ne 1}$		z ↦eiθz
Parabolic	σ = 4	k = 1		z ↦z +a
Hyperbolic	4 <σ < ∞ ${\displaystyle \sigma =2+2\cosh (\theta )}$	${\displaystyle k\in {\mathbb{R}}^{+}}$ ${\displaystyle k=e^{\pm \theta }\ne 1}$		z ↦eθz
Loxodromic	σ ∈C \ [0,4] ${\displaystyle \sigma =(\lambda +{\lambda }^{-1}{)}^2}$	${\displaystyle |k|\ne 1}$ ${\displaystyle k={\lambda }^2,{\lambda }^{-2}}$		z ↦kz

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 realconics. The terminology is due to considering half the absolute value of the trace, |tr|/2, as theeccentricity 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 tracesquared 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 theclassification of elements ofSL(2,R) (the 2-fold cover), andanalogous classifications are used elsewhere. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities.

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 itslogarithm:$${\displaystyle e^{\rho +\alpha i}=k.}$$When expressed in this way, the real numberρ becomes an expansion factor. It indicates how repulsive the fixed pointγ₁ is, and how attractiveγ₂ is. The real numberα is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise aboutγ₁ and clockwise about γ₂.

Ifρ = 0, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to beelliptic. 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 theone-parameter subgroup generated by any elliptic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes thesame 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 numberα 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 generatescontinuously moves points along the family of circular arcs suggested by the pictures.

Ifα is zero (or a multiple of 2π), then the transformation is said to behyperbolic. These transformations tend to move points along circular paths from one fixed point toward the other.

If we take theone-parameter subgroup generated by any hyperbolic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes thesame two points. All other points flow along a certain family of circular arcsaway from the first fixed point andtoward 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 numberρ 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 bothρ andα are nonzero, then the transformation is said to beloxodromic. 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". Whensailing on a constantbearing – if you maintain a heading of (say) north-east, you will eventually wind up sailing around thenorth pole in alogarithmic spiral. On themercator 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 ofk. 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 theone-parameter subgroup generated by any loxodromic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes thesame two points. All other points flow along a certain family of curves,away from the first fixed point andtoward 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 thesame axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and withρ,α determined respectively by the magnitude of the actual linear and angular velocities.

These images show Möbius transformationsstereographically projected onto theRiemann 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
Fixed points diametrically opposite
Fixed points in an arbitrary location

If a transformation${\displaystyle \mathfrak{H}}$ has fixed pointsγ₁,γ₂, and characteristic constantk, then${\displaystyle {\mathfrak{H}}^{\prime }={\mathfrak{H}}^n}$ will have${\displaystyle {\gamma }_1^{\prime }={\gamma }_1,{\gamma }_2^{\prime }={\gamma }_2,k^{\prime }=k^n}$.

This can be used toiterate 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.

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

In higher dimensions, aMöbius transformation is ahomeomorphism of⁠${\displaystyle \overline{{\mathbb{R}}^n}}$⁠, theone-point compactification of⁠${\displaystyle {\mathbb{R}}^n}$⁠, which is a finite composition ofinversions in spheres andreflections inhyperplanes.^[7]Liouville's theorem in conformal geometry states that in dimension at least three, allconformal transformations are Möbius transformations. Every Möbius transformation can be put in the form$${\displaystyle f(x)=b+\frac{\alpha A(x-a)}{|x-a{|}^{\epsilon }},}$$where${\displaystyle a,b\in {\mathbb{R}}^n}$,${\displaystyle \alpha \in \mathbb{R}}$,${\displaystyle A}$ is anorthogonal matrix, and${\displaystyle \epsilon }$ is 0 or 2. The group of Möbius transformations is also called theMöbius group.^[8]

The orientation-preserving Möbius transformations form the connected component of the identity in the Möbius group. In dimensionn = 2, the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here. The orientation-reversing ones are obtained from these by complex conjugation.^[9]

The domain of Möbius transformations, i.e.⁠${\displaystyle \overline{{\mathbb{R}}^n}}$⁠, is homeomorphic to then-dimensional sphere${\displaystyle S^n}$. The canonical isomorphism between these two spaces is theCayley transform, which is itself a Möbius transformation of⁠${\displaystyle \overline{{\mathbb{R}}^{n+1}}}$⁠. This identification means that Möbius transformations can also be thought of as conformal isomorphisms of${\displaystyle S^n}$. Then-sphere, together with action of the Möbius group, is a geometric structure (in the sense of Klein'sErlangen program) calledMöbius geometry.^[10]

An isomorphism of the Möbius group with theLorentz group was noted by several authors: Based on previous work ofFelix Klein (1893, 1897)^[11] onautomorphic functions related to hyperbolic geometry and Möbius geometry,Gustav Herglotz (1909)^[12] showed thathyperbolic motions (i.e.isometricautomorphisms of ahyperbolic space) transforming theunit 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 includeEmil Artin (1957),^[13]H. S. M. Coxeter (1965),^[14] andRoger Penrose,Wolfgang Rindler (1984),^[15]Tristan Needham (1997)^[16] and W. M. Olivia (2002).^[17]

Minkowski space consists of the four-dimensional real coordinate spaceR⁴ consisting of the space of ordered quadruples(x₀,x₁,x₂,x₃) of real numbers, together with aquadratic 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 fromspecial relativity, points withQ > 0 are consideredtimelike; in addition, ifx₀ > 0, then the point is calledfuture-pointing. Points withQ < 0 are calledspacelike. Thenull coneS consists of those points whereQ = 0; thefuture null coneN⁺ are those points on the null cone withx₀ > 0. Thecelestial sphere is then identified with the collection of rays inN⁺ whose initial point is the origin ofR⁴. The collection oflinear transformations onR⁴ with positivedeterminant preserving the quadratic formQ and preserving the time direction form therestricted Lorentz groupSO⁺(1, 3).

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

Thedeterminant of the matrixX is equal toQ(x₀,x₁,x₂,x₃). Thespecial linear group acts on the space of such matrices via

${\displaystyle X\mapsto AX{A}^{\ast }}$

1

for eachA ∈ SL(2,C), and this action ofSL(2,C) preserves the determinant ofX becausedetA = 1. Since the determinant ofX is identified with the quadratic formQ,SL(2,C) acts by Lorentz transformations. On dimensional grounds,SL(2,C) covers a neighborhood of the identity ofSO(1, 3). SinceSL(2,C) is connected, it covers the entire restricted Lorentz groupSO⁺(1, 3). Furthermore, since thekernel of the action (1) is the subgroup {±I}, then passing to thequotient group gives thegroup isomorphism

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

2

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

${\displaystyle X=\xi {\overline{\xi }}^{\text{T}}=\xi {\xi }^{\ast }.}$

3

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

The action ofPSL(2,C) on the celestial sphere may also be described geometrically usingstereographic projection. Consider first the hyperplane inR⁴ given byx₀ = 1. The celestial sphere may be identified with the sphereS⁺ of intersection of the hyperplane with the future null coneN⁺. The stereographic projection from the north pole(1, 0, 0, 1) of this sphere onto the planex₃ = 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 thecomplex coordinate$${\displaystyle \zeta =\frac{x_1+i{x}_2}{1-x_3},}$$the inverse stereographic projection gives the following formula for a point(x₁,x₂,x₃) onS⁺:

4

The action ofSO⁺(1, 3) on the points ofN⁺ does not preserve the hyperplaneS⁺, but acting on points inS⁺ and then rescaling so that the result is again inS⁺ gives an action ofSO⁺(1, 3) on the sphere which goes over to an action on the complex variableζ. 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 ofζ variable goes over to a unique Lorentz transformation onN⁺, 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 variableζ =z:w as a ratio of a pair of homogeneous coordinates for the complex projective lineCP¹. The stereographic projection goes over to a transformation fromC² − {0} toN⁺ which is homogeneous of degree two with respect to real scalings

${\displaystyle (z,w)\mapsto (x_0,x_1,x_2,x_3)=(z\overline{z}+w\overline{w},z\overline{z}-w\overline{w},z\overline{w}+w\overline{z},i^{-1}(z\overline{w}-w\overline{z}))}$

5

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

In summary, the action of the restricted Lorentz group SO⁺(1,3) agrees with that of the Möbius groupPSL(2,C). This motivates the following definition. In dimensionn ≥ 2, theMöbius group Möb(n) is the group of all orientation-preservingconformalisometries of the round sphereSⁿ to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski spaceR^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} isstable. Then he interpreted thex's ashomogeneous coordinates and {x | Q(x) = 0}, thenull cone, as theCayley 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 thatFelix Klein also wrote of this correspondence, applying stereographic projection from(0, 0, 1) to the complex plane$z=\frac{\xi +i\eta }{1-\zeta }.$ 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 groupPSL(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 whereQ = 1 in the positive light cone, which form a model ofhyperbolic 3-spaceH³, we see that the Möbius group acts onH³ 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 thePoincaré ball model, identifying the unit ball inR³ withH³, then we can think of the Riemann sphere as the "conformal boundary" ofH³. Every orientation-preserving isometry ofH³ gives rise to a Möbius transformation on the Riemann sphere and vice versa.

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