Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. 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 the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2,C). Together with its subgroups, it has numerous applications in mathematics and physics.
Möbius transformations are named in honor of August Ferdinand Möbius; they are also variously named homographies, homographic transformations, linear fractional transformations, bilinear transformations, or fractional linear transformations.
In physics, the identity component of the Lorentz group acts on the celestial 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 of twistor theory.
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 fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important discrete 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.
Möbius transformations can be more generally defined in spaces of dimension n > 2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, similarities, orthogonal transformations and inversions.
In case c ≠ 0, this definition is extended to the whole Riemann sphere by defining
Thus a Möbius transformation is always a bijective holomorphic function from the Riemann sphere to the Riemann sphere.
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
If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:
Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 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.
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.
Every non-parabolic transformation is conjugate to a dilation/rotation, i.e., a transformation of the form
From the above expressions one can calculate the derivatives of f at the fixed points:
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:
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.
In the parabolic case there is only one fixed point γ. The transformation sending that point to ∞ is
Here, β is called the translation length. The fixed point formula for a parabolic transformation is then
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:
These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation.
A Möbius transformation can be composed as a sequence of simple transformations.
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 functions g1, g2, g3, g4 such that each gi is the inverse of fi. Then the composition
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 dilation 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.
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.
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 eiθ at the point z0 can be explicitly given as
The point z∗ conjugate to z when C is the circle of radius r centered z0 can be explicitly given as
Since Möbius transformations preserve generalized circles and cross-ratios, they preserve also the conjugation.
The natural action of PGL(2,C) on the complex projective line CP1 is exactly the natural action of the Möbius group on the Riemann sphere, where the projective line CP1 and the Riemann sphere are identified as follows:
It is straightforward to check that then the product of two matrices will be associated with the composition of the two corresponding Möbius transformations. In other words, the map
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. Therefore, the fundamental group of the Möbius group is Z2.
Given a set of three distinct points z1, z2, z3 on the Riemann sphere and a second set of distinct points w1, w2, w3, there exists precisely one Möbius transformation f(z) with f(zi) = wi for i = 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 f(z) from the given sets of points.
If we require the coefficients a, b, c, d of a Möbius transformation to be real numbers with ad − bc = 1, we obtain a subgroup of the Möbius group denoted as PSL(2,R). This is the group of those Möbius transformations that map the upper half-plane H = x + iy : y > 0 to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps H → H. If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H2, the Poincaré half-plane model, and PSL(2,R) is the group of all orientation-preserving isometries of H2 in this model.
The subgroup of all Möbius transformations that map the open disk D = z : |z| < 1 to itself consists of all transformations of the form
Since both of the above subgroups serve as isometry groups of H2, they are isomorphic. A concrete isomorphism is given by conjugation with the transformation
Alternatively, consider an open disk with radius r, centered at r i. The Poincaré disk model in this disk becomes identical to the upper-half-plane model as r approaches ∞.
If we require the coefficients a, b, c, d of a Möbius transformation to be integers with ad − bc = 1, we obtain the modular group PSL(2,Z), a discrete subgroup of PSL(2,R) important in the study of lattices in the complex plane, elliptic functions and elliptic curves. The discrete subgroups of PSL(2,R) are known as Fuchsian groups; they are important in the study of Riemann surfaces.
Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic, 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.
All non-parabolic transformations have two fixed points and are defined by a matrix conjugate to
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.
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:
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.
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.
And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms. Note that in the elliptical and loxodromic images, the α value is 1/10 .
The orientation-preserving Möbius transformations form the connected component of the identity in the Möbius group. In dimension n = 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.
An isomorphism of the Möbius group with the Lorentz group was noted by several authors: Based on previous work of Felix Klein (1893, 1897) on automorphic functions related to hyperbolic geometry and Möbius geometry, Gustav Herglotz (1909) 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), H. S. M. Coxeter (1965), and Roger Penrose, Wolfgang Rindler (1984) and W. M. Olivia (2002)
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
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:
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
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.
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 R1,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.
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.