In geometry, a Möbius transformation is a function:
- f(z) = \frac
where z, a, b, c, d are complex numbers satisfying ad − bc ≠ 0. Möbius transformations are named in honor of August Ferdinand Möbius, although they are also called homographic transformations or fractional linear transformations.
A Möbius transformation is a bijective conformal map of the extended complex plane (i.e. the complex plane augmented by the point at infinity):
- \widehat = \mathbb\cup\.
The Möbius group is the automorphism group of the Riemann sphere, sometimes denoted
- \mbox(\widehat\mathbb C).
A particularly important subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations.
In physics, the identity component of the Lorentz group acts on the celestial sphere the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory.
The general form of a Möbius transformation is given by
- z \mapsto \frac
The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group. The Möbius group is usually denoted \mbox(\widehat\mathbb C) as it is the automorphism group of the Riemann sphere.
Decomposition and elementary properties
A Möbius transformation is equivalent to a sequence of simpler transformations. Let:
then these functions can be composed on each other, giving
- f_4\circ f_3\circ f_2\circ f_1 (z)= \frac.\!
This decomposition makes many properties of the Möbius transform obvious.
For example, the preservation of angles is reduced to proving the angle preservation property of circle inversion, since all other transformation are dilatations or isometries, which trivially preserve angles.
The existence of an inverse Möbius transformation function and its explicit formula is easily derived by a composition of the inverse function of the simpler transformations. That is, define functions g_1, g_2, g_3, g_4 such that g_i is the inverse of f_i, then composition g_1\circ g_2\circ g_3\circ g_4 (z) would be the explicit expression for the inverse Möbius transformation:
From this decomposition, we also see that Möbius transformation carries over all non-trivial properties of circle inversion. Namely, that circles are mapped to circles, and angles are preserved. Also, because of the circle inversion, is carried over the convenience of defining Möbius transformation over a plane with a point at infinity, which makes statements and concepts of Möbius transformation's properties simpler.
For another example, look at f_3. If ad-bc= 0, then the transformation collapses to the point 0, then f_4 moves to a/c. Collapsing to a point is not an interesting transformation, thus we require in the definition of Möbius transformation that ad-bc \ne 0.
Preservation of angles and circles
As seen from the above decomposition, Möbius transformation contains this transformation 1/z, called complex inversion. Geometrically, a complex inversion is a circle inversion followed by a reflection around the x-axis.
In circle inversion, circles are mapped to circles (here, lines are considered as circles with infinite radius), and angles are preserved. See circle inversion for various properties and proofs.
The cross-ratio preservation theorem states that the cross-ratio \frac = \frac
is invariant under a Möbius transformation that maps from z to w.
The action of the Möbius group on the Riemann sphere is sharply 3-transitive in the sense that there is a unique Möbius transformation which takes any three distinct points on the Riemann sphere to any other set of three distinct points. See the section below on specifying a transformation by three points.
Projective matrix representations
- f(z) = \frac
- \mathfrak = \begin a & b \\ c & d \end.
The usefulness of this representation is that the composition of two Möbius transformations corresponds precisely to matrix multiplication of the corresponding matrices. That is, if we define a map
- \pi\colon \mbox(2,\mathbb C) \to \mbox(\widehat\mathbb C)
The map \pi is not an isomorphism, since it maps any scalar multiple of \mathfrak to the same transformation. The kernel of this homomorphism is then the set of all scalar matrices kI, which is the center of GL(2,C). The quotient group GL(2,C)/Z(GL(2,C)) is called the projective linear group and is usually denoted PGL(2,C). By the first isomorphism theorem of group theory we conclude that the Möbius group is isomorphic to PGL(2,C). Since Z(GL(2,C)) is the kernel of the group action given by GL(2,C) acting on itself by conjugation, PGL(2, C) is isomorphic to the inner automorphism group of GL(2,C). Moreover, the natural action of PGL(2,C) on the complex projective line CP1 is exactly the natural action of the Möbius group on the Riemann sphere when the sphere and the projective line are identified as follows:
- [z_1 : z_2]\leftrightarrow z_1/z_2.
If one normalizes \mathfrak so that the determinant is equal to one, the map \pi restricts to a surjective map from the special linear group SL(2,C) to the Möbius group. The Möbius group is therefore also isomorphic to PSL(2,C). We then have the following isomorphisms:
- \mbox(\widehat\mathbb C) \cong \mbox(2,\mathbb C) \cong \mbox(2,\mathbb C).
Note that there are precisely two matrices with unit determinant which can be used to represent any given Möbius transformation. That is, SL(2,C) is a double cover of PSL(2,C). Since SL(2,C) is simply-connected it is the universal cover of the Möbius group. The fundamental group of the Möbius group is then Z2.
Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic (actually hyperbolic is a special case of loxodromic). The classification has both algebraic and geometric significance. Geometrically, the different types result in different transformations of the complex plane, as the figures below illustrate. These types can be distinguished by looking at the trace \mbox\,\mathfrak=a+d. Note that the trace is invariant under conjugation, that is,
- \mbox\,\mathfrak^ = \mbox\,\mathfrak,
and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing matrix \mathfrak has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations \mathfrak, \mathfrak' (both not equal to the identity transform) with \det \mathfrak=\det\mathfrak'=1 are conjugate if and only if \mbox^2\,\mathfrak= \mbox^2\,\mathfrak' .
In the following discussion we will always assume that the representing matrix \mathfrak is normalized such that \det=ad-bc=1 .
Parabolic transformsThe transform is said to be parabolic if
- \mbox^2\mathfrak = (a+d)^2 = 4.
A transform is parabolic if and only if it has one fixed point in the compactified complex plane \widehat=\mathbb\cup\. It is parabolic if and only if it is conjugate to
- \begin 1 & 1 \\ 0 & 1 \end.
The subgroup consisting of all parabolic transforms of this form:
- \begin 1 & b \\ 0 & 1 \end
is an example of a Borel subgroup, which generalizes the idea to higher dimensions.
All other non-identity transformations have two fixed points. All non-parabolic (non-identity) transforms are conjugate to
- \begin \lambda & 0 \\ 0 & \lambda^ \end
with \lambda not equal to 0, 1 or −1. The square k=\lambda^2 is called the characteristic constant or multiplier of the transformation.
Elliptic transformsThe transform is said to be elliptic if
- 0 \le \mbox^2\mathfrak
A transform is elliptic if and only if |\lambda|=1. Writing \lambda=e^, an elliptic transform is conjugate to
- \begin \cos\alpha & \sin\alpha \\
with \alpha real. Note that for any \mathfrak, the characteristic constant of \mathfrak^n is k^n. Thus, the only Möbius transformations of finite order are the elliptic transformations, and these only when λ is a root of unity; equivalently, when α is a rational multiple of pi.
The transform is said to be hyperbolic if
- \mbox^2\mathfrak > 4.\,
A transform is hyperbolic if and only if λ is real and positive.
Loxodromic transformsThe transform is said to be loxodromic if \mbox^2\mathfrak is not in the closed interval of [0, 4]. Hyperbolic transforms are thus a special case of loxodromic transformations. A transformation is loxodromic if and only if |\lambda|\ne 1. Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below.
Every non-identity Möbius transformation has two fixed points \gamma_1, \gamma_2 on the Riemann sphere. Note that the fixed points are counted here with multiplicity; for parabolic transformations, the fixed points coincide. Either or both of these fixed points may be the point at infinity.
The fixed points of the transformation
- f(z) = \frac
- \gamma_ = \frac = \frac.
When c = 0, one of the fixed points is at infinity; the other is given by
The transformation will be a simple transformation composed of translations, rotations, and dilations:
- z \mapsto \alpha z + \beta.\,
If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation: z \mapsto z + \beta.
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, i.e. a transformation of the form
- z \mapsto k z
- g(z) = \frac
If f has distinct fixed points (\gamma_1, \gamma_2) then the transformation gfg^ has fixed points at 0 and ∞ and is therefore a dilation: gfg^(z) = kz. The fixed point equation for the transformation f can then be written
- \frac = k \frac.
Solving for f gives (in matrix form):
- \mathfrak(k; \gamma_1, \gamma_2) =
or, if one of the fixed points is at infinity:
- \mathfrak(k; \gamma, \infty) =
From the above expressions one can calculate the derivatives of f at the fixed points:
- f'(\gamma_1)= k\, and f'(\gamma_2)= 1/k.\,
Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (k) of f as the characteristic constant of f. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:
- \mathfrak(k; \gamma_1, \gamma_2) = \mathfrak(1/k; \gamma_2, \gamma_1).
For loxodromic transformations, whenever |k|>1, one says that \gamma_1 is the repulsive fixed point, and \gamma_2 is the attractive fixed point. For |k|, the roles are reversed.
In the parabolic case there is only one fixed point \gamma. The transformation sending that point to ∞ is
- g(z) = \frac
or the identity if \gamma is already at infinity. The transformation gfg^ fixes infinity and is therefore a translation:
- gfg^(z) = z + \beta\,.
Here, β is called the translation length. The fixed point formula for a parabolic transformation is then
- \frac = \frac + \beta.
Solving for f (in matrix form) gives
- \mathfrak(\beta; \gamma) =
or, if \gamma = \infty:
- \mathfrak(\beta; \infty) =
Note that \beta is not the characteristic constant of f, which is always 1 for a parabolic transformation. From the above expressions one can calculate:
- f'(\gamma) = 1.\,
Geometric interpretation of the characteristic constant
The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case:
The characteristic constant can be expressed in terms of its logarithm:
- e^ = k \;
If \rho = 0, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptical. These transformations tend to move all points in circles around the two fixed points. If one of the fixed points is at infinity, this is equivalent to doing an affine rotation around a point.
If we take the one-parameter subgroup generated by any elliptic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the same two points. All other points flow along a family of circles which is nested between the two fixed points on the Riemann sphere. In general, the two fixed points can be any two distinct points.
This has an important physical interpretation. Imagine that some observer rotates with constant angular velocity about some axis. Then we can take the two fixed points to be the North and South poles of the celestial sphere. The appearance of the night sky is now transformed continuously in exactly the manner described by the one-parameter subgroup of elliptic transformations sharing the fixed points 0, \infty, and with the number \alpha corresponding to the constant angular velocity of our observer.
Here are some figures illustrating the effect of an elliptic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):
Iterating a transformation
If a transformation \mathfrak has fixed points \gamma_1, \gamma_2, and characteristic constant k, then \mathfrak' = \mathfrak^n will have \gamma_1' = \gamma_1, \gamma_2' = \gamma_2, k' = k^n.
This can be used to iterate a transformation, or to animate one by breaking it up into steps.
These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants.