Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form \[f(z) = \frac{a z + b}{c z + d}\] of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

Geometrically, a Möbius transformation can be obtained by first applying the inverse stereographic projection from the plane to the unit sphere, 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. 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 geometries and their transformations generalize this case to any number of dimensions over other fields.

Möbius transformations are named in honor of August Ferdinand Möbius; they are an example of homographies, linear fractional transformations, bilinear transformations, and spin transformations (in relativity theory).

Möbius transformations are defined on the extended complex plane \(\widehat{\Complex} = \Complex \cup \{\infty\}\) (i.e., the complex plane augmented by the point at infinity).

Stereographic projection identifies \(\widehat{\Complex}\) with a sphere, which is then called the Riemann sphere; alternatively, \(\widehat{\Complex}\) can be thought of as the complex projective line \(\Complex\mathbb{P}^1\). The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they are the automorphisms of \(\Complex\mathbb{P}^1\) as an algebraic variety. Therefore, the set of all Möbius transformations forms a group under composition. This group is called the Möbius group, and is sometimes denoted \(\operatorname{Aut}(\widehat{\Complex})\).

The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds.

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.

The general form of a Möbius transformation is given by \[f(z) = \frac{a z + b}{c z + d},\] where a, b, c, d are any complex numbers that satisfy ad − bc ≠ 0.

In case c ≠ 0, this definition is extended to the whole Riemann sphere by defining \[\begin{align} f\left(\frac{-d}{c}\right) &= \infin, \\ f(\infin) &= \frac{a}{c}.\end{align}\]

If c = 0, we define \[f(\infin) = \infin.\]

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

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 \(\operatorname{Aut}(\widehat{\Complex})\) as it is the automorphism group of the Riemann sphere.

If ad = bc, the rational function defined above is a constant (unless c = d = 0, when it is undefined): \[\frac{a z + b}{c z + d} = \frac{a}{c} = \frac{b}{d},\] 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 the Schwarzian derivative.

Every non-identity Möbius transformation has two fixed points \(\gamma_1, \gamma_2\) on the Riemann sphere. The fixed points are counted here with multiplicity; the parabolic 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 \[f(z) = \frac{az + b}{cz + d}\] are obtained by solving the fixed point equation f(γ) = γ. For c ≠ 0, this has two roots obtained by expanding this equation to \[c \gamma^2 - (a - d) \gamma - b = 0 \ ,\] and applying the quadratic formula. The roots are \[\gamma_{1,2} = \frac{(a - d) \pm \sqrt{(a - d)^2 + 4bc}}{2c} = \frac{(a - d) \pm \sqrt{\Delta}}{2c}\] with discriminant \[\Delta = (\operatorname{tr}\mathfrak{H})^2 - 4\det\mathfrak{H} = (a+d)^2 - 4(ad-bc),\] where the matrix \[\mathfrak{H} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\] represents the transformation. 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; in this case, one of the fixed points is the point at infinity. When a ≠ d the second fixed point is finite and is given by \[\gamma = -\frac{b}{a-d}.\]

In this case 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: \[z \mapsto z + \beta.\]

Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2: \[\chi(\hat{\Complex}) = 2.\]

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 \(\mathrm{PGL}(2,\Complex)\) 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 \((1+x) / (1-x)\) has no (real) fixed points: as a complex transformation it fixes ±i – 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 \[z \mapsto k z\] (k ∈ C) with fixed points at 0 and ∞. To see this define a map \[g(z) = \frac{z - \gamma_1}{z - \gamma_2}\] 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 \(gfg^{-1}\) has fixed points at 0 and ∞ and is therefore a dilation: \(gfg^{-1}(z) = kz\). The fixed point equation for the transformation f can then be written \[\frac{f(z)-\gamma_1}{f(z)-\gamma_2} = k \frac{z-\gamma_1}{z-\gamma_2}.\]

Solving for f gives (in matrix form): \[\mathfrak{H}(k; \gamma_1, \gamma_2) = \begin{pmatrix} \gamma_1 - k\gamma_2 & (k - 1) \gamma_1\gamma_2 \\ 1 - k & k\gamma_1 - \gamma_2 \end{pmatrix}\] or, if one of the fixed points is at infinity: \[\mathfrak{H}(k; \gamma, \infty) = \begin{pmatrix} k & (1 - k) \gamma \\ 0 & 1 \end{pmatrix}.\]

From the above expressions one can calculate the derivatives of f at the fixed points: \[f'(\gamma_1) = k\] 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: \[\mathfrak{H}(k; \gamma_1, \gamma_2) = \mathfrak{H}(1/k; \gamma_2, \gamma_1).\]

Condensed: the full section is in Wikipedia.

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

The inverse pole \(Z_\infty = \frac{a}{c}\) 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: \[\gamma_1 + \gamma_2 = z_\infty + Z_\infty.\]

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

A transform \(\mathfrak{H}\) can be specified with two fixed points γ₁, γ₂ and the pole \(z_\infty\).

\[\mathfrak{H} = \begin{pmatrix} Z_\infty & - \gamma_1 \gamma_2 \\ 1 & - z_\infty \end{pmatrix}, \;\; Z_\infty = \gamma_1 + \gamma_2 - z_\infty.\]

This allows us to derive a formula for conversion between k and \(z_\infty\) given \(\gamma_1, \gamma_2\): \[z_\infty = \frac{k \gamma_1 - \gamma_2}{1 - k}\] \[k= \frac{\gamma_2 - z_\infty}{\gamma_1 - z_\infty} = \frac{Z_\infty - \gamma_1}{Z_\infty - \gamma_2} = \frac {a - c \gamma_1}{a - c \gamma_2},\] which reduces down to \[k = \frac{(a + d) + \sqrt {(a - d)^2 + 4 b c}}{(a + d) - \sqrt {(a - d)^2 + 4 b c}}.\]

The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios \(\frac{\lambda_1}{\lambda_2}\) of \(\mathfrak{H}\) (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to \[\det (\lambda I_2- \mathfrak{H}) = \lambda^2-\operatorname{tr} \mathfrak{H}\,\lambda + \det \mathfrak{H} = \lambda^2-(a+d)\lambda+(ad-bc)\] which has roots \[\lambda_{i} = \frac{(a + d) \pm \sqrt {(a - d)^2 + 4 b c}}{2} = \frac{(a + d) \pm \sqrt {(a + d)^2 - 4(a d - b c)}}{2}=c\gamma_i+d \, .\]

A Möbius transformation can be composed as a sequence of simple transformations.

The following simple transformations are also Möbius transformations:

• \(f(z) = z+b\quad (a=1, c=0, d=1)\) is a translation.
• \(f(z) = az \quad (b=0, c=0, d=1)\) is a combination of a homothety (uniform scaling) and a rotation. If \(|a| = 1\) then it is a rotation, if \(a \in \R\) then it is a homothety.
• \(f(z)= 1/z \quad (a=0, b=1, c=1, d=0)\) (inversion and reflection with respect to the real axis)

If \(c \neq 0\), let:

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

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

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 functions g₁, g₂, g₃, g₄ such that each g_i is the inverse of f_i. Then the composition \[g_1\circ g_2\circ g_3\circ g_4 (z) = f^{-1}(z) = \frac{dz-b}{-cz+a}\] 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 \(z_1, z_2, z_3, z_4\) to four distinct points \(w_1, w_2, w_3, w_4\) respectively, then \[\frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} =\frac{(w_1-w_3)(w_2-w_4)}{(w_2-w_3)(w_1-w_4)}.\]

If one of the points \(z_1, z_2, z_3, z_4\) is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of \(z_1, z_2, z_3, \infin\) is \[\frac{(z_1-z_3)}{(z_2-z_3)}.\]

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.