Möbius transformation

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Möbius transformations should not be confused with the Möbius transform or the Möbius function.

In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i.e. the complex plane augmented by the point at infinity):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}.}

The set of all Möbius transformations forms a group under composition called the Möbius group. Möbius transformations are named in honor of August Ferdinand Möbius, although they are also called homographic transformations or fractional linear transformations.

Overview

The Möbius group is the automorphism group of the Riemann sphere, sometimes denoted

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Aut}(\widehat\mathbb C).}

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 covering group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). Möbius transformations are also closely related to isometries of hyperbolic 3-manifolds.

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.

Definition

The general form of a Möbius transformation is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \mapsto \frac{a z + b}{c z + d}}

where a, b, c, d are any complex numbers satisfying adbc ≠ 0. This definition can be extended to the whole Riemann sphere (the complex plane plus the point at infinity) with the following two special cases:

  • the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = -d/c} is mapped to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty}
  • the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=\infty} is mapped to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a/c}

We can have Möbius transformations over the real numbers, as well as for the complex numbers. In both cases, we need to augment the domain with a point at infinity.

The condition adbc ≠ 0 insures that a Möbius is invertible. The inverse transformation is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \mapsto \frac{d z - b}{-c z + a}}

with the usual special cases understood.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Aut}(\widehat\mathbb C)} as it is the automorphism group of the Riemann sphere.

Projective matrix representations

The transformation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = \frac{a z + b}{c z + d}}

can be usefully expressed as a matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.}

The condition adbc ≠ 0 is equivalent to the condition that the determinant of above matrix be nonzero (i.e. the matrix should be non-singular). Note that multiplying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak H} by any complex number λ gives rise to the same transformation. Such matrix representations are called projective representations for reasons explained below. It is often convenient to normalize Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak H} so that its determinant is equal to 1. The matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak H} is then unique up to sign.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi\colon \mbox{GL}(2,\mathbb C) \to \mbox{Aut}(\widehat\mathbb C)}

from the general linear group GL(2,C) to the Möbius group which sends the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} to the transformation f, then this map is a group homomorphism.

The map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} is not an isomorphism, since it maps any scalar multiple of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} to the same transformation. The kernel of this homomorphism is then the set of all scalar matrices Z(2,C). The quotient group GL(2,C)/Z(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). 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [z_1 : z_2]\leftrightarrow z_1/z_2.}

Here [z1:z2] are homogeneous coordinates on CP1.

If one normalizes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} so that the determinant is equal to one, the map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Aut}(\widehat\mathbb C) \cong \mbox{PGL}(2,\mathbb C) \cong \mbox{PSL}(2,\mathbb C).}

From the last identification we see that the Möbius group is a 3-dimensional complex Lie group (or a 6-dimensional real Lie group).

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.

Properties

Any Möbius transformation can be composed from the elementary transformations: dilations, translations and inversions. If we define a line to be a circle passing through infinity, then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.

The cross-ratio preservation theorem states that the cross-ratio

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} = \frac{(w_1-w_3)(w_2-w_4)}{(w_1-w_4)(w_2-w_3)} }

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.

Classification

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{tr}\,\mathfrak{H}=a+d} . Note that the trace is invariant under conjugation, that is,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{tr}\,\mathfrak{GHG}^{-1} = \mbox{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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}, \mathfrak{H}'} (both not equal to the identity transform) with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det \mathfrak{H}=\det\mathfrak{H}'=1 } are conjugate if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{tr}^2\,\mathfrak{H}= \mbox{tr}^2\,\mathfrak{H}' } .

In the following discussion we will always assume that the representing matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} is normalized such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det{\mathfrak{H}}=ad-bc=1 } .

Parabolic transforms The transform is said to be parabolic if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{tr}^2\mathfrak{H} = (a+d)^2 = 4} .

A transform is parabolic if and only if it has one fixed point in the compactified complex plane Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}} . It is parabolic if and only if it is conjugate to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}} .

The subgroup of parabolic transforms 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} not equal to 0,1 or -1. The square Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\lambda^2} is called the characteristic constant or multiplier of the transformation.

Elliptic transforms The transform is said to be elliptic if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le \mbox{tr}^2\mathfrak{H} < 4} .

A transform is elliptic if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\lambda|=1} . Writing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda=e^{i\alpha}} , an elliptic transform is conjugate to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{pmatrix}}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} real. Note that for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} , the characteristic constant of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}^n} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Hyperbolic transforms The transform is said to be hyperbolic if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{tr}^2\mathfrak{H} > 4} .

A transform is hyperbolic if and only if λ is real and positive.

Loxodromic transforms The transform is said to be loxodromic if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{tr}^2\mathfrak{H}} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\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.


Transformation Trace squared Multipliers Class representative
Elliptic Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq \sigma < 4} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k| = 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = e^{\pm i\theta} \neq 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2}\end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\mapsto e^{i\theta}z}
Parabolic Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = 4} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 1} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1 & a \\ 0 & 1\end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\mapsto z + a}
Hyperbolic Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 < \sigma < \infty} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \in \mathbb R^{+}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = e^{\pm \theta} \neq 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}e^{\theta/2} & 0 \\ 0 & e^{-\theta/2}\end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \mapsto e^\theta z}
Loxodromic Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma\in\mathbb C, \sigma \not\in [0,4]} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k| \neq 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = \lambda^{2}, \lambda^{-2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}\lambda & 0 \\ 0 & \lambda^{-1}\end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \mapsto k z}

Fixed points

Every non-identity Möbius transformation has two fixed points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = \frac{az + b}{cz + d}}

are obtained by solving the fixed point equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\gamma) = \gamma} . For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\ne 0} , this has two roots (proof):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_{1,2} = \frac{(a - d) \pm \sqrt{(a-d)^2 + 4bc}}{2c} = \frac{(a - d) \pm \sqrt{(a+d)^2 - 4(ad-bc)}}{2c}.}

Note that for parabolic transformations, which satisfy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a+d)^2 = 4(ad-bc)} , the fixed points coincide.

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=0} one of the fixed points is at infinity, the other is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma=-\frac{b}{a-d}.}

The transformation will be a simple transformation composed of translations, rotations, and dilations: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \mapsto \alpha z + \beta} .

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=d} , then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \mapsto z + \beta} .

Normal form

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, i.e. a transformation of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \mapsto k z}

with fixed points at 0 and ∞. To see this define a map

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(z) = \frac{z - \gamma_1}{z - \gamma_2}}

which sends the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\gamma_1, \gamma_2)} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,\infty)} . Here we assume that both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_2} are 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\gamma_1, \gamma_2)} then the transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gfg^{-1}} has fixed points at 0 and ∞ and is therefore a dilation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gfg^{-1}(z) = kz} . The fixed point equation for the transformation f can then be written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(z)-\gamma_1}{f(z)-\gamma_2} = k \frac{z-\gamma_1}{z-\gamma_2}.}

Solving for f gives (in matrix form):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}(k; \gamma_1, \gamma_2) = \begin{pmatrix} \gamma_1 - k\gamma_2 & (k - 1) \gamma_1\gamma_2 \\ 1 - k & k\gamma_1 - \gamma_2 \end{pmatrix}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}(k; \gamma, \infty) = \begin{pmatrix} k & (1 - k) \gamma \\ 0 & 1 \end{pmatrix}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(\gamma_1)= k\,} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(\gamma_2)= 1/k.\,}

Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (k) of f as the characteristic constant of f. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}(k; \gamma_1, \gamma_2) = \mathfrak{H}(1/k; \gamma_2, \gamma_1).}

For loxodromic transformations, whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k|>1} , one says that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1} is the repulsive fixed point, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_2} is the attractive fixed point. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k|<1} , the roles are reversed.

Parabolic case:

In the parabolic case there is only one fixed point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} . The transformation sending point to ∞ is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(z) = \frac{1}{z - \gamma}}

or the identity if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is already at infinity. The transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gfg^{-1}} fixes infinity and is therefore a translation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gfg^{-1}(z) = z + \beta\,.}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{f(z)-\gamma} = \frac{1}{z-\gamma} + \beta} .

Solving for f (in matrix form) gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}(\beta; \gamma) = \begin{pmatrix} 1+\gamma\beta & - \beta \gamma^2 \\ \beta & 1-\gamma \beta \end{pmatrix}}

or, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \infty} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}(\beta; \infty) = \begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix}}

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} is not the characteristic constant of f, which is always 1 for a parabolic transformation. From the above expressions one can calculate:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

File:Mobius Identity.jpeg

The characteristic constant can be expressed in terms of its logarithm:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\rho + \alpha i} = k \;}

When expressed in this way, the real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} becomes an expansion factor. It indicates how repulsive the fixed point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1} is, and how attractive Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_2} is. The real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1} and clockwise about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_2} .

Elliptic transformations

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0, \infty} , and with the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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):

File:Mobius Small Neg Elliptical.jpeg File:Mobius Large Neg Elliptical.jpeg
File:Mobius Small Pos Elliptical.jpeg File:Mobius Large Pos Elliptical.jpeg

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.

Hyperbolic transformations

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is zero (or a multiple of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \pi} ), 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0,\infty} , with the real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} 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):

File:Mobius Small Neg Hyperbolic.jpeg File:Mobius Large Neg Hyperbolic.jpeg
File:Mobius Small Pos Hyperbolic.jpeg File:Mobius Large Pos Hyperbolic.jpeg

It is not surprising that these pictures look very much like the field lines of bar magnets, since the circular flow lines subtend a constant angle between the two fixed points.

Loxodromic transformations

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: "loxos, slanting + 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 (ie, 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 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0, \infty} : an observer who is both rotating (with constant angular velocity) about some axis and boosting (with constant magnitude acceleration) 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0, \infty} , and with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho, \alpha} determined respectively by the magnitude of acceleration and angular velocity.

Here are some pictures illustrating the effect of a loxodromic transformation:

File:Mobius Small Loxodromic.jpeg File:Mobius Large Loxodromic.jpeg

Stereographic projection

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 to having the fixed points in an arbitrary location.

Elliptic Hyperbolic Loxodromic
One fixed point at Infinity File:Mob3d-elip-inf-200.png File:Mob3d-hyp-inf-200.png File:Mob3d-lox-inf-200.png
Full size Full size Full size
Fixed points diametrically opposite File:Mob3d-elip-opp-200.png File:Mob3d-hyp-opp-200.png File:Mob3d-lox-opp-200.png
Full size Full size Full size
Fixed points in an arbitrary location File:Mob3d-elip-arb-200.png File:Mob3d-hyp-arb-200.png File:Mob3d-lox-arb-200.png
Full size Full size Full size

Iterating a transformation

If a transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} has fixed points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1, \gamma_2} , and characteristic constant 'k, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}' = \mathfrak{H}^n} will have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

File:Mobius23621.jpeg File:Mobius23622.jpeg
File:Mobius23623.jpeg File:Mobius23624.jpeg
File:Mobius23625.jpeg


Poles of the transformation

The point

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_\infty = - \frac{d}{c}}

is called the pole of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} ; it is that point which is transformed to the point at infinity under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} .

The inverse pole

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} can be specified with two fixed points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1, \gamma_2} and the pole Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_\infty} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_\infty} given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1, \gamma_2} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_\infty = \frac{k \gamma_1 - \gamma_2}{1 - k}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_1\over \lambda_2} of the matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} }

representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{det} (\lambda I_2- \mathfrak{H}) =\lambda^2-\mbox{tr} \mathfrak{H}\,\lambda+ \mbox{det} \mathfrak{H} =\lambda^2-(a+d)\lambda+(ad-bc) }

which has roots

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_{i}=\frac{(a + d) \pm \sqrt {(a - d)^2 + 4 b c}}{2}=\frac{(a + d) \pm \sqrt {(a + d)^2 - 4(ad-b c)}}{2} \ . }

Specifying a transformation by three points

Direct approach

Any set of three points

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_1 = \mathfrak{H}(z_1), \;\; Z_2 = \mathfrak{H}(z_2), \;\; Z_3 = \mathfrak{H}(z_3)}

uniquely defines a transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} . To calculate this out, it is handy to make use of a transformation that is able to map three points onto (0,0), (1, 0) and the point at infinity.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}_1 = \begin{pmatrix} \frac{z_2 - z_3}{z_2 - z_1} & -z_1 \frac{z_2 - z_3}{z_2 - z_1} \\ 1 & -z_3 \end{pmatrix}, \;\; \mathfrak{H}_2 = \begin{pmatrix} \frac{Z_2 - Z_3}{Z_2 - Z_1} & -Z_1 \frac{Z_2 - Z_3}{Z_2 - Z_1} \\ 1 & -Z_3 \end{pmatrix} }

One can get rid of the infinities by multiplying out by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2 - z_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_2 - Z_1} as previously noted.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}_1 = \begin{pmatrix} z_2 - z_3 & z_1 z_3 - z_1 z_2 \\ z_2 - z_1 & z_1 z_3 - z_3 z_2 \end{pmatrix} , \;\; \mathfrak{H}_2 = \begin{pmatrix} Z_2 - Z_3 & Z_1 Z_3 - Z_1 Z_2 \\ Z_2 - Z_1 & Z_1 Z_3 - Z_3 Z_2 \end{pmatrix} }

The matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} to map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{1,2,3}} onto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{1,2,3}} then becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H} = \mathfrak{H}_2^{-1} \mathfrak{H}_1}

You can multiply this out, if you want, but if you are writing code then it's easier to use temporary variables for the middle terms.

Explicit determinant formula

The problem of constructing a Möbius transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}(z) } mapping a triple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z_1, z_2, z_3 )} to another triple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( w_1, w_2, w_3 )} is equivalent to finding the equation of a standard hyperbola

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, c wz -az+dw -b=0 }

in the (z,w)-plane passing through the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z_i, w_i ) } . An explicit equation can be found by evaluating the determinant

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det \begin{pmatrix} zw & z & w & 1 \\ z_1w_1 & z_1 & w_1 & 1 \\ z_2w_2 & z_2 & w_2 & 1 \\ z_3w_3 & z_3 & w_3 & 1 \end{pmatrix} \, }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\det \begin{pmatrix} z_1w_1 & w_1 & 1 \\ z_2w_2 & w_2 & 1 \\ z_3w_3 & w_3 & 1 \end{pmatrix} \, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\det \begin{pmatrix} z_1w_1 & z_1 & w_1 \\ z_2w_2 & z_2 & w_2 \\ z_3w_3 & z_3 & w_3 \end{pmatrix} \, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=\det \begin{pmatrix} z_1 & w_1 & 1 \\ z_2 & w_2 & 1 \\ z_3 & w_3 & 1 \end{pmatrix} \, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=\det \begin{pmatrix} z_1w_1 & z_1 & 1 \\ z_2w_2 & z_2 & 1 \\ z_3w_3 & z_3 & 1 \end{pmatrix} }

for the coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c,d } of the representing matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \mathfrak{H} =\begin{pmatrix} a & b \\ c & d \end{pmatrix} } . The constructed matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H} } has determinant equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 zi resp. wi are pairwise different thus the Möbius transformation is well-defined.

Remark: A similar determinant (with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle wz } replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w^2+z^2 } ) leads to the equation of a circle through three different (non collinear) points in the plane.

Alternate method using cross-ratios of point quadruples

This construction exploits the fact (mentioned in the first section) that the cross-ratio

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{cr}(z_1,z_2,z_3,z_4)= {{(z_1-z_3)(z_2-z_4)}\over{(z_1-z_4)(z_2-z_3)}} }

is invariant under a Möbius transformation mapping a quadruple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z_1,z_2,z_3,z_4) } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (w_1,w_2,w_3,w_4) } via Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_i=\mathfrak{H}z_i} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} maps a triple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z_1,z_2,z_3) } of pairwise different zi to another triple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (w_1,w_2,w_3) } , then the Möbius transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} is determined by the equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{cr}(\mathfrak{H}(z),w_1,w_2,w_3)=\mbox{cr}(z,z_1,z_2,z_3) }

or written out in concrete terms:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{(\mathfrak{H}(z)-w_2)(w_1-w_3)} \over{(\mathfrak{H}(z)-w_3)(w_1-w_2)}} ={{(z-z_2)(z_1-z_3)}\over{(z-z_3)(z_1-z_2)}}\ . }

The last equation can be transformed into

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{\mathfrak{H}(z)-w_2} \over{\mathfrak{H}(z)-w_3}} ={{(z-z_2)(w_1-w_2)(z_1-z_3)}\over{(z-z_3)(w_1-w_3)(z_1-z_2)}} \ . }

Solving this equation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}(z) } one obtains the sought transformation.

Relation to the fixed point normal form

Assume that the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2,\, z_3 } are the two (different) fixed points of the Möbius transform Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}} i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_2=z_2, \, w_3=z_3} . Write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2 =\gamma_1,\, z_3 =\gamma_2 } . The last equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{\mathfrak{H}(z)-w_2} \over{\mathfrak{H}(z)-w_3}} ={{(z-z_2)(w_1-w_2)(z_1-z_3)}\over{(z-z_3)(w_1-w_3)(z_1-z_2)}} }

then reads

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{\mathfrak{H}(z)-\gamma_1} \over{\mathfrak{H}(z)-\gamma_2}} ={{(w_1-\gamma_1)(z_1-\gamma_2)}\over {(w_1-\gamma_2)(z_1-\gamma_1)}}\cdot {{z-\gamma_1}\over {z-\gamma_2}}\ . }

In the previous section on normal form a Möbius transform with two fixed points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1, \gamma_2 } was expressed using the characteristic constant k of the transform as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{\mathfrak{H}(z)-\gamma_1} \over{\mathfrak{H}(z)-\gamma_2}} =k\,{{z-\gamma_1}\over {z-\gamma_2}}\ . }

Comparing both expressions one derives the equality

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k={{(w_1-\gamma_1)(z_1-\gamma_2)}\over {(w_1-\gamma_2)(z_1-\gamma_1)}}=\mbox{cr}(w_1,z_1,\gamma_1,\gamma_2) \ ,}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1 } is different from the fixed points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1 ,\, \gamma_2 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_1=\mathfrak{H}(z_1) } is the image of z1 under Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H} } . In particular the cross-ratio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{cr}(\mathfrak{H}(z),z,\gamma_1,\gamma_2) } does not depend on the choice of the point z (different from the two fixed points) and is equal to the characteristic constant.

See also

References

  • Hall, G. S. (2004). Symmetries and Curvature Structure in General Relativity, Singapore: World Scientific. ISBN 981-02-1051-5. See Chapter 6 for the classification, up to conjugacy, of the Lie subalgebras of the Lie algebra of the Lorentz group.
  • Needham, Tristam (1997). Visual Complex Analysis, Oxford: Clarendon Press. ISBN 0-19-853446-9. See Chapter 3 for a beautifully illustrated introduction to Möbius transformations, including their classification up to conjugacy.
  • Beardon, Alan F. (1995). The Geometry of Discrete Groups, New York: Springer-Verlag. ISBN 0-3879-0788-2.
  • Katok, Svetlana (1992). Fuchsian Groups, Chicago:University of Chicago Press. ISBN 0-226-42583-5. See Chapter 2.
  • Jones, Gareth A.; & Singerman, David (1987). Complex Functions: an Algebraic and Geometric Viewpoint, Cambridge: Cambridge Univerisy Press. ISBN 0-521-313666-X. See Chapter 2 for various isomorphisms, and for the Lorentz group viewed as a Galois group.
  • Knopp, Konrad (1952). Elements of the Theory of Functions, New York: Dover. ISBN 0-486-60154-4. See Chapters 3-5 of this classic book for a beautiful introduction to the Riemann sphere, stereographic projection, and Möbius transformations.

External links

  • A java applet allowing you to specify a transformation via its fixed points and so on can be found at [1].

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