Möbius transformation

Möbius transformations should not be confused with the Möbius transform.

Geometry

In mathematics, a Möbius transformation, also called a homographic function, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞.) It is named in honor of August Ferdinand Möbius.

The general formula is given by

<math>w = \frac{a z + b}{c z + d}<math>

almost everywhere where a, b, c, d are any complex numbers satisfying ad – bc ≠ 0. There are two special cases not covered by the formula above:

• the point <math>z = -d/c\,<math> is mapped to <math>w=\infty<math>
• the point <math>z=\infty<math> is mapped to <math>w = a/c\,<math>

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.

It can be shown that the inverse and composition of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition – called the Möbius group.

The geometric interpretation of the Möbius group is that it is the group of automorphisms of the Riemann sphere. The bilinear transform is a special case of a Möbius transformation.

Any Möbius map 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 Möbius transformation cross-ratio preservation theorem states that the cross-ratio

<math>

\frac{(w_1-w_2)(w_3-w_4)}{(w_1-w_4)(w_3-w_2)} = \frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_4)(z_3-z_2)} <math> is invariant under a Möbius transformation that maps from z to w.

Equations

The transformation

<math>w = \frac{a z + b}{c z + d}<math>

can be usefully expressed as a matrix

<math>\mathfrak{H} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}<math>

In this form, the matrix may be multiplied by any scalar λ and still represent the same transformation. This means that a Möbius transformation on C therefore has six real degrees of freedom.The matrix view of a Möbius transformation corresponds to a projectivity on the projective line over C.

Composition

Let <math>\mathfrak{H}_1, \mathfrak{H}_2<math> be two Möbius transformations:

<math>

 \mathfrak{H}_1 = \begin{pmatrix} a_1 & b_1 \\ c_1 & d_1 \end{pmatrix}, \;\;
 \mathfrak{H}_2 = \begin{pmatrix} a_2 & b_2 \\ c_2 & d_2 \end{pmatrix}<math>

If these transformations are carried out in succession, first <math>z_1 = \mathfrak{H}_1(z)<math> then <math>Z = \mathfrak{H}_2(z_1)<math> to obtain <math>Z = \mathfrak{H}_2[\mathfrak{H}_1(z)]<math>, the result can be readily seen to be another Möbius transformation <math>\mathfrak{H}_3<math> which appears as the product of the two matrices <math>\mathfrak{H}_1, \mathfrak{H}_2<math>

<math>\mathfrak{H}_3 = \mathfrak{H}_2 \mathfrak{H}_1 =

\begin{pmatrix} a_2 a_1 + b_2 c_1 & a_2 b_1 + b_2 d_1 \\ c_2 a_1 + d_2 c_1 & c_2 b_1 + d_2 d_1 \end{pmatrix} <math>

Thus, Möbius transformations form a group.

Inversion

The inverse of a Möbius transformation <math>\mathfrak{H}<math> can be derived as

<math>z = \mathfrak{H}^{-1}(Z) = \frac{d Z – b}{-c Z + a}<math>

and so

<math>\mathfrak{H}^{-1} = \begin{pmatrix} \;\;d & -b \\ -c & \;\;a \end{pmatrix}<math>

Fixed points, characteristic constant

Any Möbius transformation <math>\mathfrak{H}<math> which is not the identity mapping will have two fixed points <math>\gamma_1, \gamma_2<math>, invariant under transformation by <math>\mathfrak{H}<math>. Note that the fixed points are counted here "with multiplicity" so the fixed points may coincide. In this case the Möbius transformation is also called a parabolic transform. Either or both of these fixed points may be the point at infinity: this will happen when <math>c = 0<math>. If this is the case, then the transformation will be an affine transformation (some combination of rotation, dilation, and translation). If both fixed points are at infinity, then the transformation is a pure translation with parameters <math>a = \lambda, b=\lambda\Delta, c = 0, d=\lambda<math> i.e. the map <math> \gamma \mapsto \gamma+\Delta <math>.

Determination of fixed points

The fixed points can be derived from the fixed point equation

<math> \gamma={{a\gamma +b}\over {c\gamma +d}} <math>

as the two roots

<math>\gamma = \frac{(a – d) \pm \sqrt{(a – d)^2 + 4 c b}}{2 c}<math>

of the quadratic equation

<math>c \gamma^2 – (a – d) \gamma – b = 0 \ ,<math>

which follows from the fixed point equation by multiplying both sides with the denominator <math> c\gamma +d <math> and collecting equal powers of <math> \gamma <math>. Note that the quadratic equation degenerates into a linear equation if <math> c=0 <math>, this corresponds to the situation that one of the fixed points is the point at infinity. In this case the second fixed point is finite if <math> a-d \ne 0 <math> otherwise the point at infinity is a fixed point "with multiplicity two" (the case of a pure translation).

Construction of Möbius transformations with prescribed fixed points

Non-parabolic case:

Let us first discuss the case where the transformation has two different fixed points which are finite.

A Möbius <math>\mathfrak{H}<math> transformation is uniquely defined by its set of fixed points <math>\{\gamma_1, \gamma_2 \}<math> together with its so-called characteristic constant <math>k<math>:

<math>\mathfrak{H}(k;\gamma_1,\gamma_2) =

\begin{pmatrix}

 k \gamma_2 – \gamma_1 & (1 – k) \gamma_1     \gamma_2 \\
 k          – 1        &         \gamma_2 – k \gamma_1

\end{pmatrix} <math> The representation is unique once the two fixed points have been suitably labeled (the two matrices <math> \mathfrak{H}(k;\gamma_1,\gamma_2)<math> and <math>\mathfrak{H}({1\over k};\gamma_2,\gamma_1) <math> define the same Möbius transformation). All transformations with the same characteristic constant are similar. The above transform can be written in a normalized form by multiplying each entry with the (non-zero) scalar <math> c={{1}\over{\gamma_2-\gamma_1}}<math>. The resulting matrix <math>{\mathfrak{H}}' <math> representing the same Möbius transformation has matrix trace equal to k+1 and determinant equal to k. This implies that the characteristic polynomial of the matrix <math>{\mathfrak{H}}' <math> has roots <math> \lambda_1, \, \lambda_2 <math> equal to <math>\,\lambda_1=k <math>, <math>\, \lambda_2=1 <math>. Thus the characteristic constant k coincides with one of the two different ratios of eigenvalues <math> \lambda_2 \over \lambda_1 <math> of the matrix representing the transformation (note that each ratio is invariant under multiplication of matrices with non-zero scalars). This property shows that the characteristic constant is preserved under a similarity transformation (corresponding to a conjugation of the matrix representing the transformation).

Every transformation having two different fixed points is similar to some particular linear transformation having one fixed point at infinity and another at 0, i.e a map <math> \gamma \mapsto \lambda \gamma <math>. In the degenerate case k = 1 the transform defined above is the identity transform.

Parabolic case:

If there is only one (finite) fixed point <math> \gamma <math> then the Möbius transformation is of the form

<math>\mathfrak{H} =

\begin{pmatrix}

  1+\gamma k    & – k \gamma^2  \\
  k             &   1-\gamma k 

\end{pmatrix} <math> where k plays the role of the characteristic constant.

Geometric interpretation of the characteristic constant

The following picture depicts the two fixed points of a Möbius transformation in the non-parabolic case:

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

<math>e^{\rho + \alpha i} = k \;<math>

When expressed in this way, <math>\rho<math> becomes an expansion factor. It indicates how repulsive the fixed point <math>\gamma_1<math> is, and how attractive <math>\gamma_2<math> is. If <math>\rho = 0<math>, 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, the this is equivalent to doing an affine rotation around a point.

<math>\alpha<math> is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about <math>\gamma_1<math> and clockwise about <math>\gamma_2<math>. If <math>\alpha<math> is zero (or a multiple of <math>2 \pi<math>), 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 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.

Iterating a transformation

If a transformation <math>\mathfrak{H}<math> has fixed points <math>\gamma_1, \gamma_2<math>, and expansion and rotation factors <math>\rho<math> and <math>\alpha<math>, then <math>\mathfrak{H}' = \mathfrak{H}^n<math> will have <math>\gamma_1' = \gamma_1, \gamma_2' = \gamma_2, \rho' = n\rho, \alpha' = n\alpha<math>.

This can be used to continuously iterate a transformation.

These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants

Poles of the transformation

The point

<math>z_\infty = – \frac{d}{c}<math>

is called the pole of <math>\mathfrak{H}<math>; it is that point which is transformed to the point at infinity under <math>\mathfrak{H}<math>.

The inverse pole

<math>Z_\infty = \frac{a}{c}<math>

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:

<math>\gamma_1 + \gamma_2 = z_\infty + Z_\infty<math>

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

A transform <math>\mathfrak{H}<math> can be specified with two fixed points <math>\gamma_1, \gamma_2<math> and the pole <math>z_\infty<math>.

<math>\mathfrak{H} =

\begin{pmatrix}

 Z_\infty & – \gamma_1 \gamma_2 \\
 1        & – z_\infty

\end{pmatrix}, \;\;

 Z_\infty = \gamma_1 + \gamma_2 – z_\infty

<math>

This allows us to derive a formula for conversion between <math>k<math> and <math>z_\infty<math> given <math>\gamma_1, \gamma_2<math>:

<math>z_\infty = \frac{k \gamma_1 – \gamma_2}{1 – k}<math>

<math>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}<math>

Which reduces down to

<math>k = \frac{(a + d) + \sqrt {(a – d)^2 + 4 b c}}{(a + d) – \sqrt {(a – d)^2 + 4 b c}}<math>

The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios <math> \lambda_1\over \lambda_2<math> of the matrix

<math>\mathfrak{H} =

\begin{pmatrix}

 a & b \\
 c & d

\end{pmatrix} <math> representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to

<math>

\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) <math> which has roots

<math> \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} \ .

<math>

Specifying a transformation by three points

Direct approach

Any set of three points

<math>

 Z_1 = \mathfrak{H}(z_1), \;\;
 Z_2 = \mathfrak{H}(z_2), \;\;
 Z_3 = \mathfrak{H}(z_3)<math> 

uniquely defines a transformation <math>\mathfrak{H}<math>. 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.

<math>\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} <math>

One can get rid of the infinities by multiplying out by <math>z_2 – z_1<math> and <math>Z_2 – Z_1<math> as previously noted.

<math>\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} <math>

The matrix <math>\mathfrak{H}<math> to map <math>z_{1,2,3}<math> onto <math>Z_{1,2,3}<math> then becomes

<math>\mathfrak{H} = \mathfrak{H}_2^{-1} \mathfrak{H}_1<math>

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.

Alternate method using cross-ratios of point quadruples

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

<math>

\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)}} <math> is invariant under a Möbius transformation mapping a quadruple <math> (z_1,z_2,z_3,z_4) <math> to <math> (w_1,w_2,w_3,w_4) <math> via <math>w_i=\mathfrak{H}z_i<math>. If <math>\mathfrak{H}<math> maps a triple <math> (z_1,z_2,z_3) <math> of pairwise different z_{i to another triple <math> (w_1,w_2,w_3) <math>, then the Möbius transformation <math>\mathfrak{H}<math> is determined by the equation}

<math>

\mbox{cr}(\mathfrak{H}(z),w_1,w_2,w_3)=\mbox{cr}(z,z_1,z_2,z_3) <math> or written out in concrete terms:

<math>

{{(\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)}}\ . <math>

The last equation can be transformed into

<math>

{{\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)}} \ . <math> Solving this equation for <math> \mathfrak{H}(z) <math> one obtains the sought transformation.

References

Not to be confused with:

• Möbius transform
• Möbius function

See also

• Fuchsian group
• Hyperbolic geometry
• Inversive ring geometry
• Kleinian group
• Modular group
• Poincaré half-plane model
• Projective geometry

External link

A java applet allowing you to specify a transformation via its fixed points and so on may be found here.

This page contains material from this article and this article at PlanetMath, used under the GFDL by permission.