Ohrimenko+ / Barnsley. Superfractals

2.6 Mobius¨ transformations

Mobius¨ transformations are specified by eight real parameters. They are geometrically simple and cheap to describe, communicate and compute. Small sets of them may be used to represent apparently complex images, as we will show in Chapters 4 and 5. So here we start to explain what they are, how they act on points, sets, pictures and measures and what sorts of sets, pictures and measures are invariant under them.

Definition of a Mobius¨ transformation

Mobius¨ transformations have the quite extraordinary property that they map the set of all circles and straight lines onto the set of all circles and straight lines while, typically, substantially distorting other shapes. In addition, they preserve angles and the orientation of angles.

Various Mobius¨ transformations applied to a picture of a cyclist riding a bike are illustrated in Figure 2.31. Notice how the rims of the wheels are all nearly circular and how corresponding angles in the bike frames are all the same. But the tyres themselves are distorted and the relative sizes of the two wheels vary from bike to bike. Also, the straight lines in the bike frame are mapped onto arcs of circles.

Some other illustrations are shown in Figure 2.2, where the three large fish are each related to the small fish by a Mobius¨ transformation. Notice how the eyes of all the fish are round, how angles are preserved and how different yet fish-like all the fish look. See also Figure 2.32.

A Mobius¨ transformation is a mapping M : R2 → R2, where R2 = R2 {∞} denotes the extended real plane and ∞ is called the point at infinity. Both the domain and the range of a Mobius¨ transformation include ∞ because, as we explain in the next subsection, this point can be handled in a consistent manner, resulting in a continuous, one-to-one, onto, invertible transformation. A Mobius¨ transformation may be represented by a formula such as

This maps the unit circle C centred at the origin O = (0, 0) onto itself, maps the interior D of the unit disk centred at O onto the region outside C, maps O to ∞ and ∞ to O and involves both a reflection in the y-axis and a rotation about O. The behaviour at O and ∞ may be deduced by using continuity and taking limits.

The most general Mobius¨ transformation M : R2 → R2 may be expressed in terms of eight real parameters aR , aI , bR , bI , cR , cI , dR , dI R2, which are constrained only by the condition that

either aR dR − aI dI − bR cR + bI cI = 0 or aI dR + aR dI − bR cI − bI cR = 0. (2.6.2)

Figure 2.30 This illustrates the ‘move-three-points’ algorithm, which is defined as follows. (i) Identify a pair of points A P and A P . (ii) Apply to P the unique translation T such that T ( A ) = A . Identify a second pair of points B T (P) and B P . (iii) Apply to T (P) the unique similitude S such that S( A ) = A and S(B ) = B . Identify a third pair of points C S ◦ T (P) and C P . (iv) Apply to S ◦ T (P) the unique shear transformation F such that F( A ) = A , F(B ) = B and F(C ) = C .

The general formula is

where

A(x, y) = (aR x − aI y + bR )(cR x − cI y + dR ) + (aR y + aI x + bI )(cI x + cR y + dI ),

B(x, y) = (aR y + aI x + bI )(cR x − cI y + dR ) − (aR x − aI y + bR )(cI x + cR y + dI )

and

C(x, y) = (cR x − cI y + dR )2 + (cR y + cI x + dI )2.

142 Transformations of points, sets, pictures and measures

In order to evaluate expressions where both the numerator and denominator may vanish, limits must be taken. But these formulas are best handled using complex notation.

In this representation we have M(∞) = a/c and M(−d/c) = ∞ if c =0, and M(∞) = ∞ if c = 0.

E x e r c i s e 2.6.1 Verify that Equations (2.6.3) and (2.6.4) are equivalent.

With the aid of Equation (2.6.4) it is readily verified that the composition of two Mobius¨ transformations is also a Mobius¨ transformation; indeed,

(a1a2 + b1c2)z + (a1b2 + b1d2)

f1 ◦ f2(z) = (c1a2 + d1c2)z + (c1b2 + d1d2) . (2.6.5) Does this look familiar? Compare it with Equation (2.5.3). This means that we can use the matrix operations of complex 2 × 2 matrices to compose and invert Mobius¨ transformations.

E x e r c i s e 2.6.2 Write down the Mobius¨ transformation M in Equation (2.6.1) in complex notation. Then use matrix operations to find formulas for M−1 and

M ◦ M.

It is easy for you to check that if c =0 then the Mobius¨ transformation in Equation (2.6.4) can be written in the form

Both M1 and M3 are similitudes, which map the set of all generalized circles, namely the set of circles and straight lines, onto itself. The transformation M2 is an inversion that also maps the set of generalized circles onto itself, as we now show.

Figure 2.31 Various Mobius transformations have been applied to a picture of a person on a bike. What properties do all of the resulting bikes have in common?

for some pair of points z0, z1 C and some γ > 0, as in Exercise 2.6.3 below.

simplicity.

E x e r c i s e 2.6.3 Verify that any generalized circle C C can be expressed as in Equation (2.6.6). Hint: (x − x0)2 + (y − y0)2 = γ ((x − x1)2 + (y − y1)2).

The Riemann sphere

In order to understand how a Mobius¨ transformation handles ∞ it is natural to model R2 := R2 {∞}, or equivalently C = C {∞}, as the surface S R3

Figure 2.32 Various Mobius¨ transformations have been applied to a picture of a puffer fish. Draw a circle through any three distinctive points on one fish, and another circle through the corresponding points on a second fish. Then if one of the circles goes through another distinctive point on the first fish, the second circle will go through the corresponding point on the second fish.

of a sphere of radius 1 centred at (0, 0) R2. The surface S is also called the Riemann sphere. The mapping between the plane and the sphere is achieved by stereographic projection; see Figure 2.33. The projection mapping f : R2 → S is readily found to be given by

The point at infinity is mapped to the top of the sphere, N = (0, 0, 1). All circles and straight lines in R2 correspond to circles on S. Note that a circle on S is the

N

f

Q

O

Equator

P

P lies in the plane of the equator

Figure 2.33 Illustration of a stereographic projection f : R2 → S between the extended plane and the surface of a sphere. The point P in the plane of the equator is mapped to the point Q where the straight line from P to the north pole N first meets S. Circles and straight lines in the extended plane are mapped to circles on the sphere, and vice versa.

intersection of S with a plane in R3 that meets S in at least two points. Circles onS that go through N = (0, 0, 1) correspond to straight lines in C.

If we consider Mobius¨ transformations acting on the sphere in place of the plane, we find that the point at infinity behaves exactly like all the other points on the sphere. Any Mobius¨ transformation f ◦ M ◦ f −1 maps the sphere to itself in a one-to-one-onto continuous manner and can be expressed as a composition of rotations of the sphere and certain rescalings, corresponding to similitudes in the plane, each of which maps circles on the sphere to circles on the sphere. For example, the inversion M2(z) = 1/z becomes simply a rotation of the sphere through 180◦ about the x-axis, i.e.

It is easy to see that the most general Mobius¨ transformation for which ∞ is a fixed point is a similitude with a proper rotation; that is, it can be written in the form

M(z) = λeiθ z + t

for λ > 0, θ [0, 2π ) and t = ( f + ig) C, which is equivalent to Equation (2.5.6). For example, when λ > 1 this transformation has two distinct fixed points, one of which is ∞, and it corresponds to a rotation of the sphere about the z-axis composed with a motion away from the south pole, following longitudinal great circles, towards the north pole. Indeed, with t = 0 and λ > 1, if we make the change of coordinates provided by the inversion that interchanges 0 and ∞, the

transformation becomes

M&(z) = M2 ◦ M ◦ M−2 1(z) = λ−1e−iθ z,

which is just like the original transformation except that λ is replaced by λ−1 and the direction of rotation is reversed.

A Mobius¨ transformation that possesses two distinct fixed points either rotates points close to the fixed points in opposite directions, as illustrated in Figure 2.38, in which case it is called loxodromic, or else it does not rotate space about either fixed point. In the latter case it either expands points away from one fixed point and towards the other along arcs of generalized circles, in which case it is called hyperbolic, or else it is the identity transformation M(z) = z.

The only other possible type of Mobius¨ transformation is parabolic and possesses only one fixed point. This fixed point may be thought of as a limiting case of a family of hyperbolic transformations in which the two fixed points coalesce. As a consequence, a parabolic Mobius¨ transformation behaves in a remarkable manner: some points are repelled and some are attracted by its fixed point.

Specifically, a parabolic Mobius¨ transformation maps each circle tangent to a certain fixed line, through the fixed point, onto itself. Points are swept, along these circles, away from the fixed point on one side and towards it on the other side; the direction of this circling motion is clockwise on circles lying on one side of the fixed line and counterclockwise on circles lying on the other side.

An example of a parabolic transformation acting on a picture within a disk is illustrated in Figure 2.34; in this case the fixed point is at the top of the disk and the fixed line is tangent to the disk. Notice how colourful picture matter is maintained within each crescent, swept away from one side of the fixed point towards the other.

E x e r c i s e 2.6.4 Verify Equation (2.6.9).

We can define a metric dRiemann on R2 {∞}, or equivalently the Riemann sphere S, by

dRiemann((x1, y1), (x2, y2))

= shortest distance between f (x1, y1) and f (x2, y2) on S,

where f is given in Equation (2.6.8). Then (R2 {∞}, dRiemann) is a compact metric space. The natural topology associated with dRiemann is such that any Mobius¨ transformation M : R2 {∞} → R2 {∞} is continuous.

Figure 2.34 Illustration of a parabolic Mobius¨ transformation acting on a picture. The unique fixed point is at the top of the disk. Colourful picture material within each crescent, defined by adjacent pairs of circles, is swept round while staying within its allotted crescent. You should study carefully the two pictures, ‘before’ and ‘after’, to be sure you confirm this effect. The fixed point is both repulsive and attractive.

Fundamental theorem of Mobius¨ transformations

T h e o r e m 2.6.5 Let z1, z2, z3 and w1, w2, w3 be two sets of distinct points in the extended complex plane C = C {∞}. Then there exists a unique Mobius¨ transformation that maps z1 to w1, z2 to w2 and z3 to w3.

P r o o f This is a good exercise. Hint: Start by choosing z1 = 0, z2 = 1 and z3 = i. See [25], p. 242.

One consequence of Theorem 2.6.5 is that there are many Mobius¨ transformations that map any given generalized circle to another given generalized circle. In particular, there are many Mobius¨ transformations that map the circle C = {z C : |z| = 1} onto itself. Indeed, they are given by

where a C, with a =1, and 0 ≤ θ < 2π . It is readily verified that Ma takes three distinct points on C, such as 1, i and −1, to points on C. Notice that Ma,θ (0) = −aeiθ , so that Ma,θ maps the interior of the circle C to itself when |a| < 1 but turns the circle ‘inside out’ when |a| > 1; examples of this type of transformation applied to a flower picture are shown in Figure 2.35 and to a fish measure in Figure 2.14. For a =0, the fixed points of Ma,θ (z) lie on the circle |z| = 1. For 0 < |a| ≤ 1 and a =1, each member of this family of transformations is either parabolic or hyperbolic.

Figure 2.35 A Mobius¨ transformation of the form given in Equation (2.6.10) has been applied to the picture on the left. Notice the big buds and the curved stems in the transformed picture on the right.

Another interesting family of Mobius¨ transformations is given by

where ρ C with ρ =0, 1. Mρ (z) has two distinct fixed points, z = 0 and z = 1. It behaves like the similitude ρz near z = 0 and like the similitude 1 + ρ−1(1 − z) near z = 1: if we rotate the coordinates through 180◦ about the point halfway between the two fixed points, by means of the transformation t(z) = 1 − z, we find that

M1/ρ (z) = t ◦ Mρ ◦ t−1(z).

The transformations of this family are always either loxodromic or hyperbolic. Two examples of the transformation in Equation (2.6.11) applied to the circular

picture containing a flower in the left-hand panel of Figure 2.35 are shown in Figure 2.36. For the right-hand panel in Figure 2.36, ρ = −0.35 − 0.1i and both the domain and the visible part of the range correspond to {x + iy : −1 ≤ x, y ≤ +1}. The white disk is the image of the exterior of the original disk. For the lefthand panel in Figure 2.36, ρ = 0.3 − 0.2i and both the domain and the visible part of the range correspond to {x + i y : −2 ≤ x, y ≤ +2}.

E x e r c i s e 2.6.6 Find the unique Mobius¨ transformation M(z) that maps ∞ to 1, 0 to −i and −1 to −1. Show that this transformation maps the upper half-plane to the interior of the circle of radius 1 centred at z = 0.

Figure 2.36 Mobius¨ transformations of the form given in Equation (2.6.11) have been applied to the left-hand picture in Figure 2.35. In each of these transformed pictures the disk containing the flower has been inverted and one of the blue petals has been stretched out to infinity.

Invariant points, sets, measures and pictures for Mobius¨ transformations

Sets, pictures and measures which are invariant under certain Mobius¨ transformations that are essentially rotations are illustrated in Figures 2.6 and 2.17–2.20.

Another type of invariant set is illustrated in Figure 2.37 and is associated with a transformation of the form in Equation (2.6.11): the invariance occurs because of an underlying group structure, to be explained in Chapter 3. This example illustrates clearly that although Mobius¨ transformations map generalized circles into generalized circles, they do not preserve ellipses! A similar type of invariant picture is shown in Figure 2.38. Both these examples are interesting because the only associated invariant measures consist of point masses at the centres of the two spirals. These centres are the fixed points of the transformations. The transformations sweep all other finite measures along spiral paths away from one fixed point and in towards the other.

2.7Projective transformations

Projective transformations in two dimensions are specified by nine real parameters. They are geometrically simple and may be efficiently described, communicated and computed. Small sets of them may be used to represent apparently complex images, as we will show in Chapters 4 and 5. So in this section we start to explain what they are and how they transform sets, pictures and measures. What sorts of sets, pictures and measures do they leave invariant?

We begin straight away by introducing projective transformations informally. Examples of projective transformations are illustrated in Figures 2.39 and 2.40.