3.2: Möbius Geometry
( \newcommand{\kernel}{\mathrm{null}\,}\)
Möbius geometry provides a unifying framework for studying planar geometries. In particular, the transformation groups of hyperbolic and elliptic geometries in the sections that follow are subgroups of the group of Möbius transformations.
Möbius transformations
A Möbius transformation (also called a linear fractional transformation) of the extended complex plane ˆC is a function of the form
f(z)=az+bcz+d
where z is a complex variable, a,b,c,d are complex constants, and ad−bc≠0. To complete the definition, we make the assignments f(−d/c)=∞ and f(∞)=a/c if c≠0. If c=0, we assign f(∞)=∞.
Note on notational convention: It is customary to use capital letters such as S,T,U to denote Möbius transformations. It is also customary to omit the parentheses, and to write Tz instead of T(z) to denote the value of a Möbius transformation.
- Checkpoint 3.2.1.
-
Let f(x)=(ax+b)/(cx+d) be a function of a real variable x with real constants a,b,c,d with ad−bc≠0 and c≠0.
- Show that limx→−d/c|f(x)|=∞.
- Show that limx→∞f(x)=a/c.
- Checkpoint 3.2.2.
-
Show that the condition ad−bc≠0 is necessary and sufficient for invertibility. Find a formula for the inverse of z→(az+b)/(cz+d).
- Checkpoint 3.2.3.
-
Show that the composition of two Möbius transformations is a Möbius transformation. Suggestion: First show that the composition has the form z→rz+stz+u. Next, instead of a brute force calculation to check that ru−ts≠0, use Checkpoint 3.2.2.
Definition 3.2.4.
The set of all Möbius transformations forms a group M, called the Möbius group , under the operation of function composition. Möbius geometry is the pair (ˆC,M).
There is a natural relationship between Möbius group operations and matrix group operations. The map T:GL(2,C)→M be given by
[abcd]→[z→az+bcz+d]
is a group homomorphism. The kernel of T is the group of nonzero scalar matrices.kerT={[k00k],k≠0}.
The quotient group GL(2,C)/kerT is called the projective linear group PGL(2,C). Thus we have a group isomorphismPGL(2,C)≈M.
- Checkpoint 3.2.5.
-
Show that the map T is a group homomorphism. Show that the kernel of T is
kerT={[k00k],k≠0}.
Homotheties, rotations, translations, and inversions (see Table 3.1.4 in Section 3.1) are special cases of Möbius transformations. These basic transformations can be viewed as building blocks for general Möbius transformations, as follows.
Proposition 3.2.6.
Every Möbius transformation is a composition of homotheties, rotations, translations, and inversions.
- Proof.
-
See Exercise 3.2.6.1
- Checkpoint 3.2.7.
-
- Identify the values of the coefficients a,b,c,d in a Möbius transformation z→az+bcz+d that is a homothety, rotation, translation, and inversion, respectively.
- Write a Möbius transformation that does "clockwise rotation by one-quarter rotation about the point 2−i".
Corollary 3.2.8.
Möbius transformations are conformal.
- Proof.
-
Apply Proposition 3.2.6 and Exercise Group 3.1.4.2–5.
Next, a simple observation, in the form of the following Lemma, leads to a result called the Fundamental Theorem of Möbius Geometry.
Lemma 3.2.9.
If a Möbius transformation has more than two fixed points, then it is the identity transformation.
- Proof.
-
Hint: just solve z=az+bcz+d. You will need to consider cases.
Proposition 3.2.10. The Fundamental Theorem of Möbius Geometry.
A Möbius transformation is completely determined by any three input-output pairs. This means that for any triple of distinct input values z1,z2,z3 in ˆC and any triple of distinct output values w1,w2,w3 in ˆC, there is a unique T∈M such that Tzi=wi for i=1,2,3.
- Proof.
-
Outline: Suppose there are two such transformations, S and T. Show that S∘T−1 fixes three points. Now apply the previous Lemma.
Cross ratio
In what follows, we consider the special case of the output triple w1=1, w2=0, w3=∞. Given three distinct points z1,z2,z3 in ˆC, we write (⋅,z1,z2,z3) to denote the unique Möbius transformation that satisfies z1→1, z2→0, and z3→∞. We write (z0,z1,z2,z3) to denote the image of z0 under (⋅,z1,z2,z3). The expression (z0,z1,z2,z3) is called the cross ratio of the 4-tuple z0,z1,z2,z3. The next two propositions give important properties of cross ratio.
Proposition 3.2.11. Cross ratio is invariant.
Let z1,z2,z3 be distinct points in ˆC, let z0∈ˆC, and let T be any Möbius transformation. Then we have
(z0,z1,z2,z3)=(Tz0,Tz1,Tz2,Tz3).
- Proof.
-
The transformations (⋅,z1,z2,z3) and (⋅,Tz1,Tz2,Tz3)∘T both send z1→1, z2→0, and z3→∞, so they must be equal, by the Fundamental Theorem. Now apply both transformations to z0.
Proposition 3.2.12.
Let z1,z2,z3 be distinct points in ˆC, let T=(⋅,z1,z2,z3), and let CT=T−1(ˆR) be the inverse image of the extended real line ˆR=R∪{∞} under T. Then CT is a Euclidean circle or straight line. Furthermore, CT is the unique Euclidean circle or straight line that contains the points z1,z2,z3.
- Proof.
-
See Exercise 3.2.6.4
Corollary 3.2.13.
The cross ratio (z0,z1,z2,z3) is real if and only if z0,z1,z2,z3 lie on a Euclidean circle or straight line.
Corollary 3.2.14.
Let C be a Euclidean circle or straight line in ˆC and let T be any Möbius transformation. Then T(C) is a Euclidean circle or straight line.
A Euclidean circle or straight line is called a cline (pronounced "kline") or generalized circle. The propositions and corollaries above show that the set of all clines is a single congruence class of figures in Möbius geometry.
Symmetry with respect to a cline
Geometrically, the conjugation map z→z∗ in the complex plane is reflection across the real line. This "mirror" symmetry generalizes to symmetry with respect to any cline, as follows. Given a cline C that contains z1,z2,z3 in ˆC, let T=(⋅,z1,z2,z3). Given any point z, the symmetric point with respect to C is
z∗C=(T−1∘conj∘T)(z)
where conj:ˆC→ˆC is the extension of the conjugation map to the extended complex plane that sends ∞→∞∗=∞. The idea is to map C to the real line via T, then conjugate, then map the real line back to C. See Figure 3.2.15.
The definition of symmetric point depends only on the circle C, and not on the three points z1,z2,z3. This fact is a corollary of the following Proposition.
Proposition 3.2.16.
Let C be a cline and let S be a Möbius transformation. If z,z′ are a pair of points that are symmetric with respect to C, then Sz,Sz′ are symmetric with respect to the cline S(C). That is, we have
(Sz)∗S(C)=S(z∗C).
- Proof.
-
Let z1,z2,z3 be three points on C, so that Sz1,Sz2,Sz3 are three points on S(C). Let T=(⋅,z1,z2,z3) and let U=(⋅,Sz1,Sz2,Sz3). By invariance of the cross ratio, we have
(U∘S)z=Tz.
Thus we have(Sz)∗S(C)=(U−1∘conj∘U)(Sz)(by definition)=(S∘S−1∘U−1∘conj∘U∘S)(z)=S(S−1∘U−1∘conj∘U∘S)(z)=S(T−1∘conj∘T)(z)=S(z∗C)
as desired.
Corollary 3.2.17.
The definition of z∗C depends only on the circle C, and not on the three points z1,z2,z3 used in the definition (3.2.3).
- Proof.
-
See Exercise 3.2.6.6.
Normal forms
We conclude this section on Möbius geometry with a discussion of the normal form of a Möbius transformation. We begin with a Lemma.
Lemma 3.2.18.
If a Möbius transformation has exactly two fixed points 0 and ∞, then it has the form z→αz for some nonzero α∈C. If a Möbius transformation has a single fixed point at ∞, then it has the form z→z+β for some nonzero β∈C.
- Proof.
-
See Exercise 3.2.6.5.
Tz−pTz−q=αz−pz−q
The transformation T is called elliptic, hyperbolic, or loxodromic if U is a rotation (|α|=1), a homothety (α>0), or neither, respectively.
Finally, suppose that a Möbius transformation T has exactly one fixed point at p. Let S be given by Sz=1z−p. Again, let w=Sz and let U=S∘T∘S−1. This time, U has exactly one fixed point at ∞. Applying the Lemma, we have Uw=w+β for some nonzero β∈C. Applying both sides of S∘T=U∘S to z, we have the following normal form for T.
1Tz−p=1z−p+β
A Möbius transformation of this type is called parabolic. Here is a summary of the classification terminology associated with normal forms.
Table 3.2.19. Summary of normal forms of T∈M
normal form type | normal form | conjugate transformation type |
elliptic | Tz−pTz−q=αz−pz−q,|α|=1 | rotation |
hyperbolic | Tz−pTz−q=αz−pz−q,α>0 | homothety |
loxodromic |
Tz−pTz−q=αz−pz−q,α≠0 |α|≠1,α≯ |
composition of homothety with rotation |
parabolic | \frac{1}{Tz-p}= \frac{1}{z-p} + \beta, \beta\neq 0 | translation |
Steiner circles
Figure 3.2.20. The polar coordinate grid and Steiner circle coordinate grid
Figure 3.2.21. Degenerate coordinate grid lines and degenerate Steiner circles
The discussion of normal forms show that any non-identity Möbius transformation is conjugate to one of two basic forms, w\to \alpha w or w\to w+\beta\text{.} The natural coordinate system for depicting the action of w\to \alpha w is standard polar coordinates. See Figure 3.2.20. A homothety is a flow along radial lines and a rotation is a flow around polar circles. The natural "degenerate" coordinate system for depicting a translation w\to w+\beta is a family of lines parallel to the line that contains the origin and \beta\text{.} A translation by \beta is a flow along these parallel lines. See Figure 3.2.21.
Pulling the polar and degenerate coordinate grids back to the z-plane by S^{-1} leads to coordinate grids called Steiner circles. 1 In the case where T has two fixed points p,q\text{,} the conjugating map Sz=\frac{z-p}{z-q} takes p\to 0\text{,} q\to \infty\text{.} Therefore S^{-1} maps 0\to p and \infty \to q\text{.} The transformation S^{-1} maps radial lines in the w-plane to clines in the z-plane that contain p and q called Steiner circles of the first kind and S^{-1} maps polar circles in the w-plane to clines in the z-plane called Steiner circles of the second kind or circles of Apollonius. See Figure 3.2.20.In the case where T has one fixed point p\text{,} the conjugating map Sz=\frac{1}{z-p} sends p\to \infty\text{,} so S^{-1} maps \infty \to p\text{,} and S^{-1} maps lines in the w-plane that are parallel to the line through 0 and \beta to clines in the z-plane that contain p\text{.} Every cline in this family is tangent to every other cline in this family at exactly the one point p\text{.} Clines in this family are called degenerate Steiner circles. See Figure 3.2.21. Table 3.2.22 summarizes the graphical depiction of Möbius transformations.
Table 3.2.22. Summary of Steiner circle pictures of Möbius transformations
normal form type | graphical dynamic |
elliptic | flow along Steiner circles of the second kind |
hyperbolic | flow along Steiner circles of the first kind |
loxodromic | composition of elliptic and hyperbolic flows |
parabolic | flow along degenerate Steiner circles |
Exercises
Exercise 1
Decomposition of Möbius transformations into four basic types.
- Explain why a transformation of the form z\to az\text{,} with a any nonzero complex constant, is a composition of a homothety and a rotation.
- Explicitly identify each homothety, rotation, translation, and inversion in (3.2.6) to (3.2.9) in the derivation below for the case c\neq 0\text{.}
\begin{align} z & \to cz+d\label{mobiusdecompfirst}\tag{3.2.6}\\ &\to \frac{1}{cz+d}\tag{3.2.7}\\ &\to \frac{bc-ad}{cz+d} + a\tag{3.2.8}\\ &\to \frac{1}{c}\left(\frac{bc-ad}{cz+d} + a\right)\label{mobiusdecomplast}\tag{3.2.9}\\ & = \frac{az+b}{cz+d}\tag{3.2.10} \end{align}
- Write your own decomposition for the case c=0\text{.}
Exercise 2
Explicit form for the transformation (⋅,z1,z2,z3).
- Show that, for the special case when z_1,z_2,z_3 are complex (that is, none of the three points is \infty), we have the following.
\begin{align} (\cdot,z_1,z_2,z_3) &= \left[z\to \frac{z-z_2}{z-z_3}\frac{z_1-z_3}{z_1-z_2}\right]\label{mob10inftyform}\tag{3.2.11}\\ (z_0,z_1,z_2,z_3) &= \frac{z_0-z_2}{z_0-z_3}\frac{z_1-z_3}{z_1-z_2}\label{crossratiofracform}\tag{3.2.12} \end{align}
- Find explicit formulas for (\cdot,z_1,z_2,z_3) when z_1=\infty\text{,} then do the same for z_2=\infty and z_3=\infty\text{.}
Exercise 3
Find Möbius transformations that make the following assignments.
- \displaystyle 1\to a, 0\to b, \infty\to c
- \displaystyle a\to d, b\to e, c\to f
Exercise 4
Prove Proposition 3.2.12. Suggestion: Let Tz=\frac{az+b}{cz+d}\text{,} then manipulate Tz=(Tz)^\ast to an equation with |z|^2,z,z^\ast terms and coefficients involving a,b,c,d and their conjugates. Then use "complex completing the square" (see (1.1.24)). The hint below shows a version of a circle equation; peek if you need to, and use it to work partially forwards from Tz=(Tz)^\ast\text{,} and partially backwards from the equation in the hint.
- Hint
-
\left| z-\left(\frac{a^\ast d-bc^\ast}{ac^\ast -a^\ast c}\right)\right|^2 = \left|\frac{ad-bc}{ac^\ast -a^\ast c}\right|^2
Exercise 5
Prove Lemma 3.2.18.
Exercise 6
Symmetry with respect to a cline.
- Prove Corollary 3.2.17.
- Let C be the unit circle. Show that z^{\ast C} = 1/z^\ast\text{.} Suggestion: This is just a computation, but the choice of z_1,z_2,z_3 might make it more or less tedious. You might try \omega,\omega^2,\omega^3=1\text{,} where \omega=e^{2\pi i/3}\text{.}
- Now let C be a circle with center a and radius r\gt 0\text{.} Use Proposition 3.2.16 to show that z^{\ast C} = \frac{r^2}{(z-a)^\ast}+a\text{.}
- Let C be a straight line. Show that z^{\ast C} is the reflection of z across C\text{.}
Normal forms and Steiner circles.
Exercise 7
Find the normal form and sketch a graph using Steiner circles for the following transformations.
- \displaystyle z\to \frac{1}{z}
- \displaystyle z\to \frac{3z-1}{z+1}
Exercise 8
Let p be the single fixed point of a Möbius transformation that is conjugate to w\to w+\beta via Sz=\frac{1}{z-p}\text{.} Show that the single line in the degenerate Steiner clines through p is parallel to the direction given by \beta^\ast\text{.}
- Hint
-
Show that S^{-1}w=\frac{pw+1}{w}\text{,} so S^{-1} takes 0,\beta,\infty to \infty,\frac{p\beta+1}{\beta},p\text{.} Thus the single degenerate Steiner straight line through p is in the direction given by \frac{p\beta+1}{\beta}-p =\frac{1}{\beta}\propto \beta^\ast\text{.}
Exercise 9
Show that a (generalized) circle of Apollonius (a Steiner circle of the second kind) is characterized as the set of points of the form
\begin{equation*} C=\left\{P\in \mathbb{C}\colon \frac{d(P,A)}{d(P,B)}=k\right\} \end{equation*}
for some A,B\in \mathbb{C} and some real constant k\gt 0\text{.}