3.2: Möbius Geometry

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 \(\hat{\mathbb{C}}\) is a function of the form

\begin{equation}
f(z)= \frac{az+b}{cz+d}\label{stdlinfractransf}\tag{3.2.1}
\end{equation}

where \(z\) is a complex variable, \(a,b,c,d\) are complex constants, and \(ad-bc\neq 0\text{.}\) To complete the definition, we make the assignments \(f(-d/c)=\infty\) and \(f(\infty)=a/c\) if \(c\neq 0\text{.}\) If \(c=0\text{,}\) we assign \(f(\infty)=\infty\text{.}\)

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 \(ad-bc\neq 0\) and \(c\neq 0\text{.}\)

1. Show that \(\displaystyle \lim_{x\to -d/c}|f(x)|=\infty\text{.}\)
2. Show that \(\displaystyle \lim_{x\to \infty}f(x)=a/c\text{.}\)

Checkpoint 3.2.2.

Show that the condition \(ad-bc\neq 0\) is necessary and sufficient for invertibility. Find a formula for the inverse of \(z\to (az+b)/(cz+d)\text{.}\)

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 \(z\to \frac{rz+s}{tz+u}\text{.}\) Next, instead of a brute force calculation to check that \(ru-ts\neq 0\text{,}\) use Checkpoint 3.2.2.

Definition 3.2.4.

The set of all Möbius transformations forms a group \(\mathbf{M}\text{,}\) called the Möbius group , under the operation of function composition. Möbius geometry is the pair \((\hat{\mathbb{C}},\mathbf{M})\text{.}\)

There is a natural relationship between Möbius group operations and matrix group operations. The map \({\mathcal T}\colon
GL(2,\mathbb{C})\to \mathbf{M}\) be given by

\begin{equation}
\left[
\begin{array}{cc}a& b\\ c& d\end{array}\right]\to
\left[z\to \frac{az+b}{cz+d}\right]\label{gl2tomobgp}\tag{3.2.2}
\end{equation}

is a group homomorphism. The kernel of \({\mathcal T}\) is the group of nonzero scalar matrices.

\begin{equation*}
\ker{\mathcal T}=\left\{\left[
\begin{array}{cc}k& 0\\ 0& k\end{array}\right],
k\neq 0\right\}\text{.}
\end{equation*}

The quotient group \(GL(2,\mathbb{C})/\ker{\mathcal T}\) is called the projective linear group \(PGL(2,\mathbb{C})\text{.}\) Thus we have a group isomorphism

\begin{equation*}
PGL(2,\mathbb{C})\approx \mathbf{M}\text{.}
\end{equation*}

Checkpoint 3.2.5.

Show that the map \({\mathcal T}\) is a group homomorphism. Show that the kernel of \({\mathcal T}\) is

\begin{equation*}
\ker{\mathcal T}=\left\{\left[
\begin{array}{cc}k& 0\\ 0& k\end{array}\right],
k\neq 0\right\}\text{.}
\end{equation*}

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.

1. Identify the values of the coefficients \(a,b,c,d\) in a Möbius transformation \(z\to \frac{az+b}{cz+d}\) that is a homothety, rotation, translation, and inversion, respectively.
2. 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 \(z=\frac{az+b}{cz+d}\text{.}\) 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 \(z_1,z_2,z_3\) in \(\hat{\mathbb{C}}\) and any triple of distinct output values \(w_1,w_2,w_3\) in \(\hat{\mathbb{C}}\text{,}\) there is a unique \(T\in \mathbf{M}\) such that \(Tz_i=w_i\) for \(i=1,2,3\text{.}\)

Proof.

Outline: Suppose there are two such transformations, \(S\) and \(T\text{.}\) Show that \(S\circ
T^{-1}\) fixes three points. Now apply the previous Lemma.

Cross ratio

In what follows, we consider the special case of the output triple \(w_1=1\text{,}\) \(w_2=0\text{,}\) \(w_3=\infty\text{.}\) Given three distinct points \(z_1,z_2,z_3\) in \(\hat{\mathbb{C}}\text{,}\) we write \((\cdot,z_1,z_2,z_3)\) to denote the unique Möbius transformation that satisfies \(z_1\to
1\text{,}\) \(z_2\to 0\text{,}\) and \(z_3\to \infty\text{.}\) We write \((z_0,z_1,z_2,z_3)\) to denote the image of \(z_0\) under \((\cdot,z_1,z_2,z_3)\text{.}\) The expression \((z_0,z_1,z_2,z_3)\) is called the cross ratio of the \(4\)-tuple \(z_0,z_1,z_2,z_3\text{.}\) The next two propositions give important properties of cross ratio.

Proposition 3.2.11. Cross ratio is invariant.

Let \(z_1,z_2,z_3\) be distinct points in \(\hat{\mathbb{C}}\text{,}\) let \(z_0\in \hat{\mathbb{C}}\text{,}\) and let \(T\) be any Möbius transformation. Then we have

\begin{equation*}
(z_0,z_1,z_2,z_3)= (Tz_0,Tz_1,Tz_2,Tz_3).
\end{equation*}

Proof.

The transformations \((\cdot,z_1,z_2,z_3)\) and \((\cdot,Tz_1,Tz_2,Tz_3)\circ T\) both send \(z_1\to 1\text{,}\) \(z_2\to 0\text{,}\) and \(z_3\to \infty\text{,}\) so they must be equal, by the Fundamental Theorem. Now apply both transformations to \(z_0\text{.}\)

Proposition 3.2.12.

Let \(z_1,z_2,z_3\) be distinct points in \(\hat{\mathbb{C}}\text{,}\) let \(T=(\cdot,z_1,z_2,z_3)\text{,}\) and let \(C_T=T^{-1}\left(\hat{\mathbb{R}}\right)\) be the inverse image of the extended real line \(\hat{\mathbb{R}}=\mathbb{R}\cup \{\infty\}\) under \(T\text{.}\) Then \(C_T\) is a Euclidean circle or straight line. Furthermore, \(C_T\) is the unique Euclidean circle or straight line that contains the points \(z_1,z_2,z_3\text{.}\)

Proof.

See Exercise 3.2.6.4

Corollary 3.2.13.

The cross ratio \((z_0,z_1,z_2,z_3)\) is real if and only if \(z_0,z_1,z_2,z_3\) lie on a Euclidean circle or straight line.

Corollary 3.2.14.

Let \(C\) be a Euclidean circle or straight line in \(\hat{\mathbb{C}}\) 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 \(z\to z^{\ast}\) 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 \(z_1,z_2,z_3\) in \(\hat{\mathbb{C}}\text{,}\) let \(T=(\cdot,z_1,z_2,z_3)\text{.}\) Given any point \(z\text{,}\) the symmetric point with respect to \(C\) is

\begin{equation}
z^{\ast C}= (T^{-1}\circ conj \circ T)(z)\label{symmptdef}\tag{3.2.3}
\end{equation}

where \(conj\colon \hat{\mathbb{C}}\to \hat{\mathbb{C}}\) is the extension of the conjugation map to the extended complex plane that sends \(\infty\to \infty^\ast=\infty\text{.}\) The idea is to map \(C\) to the real line via \(T\text{,}\) then conjugate, then map the real line back to \(C\text{.}\) See Figure 3.2.15.

Figure 3.2.15. Symmetric points \(z,z^{\ast C}\) with respect to the circle \(C\)

The definition of symmetric point depends only on the circle \(C\text{,}\) and not on the three points \(z_1,z_2,z_3\text{.}\) 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 \(C\text{,}\) then \(Sz,Sz'\) are symmetric with respect to the cline \(S(C)\text{.}\) That is, we have

\begin{equation*}
(Sz)^{\ast S(C)} = S(z^{\ast C}).
\end{equation*}

Proof.

Let \(z_1,z_2,z_3\) be three points on \(C\text{,}\) so that \(Sz_1,Sz_2,Sz_3\) are three points on \(S(C)\text{.}\) Let \(T=(\cdot,z_1,z_2,z_3)\) and let \(U=(\cdot,Sz_1,Sz_2,Sz_3)\text{.}\) By invariance of the cross ratio, we have

\begin{equation*}
(U\circ S)z=Tz\text{.}
\end{equation*}

Thus we have

\begin{align*}
(Sz)^{\ast S(C)} & = (U^{-1}\circ conj\circ
U)(Sz) \;\; \text{(by definition)}\\
& = (S\circ S^{-1}\circ U^{-1}\circ conj\circ
U\circ S)(z)\\
& = S (S^{-1}\circ U^{-1}\circ conj\circ
U\circ S)(z)\\
& = S(T^{-1}\circ conj \circ T)(z)\\
& = S(z^{\ast C})
\end{align*}

as desired.

Corollary 3.2.17.

The definition of \(z^{\ast C}\) depends only on the circle \(C\text{,}\) and not on the three points \(z_1,z_2,z_3\) 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 \(\infty\text{,}\) then it has the form \(z\to \alpha z\) for some nonzero \(\alpha\in
\mathbb{C}\text{.}\) If a Möbius transformation has a single fixed point at \(\infty\text{,}\) then it has the form \(z\to z+\beta\) for some nonzero \(\beta\in \mathbb{C}\text{.}\)

Proof.

See Exercise 3.2.6.5.

Now suppose that a Möbius transformation \(T\) has two fixed points, \(p\) and \(q\text{.}\) Let \(S\) be given by \(Sz = \frac{z-p}{z-q}\text{.}\) Let \(w=Sz\) and let \(U=S\circ T\circ S^{-1}\) be the transformation of the \(w\)-plane that is conjugate to \(T\) via \(S\) (see Exercise Group 1.3.3.3–6). It is easy to check that \(U\) has exactly two fixed points \(0\) and \(\infty\text{.}\) Applying the previous Lemma, we have \(Uw=\alpha w\) for some nonzero \(\alpha\in
\mathbb{C}\text{.}\) Applying both sides of \(S\circ T=U\circ S\) to \(z\text{,}\) we have the following normal form for \(T\text{.}\)

\begin{equation}
\frac{Tz-p}{Tz-q}= \alpha
\frac{z-p}{z-q}\label{normalform2fixedpts}\tag{3.2.4}
\end{equation}

The transformation \(T\) is called elliptic, hyperbolic, or loxodromic if \(U\) is a rotation \((|\alpha |=1)\), a homothety \((\alpha >0)\), or neither, respectively.

Finally, suppose that a Möbius transformation \(T\) has exactly one fixed point at \(p\text{.}\) Let \(S\) be given by \(Sz = \frac{1}{z-p}\text{.}\) Again, let \(w=Sz\) and let \(U=S\circ T\circ S^{-1}\text{.}\) This time, \(U\) has exactly one fixed point at \(\infty\text{.}\) Applying the Lemma, we have \(Uw=w+\beta\) for some nonzero \(\beta\in
\mathbb{C}\text{.}\) Applying both sides of \(S\circ T=U\circ S\) to \(z\text{,}\) we have the following normal form for \(T\text{.}\)

\begin{equation}
\frac{1}{Tz-p}=
\frac{1}{z-p}+\beta\label{normalformfixedpt}\tag{3.2.5}
\end{equation}

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 \(T\in \mathbf{M}\)

normal form type	normal form	conjugate transformation type
elliptic	\(\frac{Tz-p}{Tz-q}= \alpha \frac{z-p}{z-q}, |\alpha|=1\)	rotation
hyperbolic	\(\frac{Tz-p}{Tz-q}= \alpha \frac{z-p}{z-q}, \alpha\gt 0\)	homothety
loxodromic	\(\frac{Tz-p}{Tz-q}= \alpha \frac{z-p}{z-q}, \alpha\neq 0\) \(|\alpha |\neq 1, \alpha\not\gt 0\)	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. 

1. 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.
2. 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}

3. Write your own decomposition for the case \(c=0\text{.}\)

Exercise 2

Explicit form for the transformation \(\)(⋅,z1,z2,z3). 

1. 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}

2. 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.

1. \(\displaystyle 1\to a, 0\to b, \infty\to c\)
2. \(\displaystyle a\to d, b\to e, c\to f\)

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. 

1. Prove Corollary 3.2.17.
2. 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 \(\omega=e^{2\pi i/3}\text{.}\)
3. Now let \(C\) be a circle with center a and radius \(r\gt 0\text{.}\) Use Proposition 3.2.16 to show that \(z^{\ast C} = \frac{r^2}{(z-a)^\ast}+a\text{.}\)
4. 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.

1. \(\displaystyle z\to \frac{1}{z}\)
2. \(\displaystyle z\to \frac{3z-1}{z+1}\)

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 \(k\gt 0\text{.}\)