Consider the function defined on $\mathbb{C}^+$ by $T(z) = (az + b)/(cz + d)$ where $a, b, c$ and $d$ are complex constants. Such a function is called a Möbius transformation if $ad - bc \neq 0\text{.}$ Transformations of this form are also called fractional linear transformations. The complex number $ad - bc$ is called the determinant of $T(z) = (az+b)/(cz+d)\text{,}$ and is denoted as Det$(T)\text{.}$

Note that in the preceeding proof we found the inverse transformation of a Möbius transformation. This inverse transformation is itself a Möbius transformation since its determinant is not 0. In fact, its determinant equals the determinant of the original Möbius transformation. We summarize this fact as follows.

If we compose two Möbius transformations, the result is another Möbius transformation. Proof of this fact is left as an exercise.

Just as translations and rotations of the plane can be constructed from reflections across lines, the general Möbius transformation can be constructed from inversions about clines.

Since Möbius transformations are composed of inversions, they will embrace the finer qualities of inversions. For instance, since inversion preserves clines, so do Möbius transformations, and since inversion preserves angle magnitudes, Möbius transformations preserve angles (as an even number of inversions).

The following “fixed point” theorem is useful for understanding Möbius transformations.

With this fixed point theorem in hand, we can now prove The Fundamental Theorem of Möbius Transformations, which says that if we want to induce a one-to-one and onto motion of the entire extended plane that sends my favorite three points ($z_1, z_2, z_3$) to your favorite three points ($w_1, w_2, w_3)\text{,}$ as dramatized below, then there is a Möbius transformation that will do the trick, and there's only one.

There is an algebraic description of the very useful Möbius transformation mapping $z_1 \mapsto 1\text{,}$ $z_2 \mapsto 0$ and $z_3 \mapsto \infty$ that arose in the proof of Theorem 3.4.8: \begin{equation*} T(z) = \frac{(z-z_2)}{(z-z_3)}\cdot \frac{(z_1 - z_3)}{(z_1-z_2)}. \end{equation*}

The reader can check that the map works as advertised and that it is indeed a Möbius transformation. (While it is clear that the transformation has the form $(az + b)/(cz + d)\text{,}$ it might not be clear that the determinant is nonzero. It is, since the $z_i$ are distinct.) We also note that if one of the $z_i$ is $\infty\text{,}$ the form of the map reduces by cancellation of the terms with $\infty$ in them. For instance, if $z_2 = \infty\text{,}$ the map that sends $z_1 \mapsto 1\text{,}$ $\infty \mapsto 0$ and $z_3 \mapsto \infty$ is $T(z) = (z_1-z_3)/(z-z_3)\text{.}$

The Möbius transformation that sends any three distinct points to 1, 0, and $\infty$ is so useful that it gets its own name and special notation.

Definition3.4.10

The cross ratio of 4 complex numbers $z,w,u,$ and $v\text{,}$ where $w,u,$ and $v$ are distinct, is denoted $(z,w;u,v)\text{,}$ and \begin{equation*} (z,w;u,v) = \frac{z-u}{z-v}\cdot\frac{w-v}{w-u}. \end{equation*}

If $z$ is a variable, and $w, u,$ and $v$ are distinct complex constants, then $T(z) = (z,w;u,v)$ is the (unique!) Möbius transformation that sends $w \mapsto 1\text{,}$ $u \mapsto 0\text{,}$ and $v \mapsto \infty\text{.}$

Example3.4.11The image of three points determines a Möbius transformation

Find the unique Möbius transformation that sends $1 \mapsto 3\text{,}$ $i \mapsto 0\text{,}$ and $2 \mapsto -1\text{.}$

One approach: Find $T(z) = (z,1;i,2)$ and $S(w) = (w,3;0,-1)\text{.}$ In this case, the transformation we want is $S^{-1} \circ T\text{.}$

To find this transformation, we set the cross ratios equal: \begin{align*} (z,1;i,2) \amp = (w,3;0,-1)\\ \frac{z-i}{z-2}\cdot\frac{1-2}{1-i} \amp = \frac{w-0}{w+1}\cdot\frac{3+1}{3-0}\\ \frac{-z+i}{(1-i)z-2+2i} \amp = \frac{4w}{3w+3}. \end{align*}

Then solve for $w\text{:}$ \begin{align*} -3zw + 3iw + 3i - 3z \amp = 4[(1-i)z-2+2i]w\\ -3z+3i \amp = [3z-3i+4[(1-i)z-2+2i]]w\\ w \amp = \frac{-3z + 3i}{(7-4i)z + (-8+5i)}. \end{align*}

Thus, our Möbius transformation is \begin{equation*} V(z) = \frac{-3z + 3i}{(7-4i)z + (-8+5i)}. \end{equation*}

It's quite easy to check our answer here. Since there is exactly one Möbius transformation that does the trick, all we need to do is check whether $V(1) = 3, V(i) = 0$ and $V(2) = -1\text{.}$ Ok... yes... yes... yep! We've got our map!

In addition to defining maps that send points to 1, 0, and $\infty\text{,}$ the cross ratio can proclaim whether four points lie on the same cline: If $(z,w;u,v)$ is a real number then the points are all on the same cline; if $(z,w;u,v)$ is complex, then they aren't. The proof of this fact is left as an exercise.

Example3.4.13Testing whether four points lie on a single cline

Take the points $1, i, -1, -i\text{.}$ We know these four points lie on the circle $|z| = 1\text{,}$ so according to the statement above, $(1,i;-1,-i)$ is a real number. Let's check: \begin{align*} (1,i;-1,-i) \amp = \frac{1+1}{1+i}\cdot\frac{i+i}{i+1}\\ \amp = \frac{2}{1+i}\frac{2i}{1+i}\\ \amp = \frac{4i}{(1-1)+2i}\\ \amp = \frac{4i}{2i}\\ \amp = 2. \tag{Yep!} \end{align*}

Another important feature of inversion that gets passed on to Möbius transformations is the preservation of symmetry points. The following result is a corollary to Theorem 3.2.12.

We close the section with one more theorem about Möbius transformations.

1

Find a transformation of $\mathbb{C}^+$ that rotates points about $2i$ by an angle $\pi/4\text{.}$ Show that this transformation has the form of a Möbius transformation.

2

Find the inverse transformation of $T(z) = \frac{3z + i}{2z + 1}\text{.}$

3

Prove Theorem 3.4.3. That is, suppose $T$ and $S$ are two Möbius transformations and prove that the composition $T\circ S$ is again a Möbius transformation.

4

Prove that any Möbius transformation can be written in a form with determinant 1, and that this form is unique up to sign. Hint: How does the determinant of $T(z) = (az+b)/(cz+d)$ change if we multiply top and bottom of the map by some constant $k\text{?}$

5

Find the unique Möbius transformation that sends $1 \mapsto i\text{,}$ $i \mapsto -1\text{,}$ and $-1 \mapsto -i\text{.}$ What are the fixed points of this transformation? What is $T(0)\text{?}$ What is $T(\infty)\text{?}$

6

Repeat the previous exercise, but send $2 \to 0\text{,}$ $1 \to 3$ and $4 \to 4\text{.}$

7

Prove this feature of the cross ratio: $\overline{(z, z_1; z_2, z_3)} = (\overline{z},\overline{z_1};\overline{z_2},\overline{z_3})\text{.}$

8

Prove that the cross ratio of four distinct real numbers is a real number.

9

Prove that the cross ratio of four distinct complex numbers is a real number if and only if the four points lie on the same cline. Hint: Use the previous exercise and the invariance of the cross ratio.

10

Do the points $2+i, 3, 5,$ and $6 + i$ lie on a single cline?

11

More on Möbius transformations.

a. Give an example of a Möbius transformation $T$ such that $\overline{T(z)} \neq T(\overline{z})$ for some $z$ in $\mathbb{C}^+\text{.}$

b. Suppose $T$ is a Möbius transformation that sends the real axis onto itself. Prove that in this case, $\overline{T(z)} = T(\overline{z})$ for all $z$ in $\mathbb{C}\text{.}$

12

Is there a Möbius transformation that sends 1 to 3, $i$ to 4, -1 to $2 + i$ and $-i$ to $4 + i\text{?}$ Hint: It may help to observe that the input points are on a single cline.

13

Find the fixed points of these transformations on $\mathbb{C}^+\text{.}$ Remember that $\infty$ can be a fixed point of such a transformation.

a. $T(z) = \frac{2z}{3z-1}$

b. $T(z) = iz$

c. $T(z) = \frac{-iz}{(1-i)z - 1}$

14

Find a Möbius transformation that takes the circle $|z| = 4$ to the straight line $3x + y = 4\text{.}$ Hint: Track the progress of three points, and the rest will follow.

15

Find a non-trivial Möbius transformation that fixes the points -1 and 1, and call this transformation $T\text{.}$ Then, let $C$ be the the imaginary axis. What is the image of $C$ under this map. That is, what cline is $T(C)\text{?}$

16

Suppose $z_1, z_2, z_3$ are distinct points in $\mathbb{C}^+\text{.}$ Show that by an even number of inversions we can map $z_1 \mapsto 1\text{,}$ $z_2 \mapsto 0\text{,}$ and $z_3 \mapsto \infty\text{,}$ in three special cases:

a. When $z_1 = \infty\text{.}$

b. When $z_2 = \infty\text{.}$

c. When $z_3 = \infty\text{.}$