Möbius Transformations

Among all the conformal maps, one family is so well-behaved that it almost runs the whole subject: the Möbius transformations (also called linear fractional transformations),

f(z) = \frac{az + b}{cz + d}, \qquad ad - bc \ne 0.

The condition ad - bc \ne 0 is exactly what stops the map collapsing to a constant: if ad - bc = 0 the numerator is a fixed multiple of the denominator and f would be the same value everywhere.

Each is a conformal bijection of the extended plane \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\} — the Riemann sphere — with the conventions f(\infty) = a/c and f(-d/c) = \infty.
They map circles-and-lines to circles-and-lines. Treating a straight line as a circle of infinite radius, every "circline" goes to a circline.
Every Möbius map is a composition of the three simplest pieces: a translation z \mapsto z + \beta, a scaling–rotation z \mapsto \alpha z, and the inversion z \mapsto 1/z.
Three points determine the map. Given any three distinct points and any three distinct target points, there is exactly one Möbius map sending one triple to the other.

Why circles go to circles

The cleanest argument is to break f into the three pieces above and note each one preserves the class of circlines.

Step 1 — divide out (the case c \ne 0). Polynomial division on \dfrac{az + b}{cz + d} peels off a constant:

\frac{az + b}{cz + d} = \frac{a}{c} + \frac{b - \frac{ad}{c}}{cz + d} = \frac{a}{c} - \frac{ad - bc}{c\,(cz + d)}.

Step 2 — read off the chain of simple maps. Building z up to that expression uses only the three primitives, in order:

z \;\xrightarrow{\ \times c,\ +d\ }\; cz + d \;\xrightarrow{\ 1/(\cdot)\ }\; \frac{1}{cz + d} \;\xrightarrow{\ \times(-\frac{ad-bc}{c}),\ +\frac{a}{c}\ }\; f(z).

Step 3 — each primitive preserves circlines. Translations and scaling–rotations obviously send circles to circles and lines to lines. The one substantive fact is that inversion z \mapsto 1/z maps circlines to circlines (a line not through the origin becomes a circle through the origin, and so on). Composing maps that all preserve circlines gives a map that preserves circlines — done.

The model example: half-plane to disk

The single most useful Möbius map in analysis is

f(z) = \frac{z - i}{z + i}.

It sends the upper half-plane \{\operatorname{Im} z > 0\} bijectively onto the open unit disk \{|w| < 1\}, carrying the boundary real axis onto the boundary unit circle. Track three boundary points and the circline rule does the rest.

Step 1 — the point 0.

f(0) = \frac{0 - i}{0 + i} = \frac{-i}{i} = -1.

Step 2 — the point 1.

f(1) = \frac{1 - i}{1 + i} = \frac{(1 - i)^2}{(1 + i)(1 - i)} = \frac{-2i}{2} = -i.

Step 3 — the point \infty. With a = b = 1, c = 1, d = i,

f(\infty) = \frac{a}{c} = \frac{1}{1} = 1.

Three points of the real axis — 0, 1, \infty — land on -1, -i, 1, three points of the unit circle. Since a circline is pinned down by three points, the whole real axis must map to the whole unit circle, and the half-plane on one side to the disk on the inside.

"Circles-and-lines to circles-and-lines" does not mean lines stay lines and circles stay circles. A perfectly straight line can map to a circle, and a circle can map to a straight line — that is exactly what the model example above does, sending the real axis (a line) onto the unit circle. The resolution is to stop treating lines and circles as separate species: in the extended plane a line is just a circle that passes through the point at infinity \infty. What a Möbius map preserves is the single family of generalized circles ("circlines") — circles or lines — not lines and circles separately. So the trap is expecting \operatorname{Im} z = t to come out straight; whenever the image circline misses \infty, your line has curled up into an honest circle.

The leap to the Riemann sphere is what makes Möbius maps so tidy. On the ordinary plane the point z = -d/c looks like a catastrophe — the denominator vanishes. Add the single point \infty and the catastrophe becomes a smooth bijection: -d/c simply goes to \infty, and \infty goes to a/c. Lines and circles stop being different species — both are just circles on the sphere, one of them passing through the north pole. The algebra of these maps is the algebra of invertible 2 \times 2 matrices \begin{pmatrix} a & b \\ c & d \end{pmatrix}, which is why composing two Möbius maps is the same as multiplying their matrices.

See lines become circles

Slide the height t to pick the horizontal line \operatorname{Im} z = t in the upper half-plane (blue). Its image under f(z) = (z - i)/(z + i) is the orange circle — always inside the unit circle. At t = 0 the line is the real axis and its image is the unit circle itself; as t grows the line climbs higher and its image shrinks toward the point f(\infty\,\text{-ish}) = 1.