Möbius Transformations — Mathematics Knowledge System

Overview

A Möbius transformation (also called a fractional linear transformation) is a map of the form f(z) = (az+b)/(cz+d), where a, b, c, d are complex constants with ad − bc ≠ 0. These maps are bijections of the extended complex plane (the Riemann sphere, ℂ ∪ {∞}) onto itself, they are conformal everywhere they are defined, and they form a group under composition. Möbius transformations send the set of 'circlines' (circles and straight lines, treating a line as a circle through ∞) to circlines, and they are uniquely determined by their action on any three distinct points. Because of this rigidity and flexibility, Möbius transformations are the standard tool for mapping one classical domain (a disk, a half-plane) conformally onto another.

Intuition

Möbius transformations are built from three elementary geometric moves composed together: translations (z ↦ z+b), rotations and scalings (z ↦ az), and inversion (z ↦ 1/z). Each of these individually sends circlines to circlines and preserves angles, so any composition of them does too — which is exactly what a general Möbius map (az+b)/(cz+d) turns out to be, once you divide through algebraically. The extra freedom of four parameters (really three, since scaling all of a,b,c,d by a constant doesn't change f) is exactly enough to send any three chosen points to any other three chosen points, which is why Möbius maps are the go-to tool for custom-fitting one domain's boundary onto another's.

Formal Definition

Definition

A Möbius transformation is defined by four complex constants a, b, c, d with ad − bc ≠ 0:

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

Möbius transformation

f(z) = \frac{dz-b}{-cz+a}

The inverse of a Möbius transformation is again a Möbius transformation

Inverse map

f'(z) = \frac{ad-bc}{(cz+d)^2}

Nonzero everywhere f is defined, since ad-bc≠0, confirming conformality

Derivative

Worked Examples

f(∞) = ∞ means cz+d has no z-term surviving in the limit unless c ≠ 0; specifically f(∞) = a/c, so a/c = -1, i.e. a = -c.
$f(\infty) = \frac{a}{c} = -1 \implies a = -c$
f(1) = ∞ means the denominator vanishes at z=1: c(1)+d=0, so d = -c.
$c + d = 0 \implies d = -c$
f(0) = b/d = 1, and d = -c, so b = d = -c. Choose c = -1 (any nonzero scalar works), giving a=1, d=1, b=1.
$a=1, b=1, c=-1, d=1$

Answer: f(z) = (z+1)/(-z+1) = (z+1)/(1-z), which sends 0→1, 1→∞, ∞→-1 as required.

Practice Problems

Difficulty 5/10

Check that ad − bc ≠ 0 for f(z) = (2z+1)/(z+1), and find f(0), f(1), and f(∞).

Difficulty 6/10

Find the inverse of f(z) = (z-1)/(z+1).

Difficulty 7/10

Explain why the Möbius transformation f(z) = (z-i)/(z+i) maps the real axis to the unit circle.

Quiz

Summary

A Möbius transformation f(z) = (az+b)/(cz+d), with ad-bc ≠ 0, is a conformal bijection of the extended complex plane (Riemann sphere) to itself.
Möbius transformations map circlines (circles and lines) to circlines and are uniquely determined by their action on any three distinct points.
They are built from translations, rotations/scalings, and inversion (z ↦ 1/z), and are the standard tool for conformally mapping one classical domain onto another.

References

WebsiteWikipedia — Möbius transformation