Möbius Transformations

Fractional linear maps on the Riemann sphere and the cross-ratio.

The idea

A Möbius transformation is a map of the form $f(z) = \frac{az + b}{cz + d}, \qquad ad - bc \neq 0.$ The condition $ad - bc \neq 0$ prevents the numerator from being a constant multiple of the denominator, which would make $f$ constant.

These maps are exactly the composites of three elementary ones: translations $z \mapsto z + b$, rotations and scalings $z \mapsto az$, and the inversion $z \mapsto 1/z$. The inversion supplies both the power of the class and its defect: it carries lines to circles, and it has nowhere to send $0$.

Enlarging the plane repairs the defect. Adjoin a single point $\infty$ to form the Riemann sphere $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$, and assign to the two exceptional points, the zero of the denominator and $\infty$ itself, the values continuity forces on them: for $c \neq 0$ the pole $z = -d/c$ goes to $\infty$, and $\infty$ goes to $a/c$. Every Möbius transformation is then a bijection of the sphere. On the sphere a line is a circle through $\infty$, so the statement that Möbius transformations carry circles to circles covers lines as well.

Ways to work on it

• Walkthrough. Evaluate a Möbius map and trace its action on the sphere's special points.
• Practice. Evaluate or invert a randomized fractional linear map.
• Hardest. Build the map from a three-point prescription and find its inverse.

Not sure where to start? Take the ten-question placement test.