Möbius Transformations

Complex Analysis · Axiom Academy

One formula, — and the remarkable way it bends every circle and line into another circle or line. 1. The Map, and What It's Made Of A Möbius transformation (or fractional linear transformation) takes a complex number z and returns The condition keeps the map non-degenerate (otherwise it would collapse the whole plane to a point). To make it a true bijection we extend the plane with a point at infinity, , so that and . The surprise is that this fraction is built from just three simple moves . Watch one shape get carried through each in turn: — turn inside-out. The one non-linear move. Stacking maps multiplies their matrices . 2. Circles and Lines Stay Circles and Lines Here is the property that makes these maps famous. A Möbius transformation sends every circle or line to another circle or line . (A line is just a "circle through " on the Riemann sphere, so really: circles go to circles.) Watch the simplest non-linear case, inversion w=1/z . On the left is the straight vertical line . As the point slides up it, its image w=1/z on the right doesn't stay on a line — it sweeps out a perfect circle : Why must this happen? Because of the building blocks from Step 1. Translations and rotations obviously keep circles as circles and lines as lines; the only thing to check is inversion — and a short calculation shows 1/z also sends circles-and-lines to circles-and-lines. Since every Möbius map is a composition of these, the property carries through.

This is the written version of the interactive lesson above. See the full Complex Analysis course.