Finding a Möbius Transformation

Complex Analysis · Axiom Academy

EXAMPLE Finding a Möbius Transformation Using the cross-ratio to build the map from three point correspondences A Möbius transformation is determined completely by where it sends three points. Find the transformation f that maps: To send , set the cross-ratio of w against the targets equal to the cross-ratio of z against the sources, then solve for w : Nice work — you built a Möbius transformation from scratch using only three point correspondences. Three points fix the map: a Möbius transformation has three independent ratios, so any three distinct source points and their three distinct images determine it uniquely. The cross-ratio is the engine: setting the w -cross-ratio equal to the z -cross-ratio and solving for w always recovers the map. is easy: when a source or target is , drop the two factors containing it — here the right side collapsed straight to z . Result: , which you can rewrite as . Check: f(0)=1 , f(1)=i , . Other ways to pin down a Möbius map Often you don't need all three points by hand — a known standard form or a fixed-point condition does the work: Whichever route you take, the same principle holds: a Möbius map is rigid — pin down enough conditions (three points, a domain pair, or fixed points plus one more) and the transformation is forced.

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