Quasiconformal mapping

Summary

A quasiconformal mapping is an orientation-preserving homeomorphism between plane domains that distorts infinitesimal circles into ellipses of bounded eccentricity. The bound $K \ge 1$ on the ratio of the axes is the maximal dilatation, and $1$-quasiconformal maps are exactly the conformal maps. Quasiconformal maps are much more flexible than conformal maps, yet they keep many of their compactness and boundary properties. They are a basic tool in Teichmüller theory, complex dynamics and the theory of Kleinian groups.