This talk is about computing (approximately) the “best” map between two polygons taking vertices to vertices. It arises out of a real-life problem, namely, surface registration. Our notion of “best” is extremal quasiconformality (least angle-distortion). I will try to keep the talk as self-contained as possible. It is based on a joint-work with M. Goswami, G. Telang, and X. Gu.