Скачать или смотреть Christopher Bishop | Mappings and Meshes, connections between continuous and discrete geometry I

5/7/2021 FRG Workshop on Geometric Methods for Analyzing Discrete Shapes

Speaker: Christopher Bishop

Title: Mappings and Meshes: connections between continuous and discrete geometry I

Abstract: I will give two lectures about some interactions between conformal, hyperbolic and computational geometry. The first lecture shows how ideas from discrete and computational geometry can help compute conformal mappings, and the second lecture reverses the direction and shows how conformal maps can give meshes of polygonal domains with optimal geometry.

Lecture 1: The conformal map from the unit disk to the interior of a polygon P is given by the Schwarz-Christoffel formula, but this is stated in terms of parameters that are hard to compute from P. After some background and motivation, I explain how the medial axis of a domain, a concept from computation geometry, can be used to give a fast approximation to these parameters, with bounds on the accuracy that are independent of P. The precise statement involves quasiconformal mappings, and proving these bounds uses a result about hyperbolic convex sets originating in Thurston's work on 3-manifolds.