Algorithms in Computational Conformal Geometry

Abstract

Computational conformal geometry is an intersectional field combining modern geometry theories from pure mathematics with computational algorithms from computer science. In the first part of this dissertation, we firstly review a powerful tool in computational conformal geometry, the discrete surface Ricci flow, which is used to conformally deform the given Riemannian metric of a surface to a Riemannian metric according to a user defined Gaussian curvature on interior points and geodesic curvature along the boundaries. Using the discrete Ricci flow to embed the high genus surface into the hyperbolic plane, we propose an efficient algorithm to compute the shortest words for loops given on triangulated surface meshes. The design of this algorithm is inspired and guided by the work of Dehn and Birman-Series. In support of the shortest word algorithm, we also propose efficient algorithms to compute shortest paths and shortest loops under hyperbolic metrics using a novel technique, called transient embedding, to work with the universal covering space. In addition, we employ several techniques to relieve the numerical errors. Experimental results are given to demonstrate the performance in practice. In the second part, we introduce two Delaunay refinement algorithms which give quality meshes on two-dimensional hyperbolic Poincar&eacute disk in computing. These two Delaunay refinement algorithms are generalizations of Chew's second algorithm and Ruppert's refinement algorithm, both of them are based on the Planar Straight Line Graph (PSLG) in Euclidean geometry. By modifying some definitions and adding new constraints, these two algorithms can be applied to surface meshes embedded in the hyperbolic Poincar&eacute disk. The generalizations will work on global meshes, and termination of these two algorithms will be given under constraints.