Computing shortest words via shortest loops on hyperbolic surfaces

Authors:
Xiaotian Yin;Yinghua Li;Wei Han;Feng Luo;Xianfeng David Gu;Shing-Tung Yau
Affiliations:
Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA;Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794-3600, USA;Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA;Mathematics Department, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA;Computer Science Department, Stony Brook University, Stony Brook, NY 11794-4400, USA;Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA
Venue:
Computer-Aided Design
Year:
2011

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Abstract

Given a loop on a surface, its homotopy class can be specified as a word consisting of letters representing the homotopy group generators. One of the interesting problems is how to compute the shortest word for a given loop. This is an NP-hard problem in general. However, for a closed surface that allows a hyperbolic metric and is equipped with a canonical set of fundamental group generators, the shortest word problem can be reduced to finding the shortest loop that is homotopic to the given loop, which can be solved efficiently. In this paper, 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.