Tracing compressed curves in triangulated surfaces

Jeff G Erickson, Amir Nayyeri

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. We describe an algorithm to "trace" a normal curve in O(min{X,n 2 log X}) time, where n is the complexity of the surface triangulation and X is the number of times the curve crosses edges of the triangulation. In particular, our algorithm runs in polynomial time even when the number of crossings is exponential in n. Our tracing algorithm computes a new cellular decomposition of the surface with complexity O(n); the traced curve appears as a simple path or cycle in the 1-skeleton of the new decomposition. We apply our abstract tracing strategy to two different classes of normal curves: abstract curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle. Our normal-coordinate algorithms are competitive with and conceptually simpler than earlier algorithms by Schaefer, Sedgwick, and Štefankovic [COCOON 2002, CCCG 2008] and by Agol, Hass, and Thurston [Trans. AMS 2005].

Original languageEnglish (US)
Title of host publicationProceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012
Pages131-140
Number of pages10
DOIs
StatePublished - Jul 23 2012
Event28th Annual Symposuim on Computational Geometry, SCG 2012 - Chapel Hill, NC, United States
Duration: Jun 17 2012Jun 20 2012

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Other

Other28th Annual Symposuim on Computational Geometry, SCG 2012
Country/TerritoryUnited States
CityChapel Hill, NC
Period6/17/126/20/12

Keywords

  • Computational topology
  • Geodesics
  • Normal coordinates

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

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