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Tracing compressed curves in triangulated surfaces

Published: 17 June 2012 Publication History

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 set{X, n2log 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 'tefankovic [COCOON 2002, CCCG 2008] and by Agol, Hass, and Thurston [Trans. AMS 2005].

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  • (2018)On the decidability of the fréchet distance between surfacesProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175342(1109-1120)Online publication date: 7-Jan-2018
  • (2013)Tracing Compressed Curves in Triangulated SurfacesDiscrete & Computational Geometry10.1007/s00454-013-9515-z49:4(823-863)Online publication date: 1-Jun-2013

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    cover image ACM Conferences
    SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry
    June 2012
    436 pages
    ISBN:9781450312998
    DOI:10.1145/2261250
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    Publication History

    Published: 17 June 2012

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    Author Tags

    1. computational topology
    2. geodesics
    3. normal coordinates

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    SoCG '12
    SoCG '12: Symposium on Computational Geometry 2012
    June 17 - 20, 2012
    North Carolina, Chapel Hill, USA

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    • (2018)On the decidability of the fréchet distance between surfacesProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175342(1109-1120)Online publication date: 7-Jan-2018
    • (2013)Tracing Compressed Curves in Triangulated SurfacesDiscrete & Computational Geometry10.1007/s00454-013-9515-z49:4(823-863)Online publication date: 1-Jun-2013

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