Nonoverlap of the star unfolding

Boris Aronov, Joseph O'Rourke

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

    Abstract

    The star unfolding of a convex polytope with respect to a point x is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: (1) It does not self-overlap: its boundary is a simple polygon. (2) The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit the conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: for constructing the ridge tree, for finding the exact set of all shortest-path "edge sequences," and for computing the geodesic diameter of a polytope. Our results suggest conjectures on "unfoldings" of general convex surfaces.

    Original languageEnglish (US)
    Title of host publicationProceedings of the Annual Symposium on Computational Geometry
    PublisherAssociation for Computing Machinery
    Pages105-114
    Number of pages10
    ISBN (Print)0897914260
    DOIs
    StatePublished - Jun 1 1991
    Event7th Annual Symposium on Computational Geometry, SCG 1991 - North Conway, United States
    Duration: Jun 10 1991Jun 12 1991

    Publication series

    NameProceedings of the Annual Symposium on Computational Geometry

    Other

    Other7th Annual Symposium on Computational Geometry, SCG 1991
    CountryUnited States
    CityNorth Conway
    Period6/10/916/12/91

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Geometry and Topology
    • Computational Mathematics

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  • Cite this

    Aronov, B., & O'Rourke, J. (1991). Nonoverlap of the star unfolding. In Proceedings of the Annual Symposium on Computational Geometry (pp. 105-114). (Proceedings of the Annual Symposium on Computational Geometry). Association for Computing Machinery. https://doi.org/10.1145/109648.109660