On polyhedra induced by point sets in space

Pankaj K. Agarwal, Ferran Hurtado, Godfried Toussaint, Joan Trias

    Research output: Contribution to journalArticle

    Abstract

    Given a set S of n ≥ 3 points in the plane (not all on a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the given points in S. For example, the shortest circuit through S is a simple polygon. In 1994, Grünbaum showed that an analogous theorem holds in R3. More precisely, if S is a set of n ≥ 4 points in R3 (not all of which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (sphere-like) polyhedron P such that the vertices of P are precisely the given points in S. Grünbaum's constructive proof may yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are star-shaped, have Hamiltonian skeletons, and admit efficient point-location queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in O (n log n) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O (n1 + ε{lunate}), for any ε{lunate} > 0.

    Original languageEnglish (US)
    Pages (from-to)42-54
    Number of pages13
    JournalDiscrete Applied Mathematics
    Volume156
    Issue number1
    DOIs
    StatePublished - Jan 1 2008

    Fingerprint

    Hamiltonians
    Polyhedron
    Point Sets
    Short circuit currents
    Stars
    Simple Polygon
    Efficient Points
    Point Location
    Coplanar
    Skeleton
    Monotonic
    Set of points
    Star
    Query
    Line
    Alternatives
    Theorem

    Keywords

    • Computational geometry
    • Polyhedra
    • Triangulation

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Applied Mathematics

    Cite this

    Agarwal, P. K., Hurtado, F., Toussaint, G., & Trias, J. (2008). On polyhedra induced by point sets in space. Discrete Applied Mathematics, 156(1), 42-54. https://doi.org/10.1016/j.dam.2007.08.033

    On polyhedra induced by point sets in space. / Agarwal, Pankaj K.; Hurtado, Ferran; Toussaint, Godfried; Trias, Joan.

    In: Discrete Applied Mathematics, Vol. 156, No. 1, 01.01.2008, p. 42-54.

    Research output: Contribution to journalArticle

    Agarwal, PK, Hurtado, F, Toussaint, G & Trias, J 2008, 'On polyhedra induced by point sets in space', Discrete Applied Mathematics, vol. 156, no. 1, pp. 42-54. https://doi.org/10.1016/j.dam.2007.08.033
    Agarwal, Pankaj K. ; Hurtado, Ferran ; Toussaint, Godfried ; Trias, Joan. / On polyhedra induced by point sets in space. In: Discrete Applied Mathematics. 2008 ; Vol. 156, No. 1. pp. 42-54.
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