Castles in the air revisited

Boris Aronov, Micha Sharir

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


    We show that the total number of faces bounding any single cell in an arrangement of n (d - 1)-simplices in IRd is O(nd-1 log n), thus almost settling a conjecture of Pach and Sharir. We present several applications of this result, mainly to translational motion planning in polyhedral environments. We then extend our analysis technique to derive other results on complexity in simplex arrangements. For example, we show that the number of vertices in such an arrangement, which are incident to the same cell on more than one 'side,' is O(nd-1 log n). We also show that the number of repetitions of a 'k-flap,' formed by intersecting d-k simplices, along the boundary of the same cell, summed over all cells and all k-flaps, is O(nd-1 log2 n). We use this quantity, which we call the excess of the arrangement, to derive bounds on the complexity of m distinct cells of such an arrangement.

    Original languageEnglish (US)
    Title of host publicationEighth Annual Symposium On Computational Geometry
    PublisherPubl by ACM
    Number of pages11
    ISBN (Print)0897915178, 9780897915175
    StatePublished - 1992
    EventEighth Annual Symposium On Computational Geometry - Berlin, Ger
    Duration: Jun 10 1992Jun 12 1992

    Publication series

    NameEighth Annual Symposium On Computational Geometry


    OtherEighth Annual Symposium On Computational Geometry
    CityBerlin, Ger

    ASJC Scopus subject areas

    • Engineering(all)


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