Castles in the air revisited

B. Aronov, M. Sharir

    Research output: Contribution to journalArticlepeer-review


    We show that the total number of faces bounding any one cell in an arrangement of n (d-1)-simplices in ℝ d is O(n d-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 than extend our analysis to derive other results on complexity in arrangements of simplices. For example, we show that in such an arrangement the total number of vertices incident to the same cell on more than one "side" is O(n d-1 log n). We, also show that the number of repetitions of a "k-flap," formed by intersecting d-k given simplices, along the boundary of the same cell, summed over all cells and all k-flaps, is O(n d-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)
    Pages (from-to)119-150
    Number of pages32
    JournalDiscrete & Computational Geometry
    Issue number1
    StatePublished - Dec 1994

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Geometry and Topology
    • Discrete Mathematics and Combinatorics
    • Computational Theory and Mathematics


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