Castles in the air revisited

Boris Aronov*, Micha Sharir

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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
Title of host publicationEighth Annual Symposium On Computational Geometry
PublisherAssociation for Computing Machinery (ACM)
Number of pages11
ISBN (Print)0897915178, 9780897915175
StatePublished - 1992
Externally publishedYes
EventEighth Annual Symposium On Computational Geometry - Berlin, Ger
Duration: 10 Jun 199212 Jun 1992

Publication series

NameEighth Annual Symposium On Computational Geometry


ConferenceEighth Annual Symposium On Computational Geometry
CityBerlin, Ger


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