On levels in arrangements of lines, segments, planes, and triangles

Pankaj K. Agarwal, Boris Aronov, Micha Sharir

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

    Abstract

    We consider the problem of bounding the complexity of the k-th level in an arrangement of n curves or surfaces, a problem dual to, and extending, the well-known k-set problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n√k+1) for the complexity of the k-th level in an arrangement of n lines. (b) We derive an improved version of Lovasz Lemma in any dimension, and use it to prove a new bound, O(n2k2/3), on the complexity of the k-th level in an arrangement of n planes in R3, or on the number of k-sets in a set of n points in three dimensions. (c) We show that the complexity of any single level in an arrangement of n line segments in the plane is O(n3/2), and that the complexity of any single level in an arrangement of n triangles in 3-space is O(n17/6).

    Original languageEnglish (US)
    Title of host publicationProceedings of the Annual Symposium on Computational Geometry
    Editors Anon
    PublisherACM
    Pages30-38
    Number of pages9
    StatePublished - 1997
    EventProceedings of the 1997 13th Annual Symposium on Computational Geometry - Nice, Fr
    Duration: Jun 4 1997Jun 6 1997

    Other

    OtherProceedings of the 1997 13th Annual Symposium on Computational Geometry
    CityNice, Fr
    Period6/4/976/6/97

    ASJC Scopus subject areas

    • Chemical Health and Safety
    • Software
    • Safety, Risk, Reliability and Quality
    • Geometry and Topology

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  • Cite this

    Agarwal, P. K., Aronov, B., & Sharir, M. (1997). On levels in arrangements of lines, segments, planes, and triangles. In Anon (Ed.), Proceedings of the Annual Symposium on Computational Geometry (pp. 30-38). ACM.