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

    Fingerprint

    Line segment
    Triangle
    Arrangement
    Dual Problem
    Three-dimension
    Lemma
    Simplify
    Curve
    Line

    ASJC Scopus subject areas

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

    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.

    On levels in arrangements of lines, segments, planes, and triangles. / Agarwal, Pankaj K.; Aronov, Boris; Sharir, Micha.

    Proceedings of the Annual Symposium on Computational Geometry. ed. / Anon. ACM, 1997. p. 30-38.

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

    Agarwal, PK, 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. ACM, pp. 30-38, Proceedings of the 1997 13th Annual Symposium on Computational Geometry, Nice, Fr, 6/4/97.
    Agarwal PK, Aronov B, Sharir M. On levels in arrangements of lines, segments, planes, and triangles. In Anon, editor, Proceedings of the Annual Symposium on Computational Geometry. ACM. 1997. p. 30-38
    Agarwal, Pankaj K. ; Aronov, Boris ; Sharir, Micha. / On levels in arrangements of lines, segments, planes, and triangles. Proceedings of the Annual Symposium on Computational Geometry. editor / Anon. ACM, 1997. pp. 30-38
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