Incidences between points and circles in three and higher dimensions

Boris Aronov, Vladlen Koltun, Micha Sharir

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

    Abstract

    We show that the number of incidences between m distinct points and n distinct circles in ℝ3 is O(m4/7n17/21 + m2/3n2/3 + m + n); the bound is optimal for m ≥ n3/2. This result extends recent work on point-circle incidences in the plane, but its proof requires a different analysis. The bound improves upon a previous bound, noted by Akutsu et al. [2] and by Agarwal and Sharir [1], but it is not as sharp (when m is small) as the recent planar bound of Aronov and Sharir [3]. Our analysis extends to yield the same bound (a) on the number of incidences between m points and n circles in any dimension d ≥ 3, and (b) on the number of incidences between m points and n arbitrary convex plane curves in ℝd, for any d ≥ 3, provided that no two curves are coplanar. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4-space, and were already used to obtain a lower bound for the number of distinct distances in a set of n points in 3-space.

    Original languageEnglish (US)
    Title of host publicationProceedings of the Annual Symposium on Computational Geometry
    Pages116-122
    Number of pages7
    StatePublished - 2002
    EventProceedings of the 18th Annual Symposium on Computational Geometry (SCG'02) - Barcelona, Spain
    Duration: Jun 5 2002Jun 7 2002

    Other

    OtherProceedings of the 18th Annual Symposium on Computational Geometry (SCG'02)
    CountrySpain
    CityBarcelona
    Period6/5/026/7/02

    Fingerprint

    Higher Dimensions
    Three-dimension
    Incidence
    Circle
    Distinct
    Convex Curve
    Coplanar
    Triangular pyramid
    Plane Curve
    Congruent
    Lower bound
    Upper bound
    Curve
    Arbitrary

    Keywords

    • Circles
    • Combinatorial geometry
    • Incidences
    • Three dimensions

    ASJC Scopus subject areas

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

    Cite this

    Aronov, B., Koltun, V., & Sharir, M. (2002). Incidences between points and circles in three and higher dimensions. In Proceedings of the Annual Symposium on Computational Geometry (pp. 116-122)

    Incidences between points and circles in three and higher dimensions. / Aronov, Boris; Koltun, Vladlen; Sharir, Micha.

    Proceedings of the Annual Symposium on Computational Geometry. 2002. p. 116-122.

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

    Aronov, B, Koltun, V & Sharir, M 2002, Incidences between points and circles in three and higher dimensions. in Proceedings of the Annual Symposium on Computational Geometry. pp. 116-122, Proceedings of the 18th Annual Symposium on Computational Geometry (SCG'02), Barcelona, Spain, 6/5/02.
    Aronov B, Koltun V, Sharir M. Incidences between points and circles in three and higher dimensions. In Proceedings of the Annual Symposium on Computational Geometry. 2002. p. 116-122
    Aronov, Boris ; Koltun, Vladlen ; Sharir, Micha. / Incidences between points and circles in three and higher dimensions. Proceedings of the Annual Symposium on Computational Geometry. 2002. pp. 116-122
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