Incidences between points and circles in three and higher dimensions

Boris Aronov, Vladlen Koltun, Micha Sharir

Research output: Contribution to conferencePaperpeer-review

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
Pages116-122
Number of pages7
DOIs
StatePublished - 2002
Externally publishedYes
EventProceedings of the 18th Annual Symposium on Computational Geometry (SCG'02) - Barcelona, Spain
Duration: 5 Jun 20027 Jun 2002

Conference

ConferenceProceedings of the 18th Annual Symposium on Computational Geometry (SCG'02)
Country/TerritorySpain
CityBarcelona
Period5/06/027/06/02

Keywords

  • Circles
  • Combinatorial geometry
  • Incidences
  • Three dimensions

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