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 language | English (US) |
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Pages | 116-122 |
Number of pages | 7 |
DOIs | |
State | Published - 2002 |
Event | Proceedings of the 18th Annual Symposium on Computational Geometry (SCG'02) - Barcelona, Spain Duration: Jun 5 2002 → Jun 7 2002 |
Other
Other | Proceedings of the 18th Annual Symposium on Computational Geometry (SCG'02) |
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Country/Territory | Spain |
City | Barcelona |
Period | 6/5/02 → 6/7/02 |
Keywords
- Circles
- Combinatorial geometry
- Incidences
- Three dimensions
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics