### Abstract

We show that the number of incidences between m distinct points and n distinct circles in ℝ^{3} is O(m^{4/7}n^{17/21} + m^{2/3}n^{2/3} + m + n); the bound is optimal for m ≥ n^{3/2}. 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 ℝ^{d}, 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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Title of host publication | Proceedings of the Annual Symposium on Computational Geometry |

Pages | 116-122 |

Number of pages | 7 |

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) |
---|---|

Country | Spain |

City | Barcelona |

Period | 6/5/02 → 6/7/02 |

### Fingerprint

### 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

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - Incidences between points and circles in three and higher dimensions

AU - Aronov, Boris

AU - Koltun, Vladlen

AU - Sharir, Micha

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Circles

KW - Combinatorial geometry

KW - Incidences

KW - Three dimensions

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UR - http://www.scopus.com/inward/citedby.url?scp=0036355627&partnerID=8YFLogxK

M3 - Conference contribution

SP - 116

EP - 122

BT - Proceedings of the Annual Symposium on Computational Geometry

ER -