### Abstract

We show that the number of incidences between m distinct points and n distinct circles in ℝ^{d}, for any d ≥ 3, is O(m ^{6/11}n^{9/11}κ(m^{3}/n) + m^{2/3}n ^{2/3} + m + n), where κ(n) = (log n)^{O(α2(n))} and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Sharir, or rather with its slight improvement by Agarwal et al., for the planar case. We also show that the number of incidences between m points and n unrestricted convex (or bounded-degree algebraic) plane curves, no two in a common plane, is O(m^{4/7}n^{17/21} + m ^{2/3}n^{2/3} + m + n), in any dimension d ≥ 3. 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 the lower bound for the number of distinct distances in a set of n points in 3-space. Another application is an improved bound for the number of incidences (or, rather, containments) between lines and reguli in three dimensions. The latter result has already been applied by Feldman and Sharir to obtain a new bound on the number of joints in an arrangement of lines in three dimensions.

Original language | English (US) |
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Pages (from-to) | 185-206 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2005 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Discrete and Computational Geometry*,

*33*(2), 185-206. https://doi.org/10.1007/s00454-004-1111-9