## Abstract

We show that the maximum number of pairwise non-overlapping k-rich lenses (lenses formed by at least k circles) in an arrangement of n circles in the plane is O(n^{3/}^{2} log(n/k^{3})k^{−}^{5/2} + n/k), and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is O(n^{3/}^{2} log(n/k^{3})k^{−}^{3/2} + n). Two independent proofs of these bounds are given, each interesting in its own right (so we believe). We then show that these bounds lead to the known bound of Agarwal et al. (JACM 2004) and Marcus and Tardos (JCTA 2006) on the number of point-circle incidences in the plane. Extensions to families of more general algebraic curves and some other related problems are also considered.

Original language | American English |
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Title of host publication | 37th International Symposium on Computational Geometry, SoCG 2021 |

Editors | Kevin Buchin, Eric Colin de Verdiere |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 35:1-35:15 |

Number of pages | 15 |

ISBN (Electronic) | 9783959771849 |

DOIs | |

State | Published - 1 Jun 2021 |

Event | 37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States Duration: 7 Jun 2021 → 11 Jun 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 189 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 37th International Symposium on Computational Geometry, SoCG 2021 |
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Country/Territory | United States |

City | Virtual, Buffalo |

Period | 7/06/21 → 11/06/21 |

### Bibliographical note

Funding Information:Funding Esther Ezra: Work partially supported by NSF CAREER under grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation. Micha Sharir: Work partially supported by ISF Grant 260/18, by grant 1367/2016 from the German-Israeli Science Foundation (GIF), and by Blavatnik Research Fund in Computer Science at Tel Aviv University. Joshua Zahl: Work supported by an NSERC Discovery Grant.

Publisher Copyright:

© Esther Ezra, Orit E. Raz, Micha Sharir, and Joshua Zahl; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).

## Keywords

- Circles
- Incidences
- Lenses
- Polynomial partitioning