On rich lenses in planar arrangements of circles and related problems

Esther Ezra*, Orit E. Raz*, Micha Sharir*, Joshua Zahl*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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(n3/2 log(n/k3)k5/2 + 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(n3/2 log(n/k3)k3/2 + 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 languageAmerican English
Title of host publication37th International Symposium on Computational Geometry, SoCG 2021
EditorsKevin Buchin, Eric Colin de Verdiere
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages15
ISBN (Electronic)9783959771849
StatePublished - 1 Jun 2021
Event37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States
Duration: 7 Jun 202111 Jun 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference37th International Symposium on Computational Geometry, SoCG 2021
Country/TerritoryUnited States
CityVirtual, Buffalo

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).


  • Circles
  • Incidences
  • Lenses
  • Polynomial partitioning


Dive into the research topics of 'On rich lenses in planar arrangements of circles and related problems'. Together they form a unique fingerprint.

Cite this