On Rich Lenses in Planar Arrangements of Circles and Related Problems

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

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Author Details

Esther Ezra
  • School of Computer Science, Bar Ilan University, Ramat Gan, Israel
Orit E. Raz
  • Institute of Mathematics, Hebrew University, Jerusalem, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel
Joshua Zahl
  • Department of Mathematics, University of British Columbia, Vancouver, Canada

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Esther Ezra, Orit E. Raz, Micha Sharir, and Joshua Zahl. On Rich Lenses in Planar Arrangements of Circles and Related Problems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Lenses
  • Circles
  • Polynomial partitioning
  • Incidences


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