2026-08-17T08:35:05Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13834 2024-03-06T10:52:50Z ddc:004 open_access

On Rich Lenses in Planar Arrangements of Circles and Related Problems Ezra, Esther Raz, Orit E. Sharir, Micha Zahl, Joshua Lenses Circles Polynomial partitioning Incidences 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Esther Ezra and Orit E. Raz and Micha Sharir and Joshua Zahl 2021 Is Part Of LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2021.35 urn:nbn:de:0030-drops-138343 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.35 eng https://creativecommons.org/licenses/by/4.0/legalcode