COUNTING and CUTTING RICH LENSES in ARRANGEMENTS of CIRCLES

Esther Ezra; Orit E. Raz; Micha Sharir; Joshua Zahl

doi:10.1137/21M1409305

COUNTING and CUTTING RICH LENSES in ARRANGEMENTS of CIRCLES

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

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the maximum number of pairwise nonoverlapping 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). The second proof gives a bound that is weaker by a polylogarithmic factor. We then show that these bounds lead to the known bound of Agarwal et al. [J. ACM, 51 (2004), pp. 139-186] and Marcus and Tardos [J. Combin. Theory Ser. A, 113 (2006), pp. 675-691] 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	English
Pages (from-to)	958-974
Number of pages	17
Journal	SIAM Journal on Discrete Mathematics
Volume	36
Issue number	2
DOIs	https://doi.org/10.1137/21M1409305
State	Published - 2022

Bibliographical note

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics

Funding

\ast Received by the editors April 1, 2021; accepted for publication (in revised form) December 20, 2021; published electronically April 11, 2022. A preliminary version of this work appeared in the Proceedings of the 37th International Symposium on Computational Geometry [7]. https://doi.org/10.1137/21M1409305 Funding: The first author's work was partially supported by NSF CAREER under grant CCF:AF-1553354 and by grant 824/17 from the Israel Science Foundation. The third author's work was partially supported by ISF grant 260/18, by grant 1367/2016 from the German-Israeli Science Foundation (GIF), and by the Blavatnik Research Fund in Computer Science at Tel Aviv University. The fourth author's work was supported by an NSERC Discovery Grant. \dagger School of Computer Science, Bar Ilan University, Ramat Gan, 5290002, Israel (ezraest@ cs.biu.ac.il). \ddagger Institute of Mathematics, Hebrew University, Givat-Ram, 919004, Jerusalem, Israel (oritraz@ mail.huji.ac.il). \S School of Computer Science, Tel Aviv University, Ramat Aviv, 6997801, Tel Aviv, Israel ([email protected]). \P Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada ([email protected]).

Funders	Funder number
Blavatnik Research Fund in Computer Science
German–Israeli Science Foundation
National Science Foundation	824/17, AF-1553354
Natural Sciences and Engineering Research Council of Canada
German-Israeli Foundation for Scientific Research and Development
Israel Science Foundation	1367/2016, 260/18
Tel Aviv University

Keywords

incidence geometry
lens cutting
polynomial partitioning

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10.1137/21M1409305

Cite this

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title = "COUNTING and CUTTING RICH LENSES in ARRANGEMENTS of CIRCLES",

abstract = "We show that the maximum number of pairwise nonoverlapping 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). The second proof gives a bound that is weaker by a polylogarithmic factor. We then show that these bounds lead to the known bound of Agarwal et al. [J. ACM, 51 (2004), pp. 139-186] and Marcus and Tardos [J. Combin. Theory Ser. A, 113 (2006), pp. 675-691] 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.",

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author = "Esther Ezra and Raz, {Orit E.} and Micha Sharir and Joshua Zahl",

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doi = "10.1137/21M1409305",

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COUNTING and CUTTING RICH LENSES in ARRANGEMENTS of CIRCLES

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