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.
Bibliographical noteFunding Information:
\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 . 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 (firstname.lastname@example.org). \P Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada (email@example.com).
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- incidence geometry
- lens cutting
- polynomial partitioning