COUNTING and CUTTING RICH LENSES in ARRANGEMENTS of CIRCLES

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

Research output: Contribution to journalArticlepeer-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 languageEnglish
Pages (from-to)958-974
Number of pages17
JournalSIAM Journal on Discrete Mathematics
Volume36
Issue number2
DOIs
StatePublished - 2022

Keywords

  • incidence geometry
  • lens cutting
  • polynomial partitioning

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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