On rich lenses in planar arrangements of circles and related problems

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

doi:10.4230/LIPIcs.SoCG.2021.35

On rich lenses in planar arrangements of circles and related problems

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

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

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/k³)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/k³)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.

Original language	English
Title of host publication	37th International Symposium on Computational Geometry, SoCG 2021
Editors	Kevin Buchin, Eric Colin de Verdiere
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages	35:1-35:15
Number of pages	15
ISBN (Electronic)	9783959771849
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2021.35
State	Published - 1 Jun 2021
Event	37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States Duration: 7 Jun 2021 → 11 Jun 2021

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	189
ISSN (Print)	1868-8969

Conference

Conference	37th International Symposium on Computational Geometry, SoCG 2021
Country/Territory	United States
City	Virtual, Buffalo
Period	7/06/21 → 11/06/21

Bibliographical note

Publisher Copyright:
© Esther Ezra, Orit E. Raz, Micha Sharir, and Joshua Zahl; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).

Keywords

Circles
Incidences
Lenses
Polynomial partitioning

Access to Document

10.4230/LIPIcs.SoCG.2021.35

Cite this

Ezra, E., Raz, O. E., Sharir, M., & Zahl, J. (2021). On rich lenses in planar arrangements of circles and related problems. In K. Buchin, & E. C. de Verdiere (Eds.), 37th International Symposium on Computational Geometry, SoCG 2021 (pp. 35:1-35:15). Article 35 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 189). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2021.35

Ezra, Esther ; Raz, Orit E. ; Sharir, Micha et al. / On rich lenses in planar arrangements of circles and related problems. 37th International Symposium on Computational Geometry, SoCG 2021. editor / Kevin Buchin ; Eric Colin de Verdiere. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2021. pp. 35:1-35:15 (Leibniz International Proceedings in Informatics, LIPIcs).

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abstract = "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(n3/2 log(n/k3)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(n3/2 log(n/k3)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.",

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Ezra, E, Raz, OE, Sharir, M & Zahl, J 2021, On rich lenses in planar arrangements of circles and related problems. in K Buchin & EC de Verdiere (eds), 37th International Symposium on Computational Geometry, SoCG 2021., 35, Leibniz International Proceedings in Informatics, LIPIcs, vol. 189, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 35:1-35:15, 37th International Symposium on Computational Geometry, SoCG 2021, Virtual, Buffalo, United States, 7/06/21. https://doi.org/10.4230/LIPIcs.SoCG.2021.35

On rich lenses in planar arrangements of circles and related problems. / Ezra, Esther; Raz, Orit E.; Sharir, Micha et al.
37th International Symposium on Computational Geometry, SoCG 2021. ed. / Kevin Buchin; Eric Colin de Verdiere. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2021. p. 35:1-35:15 35 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 189).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Zahl, Joshua

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PY - 2021/6/1

Y1 - 2021/6/1

N2 - 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(n3/2 log(n/k3)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(n3/2 log(n/k3)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.

AB - 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(n3/2 log(n/k3)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(n3/2 log(n/k3)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.

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Ezra E, Raz OE, Sharir M, Zahl J. On rich lenses in planar arrangements of circles and related problems. In Buchin K, de Verdiere EC, editors, 37th International Symposium on Computational Geometry, SoCG 2021. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2021. p. 35:1-35:15. 35. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.SoCG.2021.35

On rich lenses in planar arrangements of circles and related problems

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Bibliographical note

Keywords

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