On Triple Intersections of Three Families of Unit Circles

Orit E. Raz; Micha Sharir; József Solymosi

doi:10.1007/s00454-015-9734-6

On Triple Intersections of Three Families of Unit Circles

Orit E. Raz^*, Micha Sharir, József Solymosi

^*Corresponding author for this work

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

8 Scopus citations

Abstract

Let p₁, p₂, p₃ be three distinct points in the plane, and, for i = 1, 2, 3, let C_i be a family of n unit circles that pass through p_i. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n^11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szabó (Combinatorica 32:537–571, 2012) [and by Elekes and Rónyai (J Comb Theory Ser A 89:1–20, 2000)].

Original language	English
Pages (from-to)	930-953
Number of pages	24
Journal	Discrete and Computational Geometry
Volume	54
Issue number	4
DOIs	https://doi.org/10.1007/s00454-015-9734-6
State	Published - 13 Oct 2015
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2015, Springer Science+Business Media New York.

Keywords

Combinatorial geometry
Incidences
Unit circles

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10.1007/s00454-015-9734-6

Cite this

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title = "On Triple Intersections of Three Families of Unit Circles",

abstract = "Let p1, p2, p3 be three distinct points in the plane, and, for i = 1, 2, 3, let Ci be a family of n unit circles that pass through pi. We address a conjecture made by Sz{\'e}kely, and show that the number of points incident to a circle of each family is O(n11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szab{\'o} (Combinatorica 32:537–571, 2012) [and by Elekes and R{\'o}nyai (J Comb Theory Ser A 89:1–20, 2000)].",

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AU - Sharir, Micha

AU - Solymosi, József

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N2 - Let p1, p2, p3 be three distinct points in the plane, and, for i = 1, 2, 3, let Ci be a family of n unit circles that pass through pi. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szabó (Combinatorica 32:537–571, 2012) [and by Elekes and Rónyai (J Comb Theory Ser A 89:1–20, 2000)].

AB - Let p1, p2, p3 be three distinct points in the plane, and, for i = 1, 2, 3, let Ci be a family of n unit circles that pass through pi. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n11/6), improving an earlier bound for this problem due to Elekes et al. (Comb Probab Comput 18:691–705, 2009). The problem is a special instance of a more general problem studied by Elekes and Szabó (Combinatorica 32:537–571, 2012) [and by Elekes and Rónyai (J Comb Theory Ser A 89:1–20, 2000)].

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On Triple Intersections of Three Families of Unit Circles

Abstract

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Keywords

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