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A note on lenses in arrangements of pairwise intersecting circles in the plane

Rom Pinchasi

Abstract

Let $\F$ be a family of $n$ pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by $\F$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in $\F$ touch, then the geometric graph $G$ on the set of centers of the circles in $\F$ whose edges correspond to the lenses generated by $\F$ does not contain pairs of avoiding edges. That is, $G$ does not contain pairs of edges that are opposite edges in a convex quadrilateral. Such graphs are known to have at most $2n-2$ edges.

A note on lenses in arrangements of pairwise intersecting circles in the plane

Abstract

Let be a family of pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by is at most . This bound is tight. Furthermore, if no two circles in touch, then the geometric graph on the set of centers of the circles in whose edges correspond to the lenses generated by does not contain pairs of avoiding edges. That is, does not contain pairs of edges that are opposite edges in a convex quadrilateral. Such graphs are known to have at most edges.
Paper Structure (3 sections, 4 theorems, 10 figures)

This paper contains 3 sections, 4 theorems, 10 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem:main']}.
  3. Proof of Theorem \ref{['theorem:lenses']}.

Key Result

Theorem 1

Let ${\cal{F}}$ be a family of $n$ pairwise intersecting circles in the plane. Then ${\cal{A}}({\cal{F}})$ has at most $2n-2$ lenses. This bound is tight for $n \geqslant 4$.

Figures (10)

  • Figure 1: A family of $5$ pairwise intersecting circles with $8$ lenses.
  • Figure 2: $C_{2}$ cannot be contained in the disc bounded by $C_{1}$.
  • Figure 3: A pencil of circles. Only the two extreme circles, $C_{1}$ and $C_{5}$ in this figure, may create a lens.
  • Figure 4: The arc $s_{i}$ with its center $S_{i}$.
  • Figure 5: The impossible case where $s_{3}$ contains $s_{1}$.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2: ALPS01
  • Theorem 3: KL98V98
  • Theorem 4