On rich lenses in planar arrangements of circles and related problems

TL;DR

This paper establishes upper bounds on the maximum number and total degrees of non-overlapping k-rich lenses in circle arrangements, providing new proofs and extending results to algebraic curves and related problems.

Contribution

It introduces two independent proofs for bounds on k-rich lenses in circle arrangements and extends these bounds to algebraic curves and related geometric problems.

Findings

Maximum number of non-overlapping k-rich lenses is bounded by a specific function.

Sum of degrees of lenses is bounded by a related function.

Results lead to known bounds on point-circle incidences.

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 $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{5/2}} + \frac{n}{k} \right)$, 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 $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{3/2}} + n\right)$. 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.