On the zone of a circle in an arrangement of lines

TL;DR

This paper investigates the complexity of the zone of a circle in line arrangements, providing geometric arguments to support the conjecture that it is linear, and analyzing the combinatorial structure of related sequences.

Contribution

It introduces geometric constraints on line segments with endpoints on a circle, ruling out certain complex configurations and connecting sequence properties to zone complexity.

Findings

Certain configurations of line segments on a circle are impossible.

Hart-Sharir sequences cannot appear as subsequences in the sequence representing the zone.

A link is established between sequence properties and the linearity of zone complexity.

Abstract

Let $\mathcal L$ be a set of $n$ lines in the plane, and let $C$ be a convex curve in the plane, like a circle or a parabola. The "zone" of $C$ in $\mathcal L$, denoted $\mathcal Z(C,\mathcal L)$, is defined as the set of all cells in the arrangement $\mathcal A(\mathcal L)$ that are intersected by $C$. Edelsbrunner et al. (1992) showed that the complexity (total number of edges or vertices) of $\mathcal Z(C,\mathcal L)$ is at most $O(n\alpha(n))$, where $\alpha$ is the inverse Ackermann function. They did this by translating the sequence of edges of $\mathcal Z(C,\mathcal L)$ into a sequence $S$ that avoids the subsequence $ababa$. Whether the worst-case complexity of $\mathcal Z(C,\mathcal L)$ is only linear is a longstanding open problem. Since the relaxation of the problem to pseudolines does have a $\Theta(n\alpha(n))$ bound, any proof of $O(n)$ for the case of straight lines must necessarily use geometric arguments. In this paper we present some such geometric arguments. We show that, if $C$ is a circle, then certain configurations of straight-line segments with endpoints on $C$ are impossible. In particular, we show that there exists a Hart-Sharir sequence that cannot appear as a subsequence of $S$. The Hart-Sharir sequences are essentially the only known way to construct $ababa$-free sequences of superlinear length. Hence, if it could be shown that every family of $ababa$-free sequences of superlinear-length eventually contains all Hart-Sharir sequences, it would follow that the complexity of $\mathcal Z(C,\mathcal L)$ is $O(n)$ whenever $C$ is a circle.