Sweeping Arrangements of Non-Piercing Curves in Plane

TL;DR

The paper addresses sweeping arrangements of non-piercing curves in the plane and proves that a finite non-piercing family $\Gamma$ can be swept by any sweep curve $\gamma\in\Gamma$ while preserving the non-piercing property and keeping a designated point $P\in\tilde{\gamma}$ throughout. It introduces two conceptual tools, minimal lens bypassing and minimal triangle bypassing, and a set of discrete sweep operations (Take a loop, bypass a digon, bypass a triangle, bypass a visible vertex) to maintain the invariant. In addition, it provides a shorter proof of the Snoeyink–Hershberger result for $2$-intersecting curves and derives broad applications in computational and combinatorial geometry, including improved approximation schemes and planar supports for non-piercing regions. The results suggest that non-piercing is a natural, elementary condition under which sweep-based methods extend beyond pseudodisks, with potential impact on related packing, covering, and hypergraph problems.

Abstract

Let $Γ$ be a finite set of Jordan curves in the plane. For any curve $γ\in Γ$, we denote the bounded region enclosed by $γ$ as $\tildeγ$. We say that $Γ$ is a non-piercing family if for any two curves $α, β\in Γ$, $\tildeα \setminus \tildeβ$ is a connected region. A non-piercing family of curves generalizes a family of $2$-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG '89) proved that if we are given a family $\mathcal{C}$ of $2$-intersecting curves and a fixed curve $C\in\mathcal{C}$, then the arrangement can be \emph{swept} by $C$, i.e., $C$ can be continuously shrunk to any point $p \in \tilde{C}$ in such a way that the we have a family of $2$-intersecting curves throughout the process. In this paper, we generalize the result of Snoeyink and Hershberger to the setting of non-piercing curves. We show that given an arrangement of non-piercing curves $Γ$, and a fixed curve $γ\in Γ$, the arrangement can be swept by $γ$ so that the arrangement remains non-piercing throughout the process. We also give a shorter and simpler proof of the result of Snoeyink and Hershberger and cite applications of their result, where our result leads to a generalization.

Figures (7)

• Figure 1: Pseudocircles, non-piercing curves and piercing curves
• Figure 6: An example of non-piercing curves where the only operation available for the sweep is to bypass either the vertex labeled $a$ or the vertex labeled $b$.
• Figure 7: The operation of digon bypassing of curves $\alpha$ and $\beta$. The dotted segments are the intersection of a single curve $\delta\in\Gamma$ with the lens $L$.
• Figure 8: The operation of minimal triangle bypassing of curves $\alpha$ and $\beta$. The dotted segments are the intersection of a curve $\delta\in\Gamma$ with the triangle $T$.
• Figure 9: The operation of minimal triangle bypassing of curves $\alpha$ and $\beta$.

Theorems & Definitions (16)

• Theorem 0: Sweeping non-piercing arrangements
• Lemma 1: DBLP:journals/dcg/RoyGRR18
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 0: Support for non-piercing regions
• Lemma 0
• Corollary 0
• Lemma 0
• Lemma 0