Line Aspect Ratio

TL;DR

This paper introduces the line aspect ratio (AR) as a geometric measure to ensure the existence of finite guard sets for segment guarding in polygons under weak visibility, broadening applicability beyond previous constraints.

Contribution

It defines the line aspect ratio (AR) based on tangent lines to convex and reflex vertices, proving finite guard sets exist with size bounded by AR for polygons with constant or polynomial AR.

Findings

Finite guard sets exist when AR is constant or polynomial.

Guard set size is bounded by O(AR).

The framework generalizes previous geometric assumptions.

Abstract

We address the problem of covering a target segment $\overline{uv}$ using a finite set of guards $\mathcal{S}$ placed on a source segment $\overline{xy}$ within a simple polygon $\mathcal{P}$, assuming weak visibility between the target and source. Without geometric constraints, $\mathcal{S}$ may be infinite, as shown by prior hardness results. To overcome this, we introduce the {\it line aspect ratio} (AR), defined as the ratio of the \emph{long width} (LW) to the \emph{short width} (SW) of $\mathcal{P}$. These widths are determined by parallel lines tangent to convex vertices outside $\mathcal{P}$ (LW) and reflex vertices inside $\mathcal{P}$ (SW), respectively. Under the assumption that AR is constant or polynomial in $n$ (the polygon's complexity), we prove that a finite guard set $\mathcal{S}$ always exists, with size bounded by $\mathcal{O}(\text{AR})$. This AR-based framework generalizes some previous assumptions, encompassing a broader class of polygons. Our result establishes a framework guaranteeing finite solutions for segment guarding under practical and intuitive geometric constraints.