Catching Polygons

TL;DR

This paper investigates optimal arrangements of axis-aligned lines within a unit square to prevent non-transverse intersections with rectangles of varying aspect ratios, identifying when parallel lines or grids are optimal.

Contribution

It characterizes the optimal line arrangements based on rectangle aspect ratios, providing precise thresholds for when parallel lines or grids are most effective.

Findings

Optimal arrangements depend on rectangle aspect ratio.

Parallel lines are optimal for certain aspect ratios.

Grid arrangements are optimal for other aspect ratios.

Abstract

Consider an arrangement of $k$ lines intersecting the unit square. There is some minimum scaling factor so that any placement of a rectangle with aspect ratio $1 \times p$ with $p\geq 1$ must non-transversely intersect some portion of the arrangement or unit square. Assuming the lines of the arrangement are axis-aligned, we show the optimal arrangement depends on the aspect ratio of the rectangle. In particular, the optimal arrangement is either evenly spaced parallel lines or an evenly spaced grid of lines. We present the precise aspect ratios of rectangles for which each of the two nets is optimal.