Catching a Polygonal Fish with a Minimum Net

TL;DR

This paper investigates the problem of designing a minimal set of lines within a unit square that guarantees intersecting a polygon that can be translated, rotated, and scaled arbitrarily, and proves the optimal solutions are regular grids or equidistant lines.

Contribution

It characterizes the optimal line configurations for guaranteed intersection with any transformed polygon within a unit square, showing they are regular grids or equidistant lines.

Findings

Optimal line sets are regular grids or equidistant lines.

The number of lines is minimized based on the polygon's properties.

The solution adapts to any polygon via transformations.

Abstract

Given a polygon $P$ in the plane that can be translated, rotated and enlarged arbitrarily inside a unit square, the goal is to find a set of lines such that at least one of them always hits $P$ and the number of lines is minimized. We prove the solution is always a regular grid or a set of equidistant parallel lines, whose distance depends on $P$.