Catching a Polygonal Fish with a Minimum Net

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 $P$.

The original problem gives the number of lines (k) and the shape as the input and asks for the smallest scaling of the shape that is always stabbed. For parallel lines, the result was already known for squares and disks. We discussed axis-parallel solutions but we did not find the rotation. For k=1 and a rectangle, the optimal solution is the diameter