Line-of-Sight Pursuit in Monotone and Scallop Polygons

We study a turn-based game in a simply connected polygonal environment $Q$ between a pursuer $P$ and an adversarial evader $E$. Both players can move in a straight line to any point within unit distance during their turn. The pursuer $P$ wins by capturing the evader, meaning that their distance satisfies $d(P, E) \leq 1$, while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader $E$ always knows the location of $P$. Meanwhile, $P$ only has line-of-sight visibility: $P$ observes the evader's position only when the line segment connecting them lies entirely within the polygon. Therefore $P$ must search for $E$ when the evader is hidden from view. We provide a winning strategy for $P$ in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line $L$ can be moved continuously over $Q$ so that (1) $L \cap Q$ is a line segment and (2) every point on the boundary $\partial Q$ is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook's strategy during its pursuit phase, rather than the well-known lion's strategy. The rook's strategy is crucial for obtaining a capture time that is linear in the area of $Q$. For both monotone and scallop polygons, our algorithm has a capture time of $O(n(Q) + \mbox{area}(Q))$, where $n(Q)$ is the number of polygon vertices.