Line-of-Sight Pursuit in Monotone and Scallop Polygons
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Bibliographic record
Abstract
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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it