Periodic orbits for square and rectangular billiards

The paper’s Theorem 3.11 classifies, for fixed tan(α0)=p/q with gcd(p,q)=2, all trajectories generated by varying P0: singular exactly when P0=2ℓ/p; otherwise periodic of type (p,q) with period K=p+q; and in the singular case the orbit lies on a generalized diagonal of length (p+q)/2−2 with the stated split of collisions. This matches the model’s three-part claim point-for-point. The paper proves the singular set via translating intersection points along the unfolded generalized diagonal, while the model uses a diophantine p a − q b = p P0 argument; both rely on the same unfolding framework and earlier results establishing K=p+q and the generalized-diagonal length and counts, so both are correct though the proofs differ in style .