Half-Angle Sines, Steiner Pedal Areas and Evolute Areas of N-Periodics in the Elliptic Billiard

Abstract

Reznik, Garcia and Koiller listed many quantities that they found experimentally to be constant along the families of N-periodic billiard trajectories in an ellipse with a confocal elliptic caustic. We treat four rows of their tables (records AMR-050-0002, -0028, -0039 and -0032 of the UnsolvedMath collection) for trajectories of least period N and every winding number, with signed areas; the source's families are simple, and our statements include them. Let A and A′ be the areas of the trajectory and of its outer polygon and θ_i the angles of the trajectory. Row k108 states that (A′/A)/∏_i sin(θ_i/2) is constant for N ≡ 2 (mod 4). A. Kriebel showed in a Zenodo deposit, with an exact six-periodic example, that this is false; we prove that the product (A′/A)∏_i sin(θ_i/2) is constant on every family with N ≡ 2 (mod 4), N ≥ 6, and that the printed quotient is constant on none. Row k502 states that A′/A′_K′ is constant for odd N, where A′_K′ is the area of the pedal polygon of the outer polygon with respect to its Steiner curvature centroid; we prove that it equals c²σ/(2(π_g − π_f)), with three further constants σ, π_f, π_g of the family, wherever it is defined, while its definedness on every family is not proved. Row k701 states that A/A_ev is constant for N > 4, A_ev being the area of the evolute polygon; we prove A_ev = νA for every N ≥ 5 with an explicit constant ν, so the ratio is constant if ν ≠ 0, which is observed numerically but not proved. For N = 6, A_ev = −c⁴A/(4a²b²) for every centrally symmetric hexagon inscribed in the ellipse with semi-axes a > b, c² = a² − b², and for N = 4 the ratio is not constant. Row k603, on focal antipedal distances, was proved by Kriebel in another deposit; we only record a confirmation. The proofs use Stachel's parametrisation by Jacobi elliptic functions, the zeros and poles of the quantities on the period torus, and Liouville's theorem. No printed or deposited proof of the product identity, of row k502 or of row k701 was found in the sources searched. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifiers: AMR-050-0002, AMR-050-0028, AMR-050-0039, AMR-050-0032