Pedal Areas of Periodic Elliptic Billiard Trajectories with Respect to an Arbitrary Point

Abstract

Reznik, Garcia and Koiller listed many quantities that are experimentally constant along the families of N-periodic billiard trajectories in an ellipse whose caustic is a confocal ellipse. We treat three of their rows on the signed areas of pedal polygons with respect to an arbitrary point M. Let A be the signed area of the trajectory, A′ that of its outer polygon, and A_M, A′_M the signed areas of their pedal polygons with respect to M. The rows state that A·A_M is constant for N ≡ 0 (mod 4) (row k203,a), A/A_M for N ≡ 2 (mod 4) (row k204) and A′/A′_M for N ≡ 0 (mod 4) (row k304). We prove them for trajectories of least period N, every winding number and every point M, real or complex (the two ratios wherever they are defined). For k203,a we use a Jacobi parametrisation of the family, in which the product is an elliptic function of the phase. Its possible poles lie at the poles of the vertices and at the points where a side normal is isotropic; these two sets are disjoint exactly when N is odd or divisible by 4, an order-reversing symmetry makes each factor vanish at the poles of the other, and Liouville's theorem concludes. For N ≡ 2 (mod 4) we prove A_M = κ(M)A, and for the outer polygon with N ≡ 0 (mod 4) we prove A′_M = κ′(M)A′, where κ and κ′ are quadratic in M with positive constant terms. So the ratios are constant wherever κ(M) ≠ 0, resp. κ′(M) ≠ 0; the exceptional set can be a real circle in star families (an exact example with N = 8 for the outer polygon). For even N the part of A_M that is linear in M vanishes. The rows are due to Reznik, Garcia and Koiller, and the parametrisation is Stachel's. The case M = O of k203,a is in an earlier note of the author, and the dependence of A′_M on M for every even N was proved by Kriebel; for k304 only the constancy of A′/A′_O for N ≡ 0 (mod 4) is new. 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-050-0011, AMR-050-0013, AMR-050-0016 (Reznik–Garcia–Koiller, Eighty new invariants of N-periodics in the elliptic billiard, arXiv:2004.12497v11, rows k203,a, k204 and k304; journal version Arnold Math. J. 7 (2021)).