Nonconstant Antipedal Area Ratios in Four-Periodic Elliptic Billiards

Alper Ferudun

For a periodic elliptic billiard, let A' be the area of the polygon formed by consecutive tangents to the ellipse, and let A*_M be the antipedal area of the original orbit with respect to a fixed point M. We give an exact counterexample to the claimed constancy of A'/A*_M for periods divisible by four, listed as k402 in Table 5 of Reznik, Garcia and Koiller's 2021 article Fifty New Invariants of N-Periodics in the Elliptic Billiard. The denominator is unprimed and the printed point condition is All; taking M to be the center O is allowed.

On a standard connected family of convex primitive four-periodic orbits in a fixed noncircular ellipse with one fixed confocal elliptic caustic, put s=a^2+b^2, D=a^2 b^2 and d(t)=(b^2-a^2) sin(t) cos(t). We derive the central-antipedal area directly from its defining lines and obtain A'/A*_O=D(D+d(t)^2)/(D^2+s^2 d(t)^2). Its sharp range on the displayed family is [D s^2/(s^2-2D)^2, 1], with distinct endpoints for every a>b>0. For a=4,b=3, an axis diamond and an axis-aligned rectangle share the caustic with semiaxes 16/5 and 9/5. They give the ratios 1 and 90000/113569, respectively. The continuous-family certificate proves reflection and strictly interior caustic tangency; all original, outer and antipedal polygons in the witnesses are finite, convex and positively oriented. No zero denominator, hyperbolic caustic, shorter repeated period or sampled numerical inference is used.

The simple four-periodic family, its caustic and its outer-polygon geometry are established material and are credited to prior literature. This note concerns the literal k402 assertion retained in the final 2021 table and frozen dataset record AMR-050-0019 (raw ID 5100019), not every source invariant or an author-confirmed intended correction. A bounded primary-source search did not identify a correction of this exact row, but does not certify novelty or absolute priority. Two standalone standard-library exact rational checkers and their actual results accompany the analytic proof. This AI-assisted manuscript is unrefereed; no independent human review or proof-assistant verification is claimed.

Author affiliation and contact: Mercury Software GmbH; alper@mercurycodelab.com; https://github.com/AlperTheKing.
