Polar, Dual and Inversive Polygon Areas of N-Periodics in the Elliptic Billiard

Reznik, Garcia and Koiller listed many quantities found experimentally to be constant along the families of \(N\)-periodic billiard trajectories in an ellipse. We treat five rows of their table of inversive objects (records AMR-050-0056, -0057, -0058, -0059 and -0061 of the UnsolvedMath collection), built from a trajectory \(P\) and one focus \(f\): the inversion \(P^\dagger\) of \(P\) in the unit circle at \(f\), the polar polygon (the antipedal polygon of \(P^\dagger\)), its dual polygon, and the focal pedal and antipedal polygons. For least period \(N\), any winding number, a confocal elliptic caustic and signed areas we prove that the ratios of the polar area to the inversive area (row \(k_{813}\)) and to the dual area (row \(k_{814}\), credited by the source to A. Akopyan) are constant for every \(N\), the dual being the polar reciprocal of the polar polygon about the centre of the circle touched by its sides, as in the source's figure, or its point-wise inversion; for \(N=4\) the constants are \(2a^4/b^4\) and \(2a^6/b^{10}\). Rows \(k_{815}\) (odd \(N\)) and \(k_{816}\) (even \(N\)) use a symbol that the source does not define. Read as the area of the pedal polygon of \(P^\dagger\), as in the source's row \(k_{908,b}\), row \(k_{815}\) is false (exact certificate for \(N=3\)) and \(k_{816}\) is false for \(N=4\) (exact certificate) but true for every even \(N\ge6\); read as the area of the inversion of the focal pedal polygon, \(k_{815}\) is false and \(k_{816}\) true; read as the focal pedal area, both hold. Row \(k_{818}\) (\(N\equiv2\pmod4\)) follows from two earlier deposits of the author, doi:10.5281/zenodo.23094118 and doi:10.5281/zenodo.23079799; we reprove it, and its quotient is undefined on the families where the focal antipedal area vanishes identically, which are recorded in earlier deposits (for \(N=6\): exactly \(b/a=\frac12\)). The method is Stachel's Jacobi parametrisation, the phases at which a side of \(P\) is an isotropic line through \(f\), and Liouville's theorem. This is an unrefereed note.