Midpoint Polygons and Four Inversive Area Invariants of N-Periodics in the Elliptic Billiard

Reznik, Garcia and Koiller listed many experimentally found invariants of the families of \(N\)-periodic billiard trajectories in an ellipse. Let \(A\) and \(A'\) be the signed areas of the trajectory and of its outer polygon, and \(A^\otimes\) and \(A'^\ominus\) those of the polygons obtained by inverting the vertices of the trajectory in the caustic and those of the outer polygon in the billiard ellipse. Their rows \(k_{807}\) and \(k_{809}\) state that \(AA^\otimes\) and \(A'A'^\ominus\) are constant along the family for even \(N\), and \(k_{808}\) and \(k_{810}\) that \(A/A^\otimes\) and \(A'/A'^\ominus\) are constant for odd \(N\) (records AMR-050-0049 to AMR-050-0052 of the UnsolvedMath collection). We prove all four, for every ellipse with semi-axes \(a>b>0\), every confocal elliptic caustic and every least period \(N\) of the stated parity, for convex and for self-intersecting trajectories. The inverted polygons are midpoint polygons, so their areas involve the area of the polygon of diagonals. In Stachel's canonical parametrisation by Jacobi elliptic functions, and by Stachel's conjugate-billiard relation, all areas are expressed by two elliptic functions \(F\), \(G\) of the phase. For even \(N\), \(F(u)F(u+\Delta)\) and \(F(u)G(u+\Delta)\) are constant by Liouville's theorem, the poles of one factor meeting zeros of the other that are forced by a folding symmetry. For odd \(N\), \(G=\gamma F\), and \(k_{808}=k_{810}=4\kappa/(2+\gamma)\) with \(\kappa=ab/(a''b'')\), where \(a''\), \(b''\) are the semi-axes of the caustic and \(\gamma\) is an explicit algebraic function with \(2+\gamma>0\). For \(N=4\) the constants are \(4a^4b^4/(a^2+b^2)^2\) and \(4a^2b^2\). The argument also gives the row \(k_{110}\), that \(AA''\) is constant for even \(N\), \(A''\) being the area of the inner polygon (record AMR-050-0004); this row is not new, as it follows at once from Chavez-Caliz's theorem on \(AA'\) and Stachel's relation \(A'=\kappa^2A''\). The cases \(N=3\) of \(k_{808}\) and \(k_{810}\) follow from a theorem of Akopyan, Schwartz and Tabachnikov and Stachel's relation, and the cases \(N=4\), \(6\) of \(k_{807}\) and \(k_{809}\) from Chavez-Caliz's theorem. This is an unrefereed note.