Five Focal Area Invariants of N-Periodics in the Elliptic Billiard

Reznik, Garcia and Koiller listed many quantities that are experimentally constant along the families of \(N\)-periodic billiard trajectories in an ellipse with a confocal elliptic caustic. We prove five of their rows on areas of polygons built from the trajectory and the two foci, for trajectories of least period \(N\), every winding number and signed areas. For every \(N\), the area of the image of the outer polygon under the inversion in a unit circle centred at a focus is a constant multiple \(\kappa>0\) of the area of the image of the trajectory, with the same \(\kappa\) for both foci (row \(k_{806,a}\)); \(\kappa=\frac12\) for \(N=4\), where the source prints \(2\). For odd \(N\), the products over the two foci of these inversive areas of the outer polygon (row \(k_{904,a}\)), of the focal pedal areas of the outer polygon (row \(k_{605,a}\)) and of the areas of the focal dual polygons (row \(k_{907,a}\)) are constant; the dagger printed in row \(k_{907,a}\) is not defined in the source, and we read the row without it, in two readings of “dual”. For \(N\equiv2\pmod4\), the focal antipedal area divided by the focal pedal area is constant (row \(k_{404}\)). The focal pedal area is positive at every trajectory, so this ratio is always defined; its numerator can vanish identically, and for \(N\equiv0\pmod4\) the ratio is constant only on the families where it does. The proofs use Stachel's parametrisation by Jacobi elliptic functions, in which every area is an elliptic function of the phase; a list of its possible poles and two symmetries reduce each row to Liouville's theorem or to a comparison of residues. The rows are due to Reznik, Garcia and Koiller. Rows \(k_{904,a}\) and \(k_{605,a}\) were first proved by Kriebel in two deposits of 15 October 2026; our proofs of them are independent, and we claim no novelty for these two rows. The case \(N=4\) of row \(k_{806,a}\) is in an earlier note of the author, and the ratio of row \(k_{404}\) also follows from two earlier notes of the author together with the positivity of the focal pedal area, which Kriebel proved. To our knowledge, row \(k_{806,a}\) for \(N\ne4\), row \(k_{907,a}\) and the statement of row \(k_{404}\) as a ratio defined at every trajectory are new. This is an unrefereed note.