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

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 \(N\equiv0\pmod 4\) (row \(k_{203,a}\)), \(A/A_M\) for \(N\equiv2\pmod4\) (row \(k_{204}\)) and \(A'/A'_M\) for \(N\equiv0\pmod4\) (row \(k_{304}\)). 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 \(k_{203,a}\) 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 \(N\equiv2\pmod4\) we prove \(A_M=\kappa(M)A\), and for the outer polygon with \(N\equiv0\pmod4\) we prove \(A'_M=\kappa'(M)A'\), where \(\kappa\) and \(\kappa'\) are quadratic in \(M\) with positive constant terms. So the ratios are constant wherever \(\kappa(M)\ne0\), resp. \(\kappa'(M)\ne0\); 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 \(k_{203,a}\) 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 \(k_{304}\) only the constancy of \(A'/A'_O\) for \(N\equiv0\pmod4\) is new. This is an unrefereed note.