Caustics of tables of constant width

If is an invariant curve of the billiard map, then each trajectory belonging to an orbit on both leaves and hits the table orthogonally. The period of the orbit is then 2. Because each of these orbits is a critical point of the length functional, the distance between the two impact points is constant and the table T is consequently a curve of constant width. It is more convenient to parametrize such a table by the angle of the tangent to the boundary instead of using the arc length s of the table. The table is of constant width if satisfies . If , then . In the case, when the invariant curve satisfies , the formula in Lemma 3.1 simplifies to

 

which is the formula for an evolute of the table T (see [5])). If we plug in the Fourier series , we get

We need to assure that the table is closed. The condition holds if for . For to be real, we have to require also . We summarize:

From formulas (1) and (2), we get which shows that has the same critical points as . In other words, the vertices of T correspond to cusps of .

The caustics in Figures 6,7 were drawn using trigonometric polynomials . While the functions were real analytic, in general, the image of has singularities. Indeed, the function has at least N zeros if it is a trigonometric polynomial of degree N [22]. If the zeros are simple, the curve has at least N cusps counted with multiplicity. Note that for the caustics belonging to the invariant curve , the map is 2 to 1.