For the illustrations in Figure 3 and Figure 4, we constructed the map by finding so that is real. This problem is equivalent to solving
Numerically, the second root of this equation can be found efficiently with a Newton method. (For three-dimensional billiards, the situation is similar but slightly more complicated). In the simplest case , one has to solve
for . We do not know, whether the second
root is an algebraic expression in and .
One has the problem to find roots of a specific class of polynomials of
degree eight. More generally, one can ask, whether there exists a table of
constant width different from the circle, for which the return map is
algebraic in some coordinates.
We do not know of any smooth convex billiard table different from an ellipse,
where the return map is an algebraic expression
in some coordinates. This question is related to the open Birkhoff-Poritsky
conjecture  which claims that the ellipse
is the only integrable smooth convex billiard.
it was suggested that curves of equal thickness
might provide counter examples to the Birkhoff-Poritsky conjecture.
Acknowledgments. I want to thank E. Gutkin for discussions on billiards and caustics, P. Gruber for literature references, M. Wojtkowski for useful remarks on an earlier version of the manuscript and E. Amiran for helpful suggestions on the final draft. A first version of this paper was written at Caltech