Next: Nowhere differentiable caustics Up: On nonconvex caustics of Previous: Caustics of convex billiards

Caustics of tables of constant width

If tex2html_wrap_inline1150 is an invariant curve of the billiard map, then each trajectory belonging to an orbit on tex2html_wrap_inline878 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 tex2html_wrap_inline1158 of the tangent to the boundary instead of using the arc length s of the table. The table is of constant width if tex2html_wrap_inline1030 satisfies tex2html_wrap_inline1164 . If tex2html_wrap_inline1166 , then tex2html_wrap_inline1168 . In the case, when the invariant curve tex2html_wrap_inline878 satisfies tex2html_wrap_inline1172 , the formula in Lemma 3.1 simplifies to

  equation179

which is the formula for an evolute of the table T (see [5])). If we plug in the Fourier series tex2html_wrap_inline1176 , we get

displaymath1178

We need tex2html_wrap_inline1180 to assure that the table is closed. The condition tex2html_wrap_inline1182 holds if tex2html_wrap_inline1184 for tex2html_wrap_inline1186 . For tex2html_wrap_inline1030 to be real, we have to require also tex2html_wrap_inline1190 . We summarize:

propo199

tex2html_wrap1286 tex2html_wrap1288
tex2html_wrap1290 tex2html_wrap1292
From formulas (1) and (2), we get tex2html_wrap_inline1228 which shows that tex2html_wrap_inline1030 has the same critical points as tex2html_wrap_inline880 . In other words, the vertices of T correspond to cusps of tex2html_wrap_inline880 .

The caustics in Figures 6,7 were drawn using trigonometric polynomials tex2html_wrap_inline1238 . While the functions tex2html_wrap_inline880 were real analytic, in general, the image of tex2html_wrap_inline880 has singularities. Indeed, the function tex2html_wrap_inline1244 has at least N zeros if it is a trigonometric polynomial of degree N [22]. If the zeros are simple, the curve tex2html_wrap_inline880 has at least N cusps counted with multiplicity. Note that for the caustics belonging to the invariant curve tex2html_wrap_inline1254 , the map tex2html_wrap_inline880 is 2 to 1.

tex2html_wrap1294 tex2html_wrap1296

tex2html_wrap1298 tex2html_wrap1300

Next: Nowhere differentiable caustics Up: On nonconvex caustics of Previous: Caustics of convex billiards

Oliver Knill
Fri Jun 12 13:34:37 CDT 1998