  In a recent paper published in Nature, Y.I. Sobolev et al. introduced the
concept of trajectoids: convex, rigid objects, which roll without slip or spin
on a flat plane along a prescribed periodic, unbounded planar path. A geometric
construction method applicable to many paths was introduced, and the theory was
experimentally verified using objects rolling downwards on slightly inclined
planes. The construction method was applicable to many but not all curves. A
possible extension of the method (referred to as period-n trajectoids) was also
proposed, but the limits of applicability were not clarified. Here, a geometric
proof is given for the existence of period-n trajectoids for any sufficiently
smooth prescribed curve. A somewhat different proof was recently proposed by O.
Muller independently from this work. We also highlight some related geometry
problems.
