Legendre Collapse
This week we looked at evolutes e(t) = r(t) + n(t)/k(t) of simple closed convex curves r(t). The 4-vertex theorem means that the evolute has at least 4 cusps. The evolute is the singularity locus or caustic of the Legendre collapse of the waves starting at the boundary of the curve. In 1998, I wrote a paper about billiards in curves of equal with, especially in situations where k(t) is not regular so that evolute becomes a "fractal hedgehog": https://people.math.harvard.edu/~knil...
What is nice about curves of constant width that the evolute now belongs to an invariant curve of the billiard map where every orbit has period 2. In my paper, I also constructed related "fractal" lines of striction of ruled surfaces or caustics of a point in a Riemannian 2-manifold which are "fractal". I loved curves of cconstant width as one can explicitly give from the Fourier expansion of the curve the Fourier expansion of the curvature and so a Fourier expansion of the evolute. One mathematical motivation is the open problem whether there exists a convex table different from an ellipse for which the billiard return map is given explicitly as an algebraic map in some coordinates. I also had hoped to use such examples to construct smooth billiards with positive entropy. Nobody has found a smooth convex billiard with positive metric entropy! My research had been at a time, when I had tried to use subharmonic techniques for establishing positive Lyapunov exponents, a technique pioneered by Michael Herman.