Paper detail

ccc-Autoevolutes

ccc-Autoevolutes are closed constant curvature space curves which are their own evolutes. A modified Frenet equation produces constant curvature curves such that the curve on $[0, π]$ is congruent to the evolute on $[π, 2π]$ and vice versa. Closed curves are then congruent to their evolutes. If the ruled surface spanned by the principal normals between curve and evolute is a Möbius band then the curve is its own evolute. We use symmetries to construct closed curves by solving 2-parameter problems numerically. The smallest autoevolute which we found is a trefoil knot parametrized by three periods $[0, 6π]$.Our smallest closed solution of the ODE is parametrized by two periods.