ccc-Autoevolutes

TL;DR

This paper introduces ccc-Autoevolutes, special closed constant curvature curves that are their own evolutes, constructed using symmetries and numerical methods, with the smallest example being a trefoil knot.

Contribution

It defines a new class of self-evolute curves, develops a modified Frenet equation for their construction, and numerically finds explicit examples including a trefoil knot.

Findings

Smallest autoevolute is a trefoil knot

Constructed closed curves using symmetry and numerical solutions

Identified conditions for curves to be their own evolutes

Abstract

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 $[0, \pi]$ is congruent to the evolute on $[\pi, 2\pi]$ 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 M\"obius 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 $[0, 6\pi]$.Our smallest closed solution of the ODE is parametrized by two periods.