2021/02/01 by Karcher, Hermann, Tjaden, Ekkehard-H.
#Differential Geometry #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2102.00832
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.