Curves of Constant Curvature and Torsion in the 3-Sphere

Abstract

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the three-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion.