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 3-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion.
"Curves of constant curvature and torsion in the 3-sphere." Involve 12 (2) 235 - 255, 2019. https://doi.org/10.2140/involve.2019.12.235