Curves of constant curvature and torsion in the 3-sphere

Debraj Chakrabarti, Rahul Sahay, Jared Williams

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish
Pages (from-to)235-255
Number of pages21
Issue number2
StatePublished - 2019


  • 3-sphere
  • Frenet–Serret equations
  • constant curvature and torsion
  • curves in the 3-sphere
  • geodesic curvature
  • helix


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