Curves of constant curvature and torsion in the 3-sphere. (English) Zbl 1402.53009

Summary: 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.


53A35 Non-Euclidean differential geometry
Full Text: DOI arXiv


[1] 10.1023/B:GERG.0000022580.08717.40 · Zbl 1055.83005
[2] 10.1090/S0002-9939-97-03692-7 · Zbl 0876.53035
[3] 10.1017/CBO9780511998188
[4] 10.1007/BF01389060 · Zbl 0585.53051
[5] ; Stoker, Differential geometry. Pure and Applied Math., 20 (1969)
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