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Abstract
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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.
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Keywords
Frenet–Serret equations, constant curvature and torsion,
geodesic curvature, helix, 3-sphere, curves in the 3-sphere
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Mathematical Subject Classification 2010
Primary: 53A35
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Milestones
Received: 23 June 2017
Revised: 13 October 2017
Accepted: 22 April 2018
Published: 8 October 2018
Communicated by Colin Adams
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