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Curves of
constant curvature and torsion in the 3-sphere
Debraj Chakrabarti, Rahul Sahay and Jared Williams
Vol. 12 (2019), No. 2, 235–255
DOI: 10.2140/involve.2019.12.235
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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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Authors |
| Debraj Chakrabarti | |
| Department of Mathematics Central Michigan University Mt. Pleasant, MI United States |
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| Rahul Sahay | |
| University of California Berkeley, CA United States |
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| Jared Williams | |
| Department of Physics and
Astronomy University of Missouri Columbia, MO United States |
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