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State and prove necessary and sufficient conditions for a Frenet curve $\gamma :I \to \mathbb{R}^3$ to be contained in some $2$-sphere $S^2=\{x \in \mathbb{R}^3; ||x−c||=r\}$ of centre $c$ and radius $r>0$.

The only one I can think of is the torsion needs be equal to $0$ as only a plane curve can lie on the surface of a two sphere. Would one of the conditions also be that the inner product of the tangent and position vectors must be $0$ (i.e. they are orthogonal) as any bit of tangent vector in the direction of the position would push the curve off the surface?

Ethan Bolker
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Numerical Disintegration
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  • Concerning the tangent & the position vectors: your assertion is basically correct, and can be proven rigorously by taking the derivative of $(\vec{\gamma}(s) - \vec{c}) \cdot (\vec{\gamma}(s) - \vec{c})$ with respect to $s$. – Michael Seifert Mar 23 '17 at 20:15
  • Only a planar curve can lie on the surface of a sphere? Huh? ... You need to start searching this site more carefully. This question has been answered numerous times. For example: There's this. This is also an exercise in my text. :) – Ted Shifrin Mar 23 '17 at 20:22
  • Your text seems to have it all Mr. Shifrin! I shall check out the post you've linked and poke around more (after doing some looking in it I think I've reconstructed the curve from Curvature and Torsion -- so thank you!). @MichaelSeifert thank you for your insight as well, I shall make sure to prove it rigorously! – Numerical Disintegration Mar 23 '17 at 20:50

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