(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]\sigma:I\rightarrow R^{3}[/tex] be a non-degenerate unit speed curve, and [tex]R[/tex] be a real number [tex]>0[/tex]. Fix a value [tex]s_{0}\in I[/tex]. Prove that:

(There exists a center [tex]\vec{p}\in R^{3}[/tex] such that [tex]\sigma(I)\subset S_{R}(p)[/tex])[tex]\iff[/tex] (There exists an angle [tex]\phi\in R[/tex] such that, for all [tex]s\in I[/tex], [tex]\frac{1}{\kappa(s)}=R\cos(\phi+\int_{s_{0}}^{s}\tau(\lambda)d\lambda)[/tex]).

2. Relevant equations

I know all of the equations for Frenet, but I'm not sure how to apply them.

3. The attempt at a solution

No idea where to start...I have been staring at this problem for many days now, and I haven't a clue what to do. Please help!

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# Differential Geometry Problem

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