Can Differential Geometry Solve This Challenging Curve Containment Problem?

[Dahaka14]

Homework Statement

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

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

Homework Equations

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

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!

 

[lanedance]

might be a start that the curve is constrained to a sphere so t will be tangent to the sphere

define
r = sigma-p

then
t.r = 0

differentiating and some frenet substitution gets to
1/k(s) = -n.r

this is a step closer to the equation...

 

[lanedance]

also worth thinking about this physically.. when the torsion is zero, the curve can be contained in a plane, and a plane intersecting a sphere gives a circle

the largest circle is a great circle of the sphere, & on this path the normal will point towards the centre of the circle

so where does the normal point as the radius of the circle is made smaller? and how does the torsion relate to a change in radius?