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The question is to find all possible curvature for any smooth curve in a donut. I parametrize the donut as $$((3+\sin v)\sin u, (3+\sin v)\cos u, \cos v)$$
My thought is, since if we keep shrinking a circle, curvature goes to infinity, we only need to find the minimum curvature of any normal section. But I have trouble finding that...
If instead of unit normal curvature we have a different normal curvature of the donut ( produced by tightening a string around the tube meridional section for instance ) then its parameterization would be:
$$ [ (3+ \frac{\sin v}{kn})\sin u, (3+ \frac{\sin v}{kn} )\cos u, \frac{\cos v}{kn}]$$
Yes, when $ kn \to \infty $ we are left with a thin circle around the middle of torus.
It is then no more a 2D surface embedded in 3-space, but just a line in 3-space.
