# How to find all possible curvatures for any smooth curve in a donut?

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...

• All curvatures? There are many curvatures : Normal curvature, principal normal curvatures, geodesic curvature, Gaussian curvature, Mean curvature, Integral curvature, geodesic torsion.which are some common curvatures. – Narasimham 2 days ago

$$[ (3+ \frac{\sin v}{kn})\sin u, (3+ \frac{\sin v}{kn} )\cos u, \frac{\cos v}{kn}]$$
Yes, when $$kn \to \infty$$ we are left with a thin circle around the middle of torus.