I am trying to understand the properties of curves (especially closed curves) that live on the surface of a sphere. What are the implications on the curvature and torsion of a curve, of living at a constant distance from a fixed point ? Also, how does one connect pure space curve properties ($\kappa$ and $\tau$) with the properties of a curve lying on a surface($\kappa_g$ and $\kappa_n$)? I would also really appreciate any intuition behind any results.
What I have thought of/found till now:
- Being embedded on a spherical surface, the curve should have constant normal curvature: $\kappa_n = \rm const$.
- I also found the following question which proves a local constraint between $\kappa$ and $\tau$ for the space curve: Prove that a curve is spherical iff it satisfies the relation. But this relation does not give my any intuition.
Any references would be much appreciated. Thank you!
