Consider the tube defined by
$$ α(s,v) = c(s) + r\big( \cos(v)\,b(s) + \sin(v)\,n(s)\big) , \quad r > 0. $$
Here $c$ is a Frenet curve with curvature $k>0$, torsion $\tau$ and $(t ,n, b) $ is the Frenet frame. $r$ is the radius of (toroidal) tube.
i) Characterize the curves $c$ such that the tube defined above is a regular surface.
ii) Analyze the geometry of the tube identifying elliptic, parabolic and hyperbolic points and also singular points.
iii) Analyze the principal curvature and asymptotic lines of the tube surface.
I'm thinking about this issue for more than a day,cannot resolve it. I need some idea to present it in about 2 days as I still have no idea how even to start the question. Can someone give me any tips about these items, please?