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?
Best Answer
In English, these things are often called canal surfaces. They even have their own Wikipedia page here. If you google this term, you will find quite a lot of material.
There's some info here. It's in French, but judging by your name, peut-etre cela n'est pas un problem.
There is discussion of their lines of curvature here.
See also this question.
Such a surface can be regarded as the envelope of a family of spheres moving along the "centerline" curve $c$. If $c$ is the intersection of two offset surfaces, then the moving sphere is tangent to the base surfaces of these offset surfaces, so it forms a "fillet" between these two base surfaces. For this reason, canal surfaces are often used to perform filleting (or rounding or blending) in CAD systems.