Parameterization of tubular surface and analysis of its geometry

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?

• Someone can talk me when find topics that help answer? Mar 27 '16 at 19:43
• When $r=0$ it is space curve. If $r >0$ it becomes thick tube /tubular surface. Please check if $r(cos(v) b(s) + sin(v)n(s))$ needs to be considered. Mar 27 '16 at 21:58
• I believe that has to be considered, sought things related to this topic and this part $r(cos(v)b(s) + sin(v)n(s))$ causes it to create the surface of the tube votes curve $c$ Mar 27 '16 at 22:06
• Thank you Narasimham!! My writing was very bad and I myself did not understand the data, the editing in question was good. Mar 27 '16 at 22:19
• If $r=0$, it doesn't make much sense to talk about hyperbolic points, principal curvatures, etc. So, it seems clear to me that we should assume $r > 0$ Mar 28 '16 at 13:43

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.

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.
• Not related to $Tubes$ (Hermann Weyl, Alfred Gray) , I hope. Mar 28 '16 at 13:48