Normal along the curve on surface of revolution  
I am struggling with the calculation of normal along the curve on a revolution surface.
Assume a revolution surface is constructed as:
S(t, Ɵ) = (L(t), R(t)cos(Ɵ), R(t)sin(Ɵ));
I apply the equation to calculate normal at a point as:
N(t, Ɵ) = dS/dt x dS/dƟ;
I am able to see that if the generatrix is a single curve as curve (1) then Normal is corrected.
However, when the generatrix is turned or passes the center line as curve (2) or (3), the normal changes the direction suddenly at some points (Nx = +/-1). They look not consistent along the curve, some normal point outward some normal point to inward as attached.


*

*I have tried to add term sign (dL), sign(dR) in equation of nomal to correct the normal direction but it does not work for cases generatrix passes the center line or turned (generatrix is not a single-valued function).

*But I found an interesting thing to decide the direction of normal of surface is:


*

*calculate geodesic curvature of curve:
Ng = dT/dt (t is arc length)

*Chose the direction of surface normal that makes angle with Ng an angle less 90 degree.
and correct: if (dot(N, Ng) <0) then N = -N.

*I tested, and results for any shape of generatrix (passes center line or turned) the normal along the curve are always consistent, no sudden change happened. So I suspected that, the Normal of revolution surface and geodesic-normal of curve (defined in Frenet curve dNg = dT/ds). Always make an angle less than 90 degree. (I cannot prove it). Assume Ɵ always increases along the curve.



My Question is:
- Can you help me give an equation the normal surface along the curve will always point out even in case generatrix passes the centerline or bended as curve 3. 
- Or can you help me confirm my argument about the angle between geodesic normal of curve and normal surface along a curve, if not can you give an counter example?.     
Thank you all so much.


 A: Basically, you've observed something that you might discover by looking at the hypotheses of the theorem for the Frenet-Serret formulas: they really only apply when the curvature is nonzero. Others have noted (and been frustrated by) this over the years, particularly several of my friends in computer graphics, where the sudden switch of normal vectors (which are often used to control camera orientation) leads to a very unpleasant viewing experience. 
Lots of folks have attempted to "fix" the frenet formulas in various ways; your fix might actually be a good one...who knows? But your "assume $\theta$ always increases along the curve" sounds like a warning bell to me -- an opportunity to craft an example that breaks your trick...but I haven't worked through it. 
One place to look for more information on this, by someone who thought about it a good deal, is an article by Bishop (1975) called "There's more than one way to frame a curve" (https://www.researchgate.net/publication/210222830_There_is_More_than_One_Way_to_Frame_a_Curve). 
I know this isn't an answer to the specific questions you asked, but I hope that it provides some insight into the area of the question you're asking. 
