$ \kappa = \cos s, \tau = \sin s $ and passing through (1,0,0), TNB triad identity matrix?
When numerically computed it looks like a catenoid surface of revolution for all appearances. ( unit curvature at extremes and like an symptote at symmetrical plane). Its parametrization is not mentioned in Differential Geometry books that I could come across.
Can it at least be proved that the line lies on a surface of revolution?
In general for a surface of revolution, what functional relation/s can exist between $ \kappa, \tau $ ?
Initial values for (T,N,B) taken as an identity matrix to fix rotation.
The lines given by achille hui are described on an equiangular rotational hyperboloid inclined at $\pi/4$ to one axis, result of initial point $(0,0,0)$ and starting $T,N,B$ an identity matrix.
May I request someone to extend it further or integrate the locus incorporating constants for generality ? like
$$ (a \kappa, a \tau) = ( \cos s/b , \sin s/b) $$
An image of a hyperboloid of revolution for $ (a= 1/6, b =2 ) $:
Motivation of this post was in fact to readily plot these lines on a surface of revolution using scalar curvatures properly.
In hindsight it slightly puzzles (me)... Although we started off with two intrinsic curvature invariants in 3-space, when combined it forces the arc to belong to 1- sheet hyperboloid $ of \, revolution $ in $ \mathbb R^2 $ that was neither specified in the input nor expected (by me). Is there some explanation or insight for it to so happen?