For time being I define a class of parallel lines in $ E^3 $ as lines with constant minimum distance along their common normal.

Apart from helices with parametrization $ (x,y,z) = (a \cos (u) , a \sin (u),b u) $ having constant normal distance $a$ with $z$-axis $ a=0 $ what other space curve pairs are known which can be generated from a single parametrization ?

Firstly, can it be stated that curves of constant torsion ( rotation of normal in Frenet frame per unit arc length) erected perpendicularly on an arbitrary zero torsion planar curve with a common constant length normal belong to a parallel line set ?

Secondly, how is the Gauss curvature ( negative for a ruled surface) expressed in terms of local curvature and torsion?


This is a very late response and you might have already found the answer. But I thought I will give it a go and I presume you meant axis $z=0$ with distance $a$.

Anyways, if you are looking at a paired curve system with a constant distance maintained along the common normals, then you might want to look at the class of curves called Bertrand curves which are "offset'' curves that have coinciding normals and are at a constant distance from one another throughout. Given a curve $\gamma(s)$, the normal offset would be $$\gamma_1(s_1) = \gamma(s) + \lambda \mathbf{N}(s),$$ $\lambda$ being the constant distance.

In particular a curve $\gamma$ admits an offset $\gamma_1$ such that the normals of $\gamma$ and $\gamma_1$ are the same iff the curvature($\kappa$) and torsion($\tau$) of $\gamma$ satisfy the relationship $$\alpha\kappa + \beta\tau=1,$$ for some constants $\alpha, \beta$.

Curves of constant torsion then are a special case of this class of curves as you can see. There are also other possible paired curve systems I believe, but I am not too familiar with the literature on them.

As for your last question, I thought once you parametrise the ruled surface in terms of the generator and the directrix, it is possible to get the Gaussian curvature once you find the fundamental forms. Of course this may be easier if the parametrisation is in terms of the striction curve.


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