Possible equation of a helix like curve with axis as tangent vector of another curve I am looking for a parametric equation of a curve with certain helix like attributes. A helix is generated by the rotational and translational motion of a line along a fixed axis. But if I were to allow the axis itself to rotate, what would happen??
One example I had in mind was taking the tangent vectors of a second space curve as different positions of the rotating axis. The position vector of the original curve maintains a fixed distance from the position vector of the second one.  But I am not sure whether the two notions are the same.
In terms of curvature and torsion functions, the helix has 
$$\kappa = \dfrac{ \cos^2(\theta)}{r^2},\;\;\;\; \tau = \dfrac{\cos(\theta)\sin(\theta)}{r^2}$$
respectively where $\theta$ is the constant angle made by the tangent vector of the helix with the axis. In my case, I am looking at $\theta$ to be a $\mathbf{non-constant}$ function of say, arc-length $s$ i.e. $\theta(s)$.
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$If $c$ is a (sufficiently smooth) curve in $\Reals^{3}$ and if $\Basis_{1}$ and $\Basis_{2}$ are orthonormal unit normal vector fields along $c$ (that is, $\Brak{c'(s), \Basis_{j}(s)} = 0$ and $\Brak{\Basis_{i}(s), \Basis_{j}(s)} = \delta_{ij}$ for all $s$), then for a sufficiently small $r > 0$ and a sufficiently smooth function $\theta$, the curve
$$
h(s) = c(s) + r\bigl(\Basis_{1} \cos \theta(s) + \Basis_{1} \sin \theta(s)\bigr)
$$
is a curve of the type you want (assuming I understand the question).
A couple of natural ways to construct fields $\Basis_{j}$ come to mind:


*

*If $c$ has non-vanishing curvature at each point, you could take $\Basis_{1}$ to be the principal normal field, and $\Basis_{2} = \frac{c'}{\|c'\|} \times \Basis_{1}$ to be the binormal field.

*If $c'$ is never parallel to the $(x, y)$-plane, you could define $\Basis_{j}(s)$ by doing Gram-Schmidt on the ordered triple with elements $c'(s)$, $1, 0, 0)$, $(0, 1, 0)$.
Generally, if $c'$ is never parallel to some plane $P$ through the origin, you could fix an orthonormal basis of $P$ and do Gram-Schmidt as in the preceding paragraph.
Here's a curve of the first type: The "core" (black) is a right circular helix equipped with its Frenet frame; the blue curve winds around the helical core with variable pitch.

