A helicoid has the following parametric equation:
$ r(u,v)=\left \langle v\cos u,v\sin u,cu \right \rangle,\qquad u,v,c\in\mathbb{R}. $
In ruled form,
$r(u,v)=\alpha(u)+v\Lambda(u),$
it has the directrix and the rulings
$\begin{align} \alpha(u)&=\left \langle 0,0,cu \right \rangle,\\ \Lambda(u)&=\left \langle \cos u,\sin u,0 \right \rangle, \end{align}$
respectively.
I want to create a circular helicoid whose directrix is not a vertical line but a circle,
$ \alpha(u)=\left \langle R\cos u,R\sin u,0 \right \rangle, $
and whose rulings rotate on the plane spanned by $\left \langle -\cos t,-\sin t,0 \right \rangle$ and $\left \langle 0,0,1 \right \rangle$ and not on the plane spanned by $\left \langle 1,0,0 \right \rangle$ and $\left \langle 0,1,0 \right \rangle$.
I need a hint on how to begin approaching this problem.