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.
Best Answer
Note that for each $v$, the intersection of the helicoid with a coaxial cylinder (spiral) in cylindrical coordinates $(v,u,z)$ has the equation $z=u$, or more generally $z=\alpha u$; i.e., a "straight line through the origin".
Therefore, it makes sense to define a "circular helicoid" in toroidal coordinates $(r,u,v)$ by the equation $v=\alpha u$.
$$\begin{aligned}x= & \left(R+r\cos\alpha u\right)\cos u\\ y= & \left(R+r\cos\alpha u\right)\sin u\\ z= & r\sin\alpha u \end{aligned}$$