Surface equation between two curves

Question

I am faced to a problem where I need to get the parametric equation of the surface generated by 2 curves.

For example, I have the equation of a circle $(x, y, z) = (R\cos(t), R\sin(t), 0)$ and the equation of a line. I want to get the surface between them to have a special cylinder like the photo below :

Special cylinder

Is it possible to compute the parametric equation of this surface ?

Thanks

Answer 1

Something like this?

\begin{align} L=6\,R ,\\ S(u,v)&= (R\cos u,\, R\,(1-\tfrac{v}L)\sin u,\,v) ,\quad u\in[0,2\pi],\ v\in[0,L] . \end{align}