Describe a twisted parabolic trough I want to describe a parabolic trough of the form $z=x^2$ and give it a twist, like a torsion in $y$ direction. Does anybody know how I can do that?

Imagine this is the trough and the $z$ direction would be my $y$ direction. Thats the best image I could find to make it clear. It doesn't matter what direction the twist is in, really as long as it looks like that.
 A: If the parabola is to be rotated about its focus, then we must have parabolic sections of the form $$z = ax^2 - \frac{1}{4a}, \quad a \ne 0.$$  Without loss of generality we may suppose $a > 0$, and suppose that for each unit increase in $y$, the parabola is rotated by some angle $\theta$.  This yields the parametrization $$\begin{align*} x(u,v) &= u \cos (\theta v) + \left( au^2 - \frac{1}{4a} \right) \sin (\theta v) \\ y(u,v) &= v \\ z(u,v) &= \left( au^2 - \frac{1}{4a}\right) \sin (\theta v) - u \sin (\theta v). \end{align*}$$
The below animation corresponds to varying $\theta \in [0,2]$ for a fixed $a = 0.5$:

The below animation corresponds to varying $a \in [0.05, 2]$ for a fixed $\theta = 1$:

For some reason, now I have a craving for pasta.
A: If the parabola has equation $y = x^{2} - a$, and is rotated about the origin at angular speed $k$ as the "horizontal" section moves along the $z$-axis, the resulting surface may be given the parametric description
\begin{align*}
x(u, v) &= u\cos(kv) - (u^{2} - a)\sin(kv), \\
y(u, v) &= u\sin(kv) + (u^{2} - a)\cos(kv), \\
z(u, v) &= v.
\end{align*}
The plot below shows $a = 1$ and $k = \pi/4$, for $-1 \leq u, v \leq 1$.
(This gives a couple of parameters to play around with, and should suggest how to rotate at non-uniform speed, or have the shape of the section change with height, or....)

