Parametrisation of an elliptical paraboloid

Question

Is there a way to parameterise $x+y^2+2z^2=4$ in terms of $r$ and $\theta$ (i.e. similar to cylindrical coordinates)?

Answer 1

Sure, you can use

$$\begin{cases}x=4-(r\cos\theta)^2-2(2r\sin\theta)^2\\y=r\cos\theta\\z=2r\sin\theta\end{cases}$$

with $r\ge0$ and $\theta\in[0,2\pi]$.