How to draw a z-simple paraboloid in a “clean” way

Question

I'd like to draw z=x^2+y^2 on the interval 0<z<9. So normally I've just been using pgfplots with the addplot3 command, and for rectangular domains that doesn't seem to cause too much trouble. Example (from Is there any easy way to draw a ruled surface like a hyperbolic paraboloid in TikZ?):

In my case, however, I'd like to draw a paraboloid and I'd like the edge of the curve to be at r=3 in the z=9 plane. My first attempt was something like this (adapted from previous question):

\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xmin=-3,ymin=-3,zmin=0,xmax=3,ymax=3,zmax=10]
\addplot3 [surf,draw=none,restrict z to domain=0:9] {x^2+y^2};
\end{axis}
\end{tikzpicture}
\end{document}

Clearly the domain causes a problem: the paraboloid doesn't print pretty at all because the last z value evaluated is not at z=9, it is rather somewhere close to that below z=9, varying for each pair of coordinates (x,y).

I could set samples y=300, but the compilation time will be insane. Also, it limits the possibilities of the document, since I'm consuming a lot of stack size.

\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xmin=-3,ymin=-3,zmin=0,xmax=3,ymax=3,zmax=10]
\addplot3 [surf,draw=none,restrict z to domain=0:9,samples y=300] {x^2+y^2};
\end{axis}
\end{tikzpicture}
\end{document}

Surely there's a better way to do this?

Answer 1

As percusse said in the comment, you could use polar coordinates:

\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[xmin=-3,ymin=-3,zmin=0,xmax=3,ymax=3,zmax=10]
\addplot3 [surf,draw=none,restrict z to domain=0:9, data cs=polar, domain=0:360, y domain=0:3] (x, y, y^2);
\end{axis}
\end{tikzpicture}
\end{document}