# How do I make a 3DPlot using cylindrical coordinates?

I'm trying to plot a function of the form $z(r,\theta)$ where $r \in [0, R]$ for a finite R, $\theta \in [0,2\pi[$, and z is the third coordinate, a function of the first two. I couldn't find anything to do it natively, so I went back to Cartesian coordinates. But the result does not satisfy me, because the range of x is a function of y, a consequence of the constraint $x^2+y^2 < R^2$. Is there already something in Mathematica to handle this kind of plot?

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Have you tried RevolutionPlot3D? –  David Skulsky Feb 5 '13 at 14:59
@David That's a nice idea when $z$ is independent of $\theta$, but how do you propose using RevolutionPlot3D when $z$ does vary with $\theta$? –  whuber Feb 5 '13 at 16:26
@whuber: Of course you're right. I didn't read the question too carefully. –  David Skulsky Feb 5 '13 at 20:49
@whuber, you can still use RevolutionPlot[] for the purpose. Witness for instance RevolutionPlot3D[r^2 Cos[3 t], {r, 0, 1}, {t, 0, 3 π/2}]. Of course, it's more enlightening to use ParametricPlot3D[] instead, as in your answer. –  Ｊ. Ｍ. Feb 7 '13 at 3:12

Do it parametrically. Here's a generic implementation:

cylinderPlot3D[f_, {rMin_, rMax_}, {tMin_, tMax_}, opts___] :=
ParametricPlot3D[{r Cos[t], r Sin[t], f[r, t]}, {r, rMin, rMax}, {t, tMin, tMax}, opts]


For example,

f[r_, t_] := r^2 Cos[3 t]];
cylinderPlot3D[f, {0, 1}, {0, 2 Pi}, Mesh->None, Boxed->False]


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Use

RevolutionPlot3D[ f[r,t], {r, rmax, rmin}, {t, tmax, tmin}]

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