List version of SphericalPlot3D

Question

could you give me some input on how to create a list version of

SphericalPlot3D[r[theta,phi],{theta,min,max},{phi,min,max}]

When r[] is an explicit analytical function it's not an issue, but what if one numerically generates a list {{r1,theta1,phi1},...,{rn,thetan,phin}} and wants to visualize it in a similar manner? Any ideas?

Answer 1

A quick way of doing this would be to take the data and apply an Interpolation to create a function that can be plotted with SphericalPlot3D. Here is an example:

data = Flatten[
   Table[{{theta, phi}, 
     N@Re@SphericalHarmonicY[2, 1, theta, phi]}, {theta, Pi/10, Pi, 
     Pi/10}, {phi, Pi/10, 2 Pi, Pi/10}], 1];

f = Interpolation[data];

SphericalPlot3D[f[theta, phi], {theta, 0, Pi}, {phi, 0, 2 Pi}]

Here you get a warning because extrapolation was used since the data range is smaller than the angle range I'm plotting. But that's normal.

Note that the data has to be arranged differently from the structure you gave in the question: It's of the form {{{theta1, phi1}, f1}, {{{theta2, phi2}, f2}, ...}

Answer 2

I was considering to advise the use of ListSurfacePlot3D to do this, but after tinkering with it for some while, I found it is awful for reconstruction of spherical plots.

Let's take an example from the documentation:

SphericalPlot3D[1 + 2 Cos[2 θ], {θ , 0, Pi}, {ϕ, 0, 2 Pi}]

The surface could be sampled like this:

data = Flatten[Table[{1 + 2 Cos[2 θ], θ, ϕ}, {θ, 0, Pi, Pi/30}, {ϕ, 0, 2 Pi, 2 Pi/30}], 1];

In v9 we have got an easy way to transform coordinates form spherical to Cartesian:

mapping =  CoordinateTransformData["Spherical" -> "Cartesian", "Mapping"]

{Cos[#1[[3]]] Sin[#1[[2]]] #1[[1]], Sin[#1[[2]]] Sin[#1[[3]]] #1[[1]], Cos[#1[[2]]] #1[[1]]} &

Those without v9 can use the pure function in the output above.

This works fine withListPointPlot3D:

ListPointPlot3D[mapping /@ data, BoxRatios -> Automatic]

However, it gets absolutely horrible results with ListSurfacePlot3D

ListSurfacePlot3D[mapping /@ data, BoxRatios -> Automatic]

Increasing sampling density doesn't help at all. Neither does the application of the MaxPlotPoints->Infinity option. So, it looks like that Jens' answer is the best solution in this case.