ListPlot3D of a set of points in a non-convex region

I would like to plot a set of points in the same style as ListPlot3D does, but only in a non-convex region that I can specify (e.g. using a RegionFunction). Is there a simple solution for this?

Let me illustrate with an example.

Let's generate some sample points ...

Clear[regionFun]
regionFun[{x_, y_}] := y < x^2

fun[{x_, y_}] := -Norm[{x, y} - {.5, .5}]

pts = Select[Tuples[Range[0, 1, 0.01], 2], regionFun];

... and try to plot then in the region we're interested in:

ListPlot3D[
Append[#, fun[#]] & /@ pts,
RegionFunction -> Function[{x, y}, regionFun[{x, y}]],
InterpolationOrder -> 1
] Notice that the plot occupies the complete convex hull of the points because of how the Delaunay triangulation was constructed for the interpolation.

What I would like to see instead is this:

Plot3D[fun[{x, y}], {x, 0, 1}, {y, 0, 1},
RegionFunction -> Function[{x, y}, regionFun[{x, y}]]] The only possible solution I see at the moment is to use an external tool to "manually" construct a Delaunay triangulation which is confined within a region and build the plot from that data. This, however, is a lot of work, so I thought I'd ask first if there's a simple solution.

Note that this is just a generated example dataset, not my real data. I can't use Plot3D for my real application, only ListPlot3D and related functions.

• Interpolation + Plot3D works.
– rm -rf
Nov 10 '13 at 20:55
• @rm-rf For a arbitrary point cloud on an unstructured grid we will have to accept the Interpolation::udeg message too! Interpolationorder will be constrained to one. Nov 10 '13 at 21:24
• @PlatoManiac Right, but Szabolcs used order 1 anyway, so that shouldn't be a problem here. Szabolcs, I don't understand why you said you don't get nice plots... with ifun = Interpolation[Append[#, fun[#]] & /@ pts, InterpolationOrder -> 1, "ExtrapolationHandler" -> {(Indeterminate &), "WarningMessage" -> False}]; and MaxRecursion -> 6, I get a plot that's as good as the native Plot3D.
– rm -rf
Nov 10 '13 at 21:29
• @PlatoManiac All built-in List* plotting functions that work on 2D data will only do linear interpolation for unstructured grids, so it's the same thing. Nov 10 '13 at 21:31
• @rm-rf I also don't understand why I don't get nice plots for my real data, but I don't. This example I posted works perfectly. I guess my real data is nastier: dropbox.com/s/4xn79zcpnz81711/… Regardless, I'd accept your answer if you posted it as I'm using it now. (I did increase PlotPoints and MaxRecursion --- I think somehow the higher-than-order-1 adaptive sampler of Plot3D doesn't work well with the order-1 interpolation) Nov 10 '13 at 21:33 