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

Question

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.

Answer 1

Using Interpolation with Plot3D and slightly increasing the MaxRecursion gives you a nice plot. Since you have an unstructured grid, the interpolation order will be restricted to 1.

ifun = Interpolation[Append[#, fun[#]] & /@ pts, InterpolationOrder -> 1, 
   "ExtrapolationHandler" -> {(Indeterminate &), "WarningMessage" -> False}];
Plot3D[ifun[x, y], {x, 0, 1}, {y, 0, 1}, MaxRecursion -> 6,
    RegionFunction -> Function[{x, y}, regionFun[{x, y}]]]