How can I make ListDensityPlot show up only in a specified region?

Question

Disclaimer: I am new to Mathematica and this is my first post here! I am trying to plot EEG data of different intensities of brain waves from an FFT to areas of the brain where they were collected. I am trying to find the right code that will enable the data to be represented within a region that is horseshoe-shaped. The data was collected along on the outer edge of the horseshoe shape. So this is the what I tried to do:

ListDensityPlot[{{-1.2, 0, 22}, {1.2, 0, 16}, {0, 1.5, 34}},

RegionFunction -> Function[{x, y}, 0.5 < (x^2)/2 + (y^2)/3 < 1]] 

However, the output is a triangle formed by the points where the data was collected.

Can anyone think of a way to make the density plot appear inside of the horseshoe-shaped region? Maybe I could use an image in Mathematica that the data could be plotted over? I really don't know what would be the best way to do this. Any help would be appreciated! Thank you!

Answer 1

The sparse data provided in the question yields the density plot,

dp = ListDensityPlot[{{-1.2, 0, 22}, {1.2, 0, 16}, {0, 1.5, 34}}]

There is insufficient data to create a density plot by interpolation in the white areas. The question then asks how to plot only that part of the density plot in the red region,

ContourPlot[(x^2)/2 + (y^2)/3, {x, -1.2, 1.2}, {y, 0, 1.5}, 
    ContourStyle -> White, Contours -> {1/2, 1}, ContourShading -> {White, Red}]

This contour plot can be converted into a stencil of sorts by replacing Red by None (i.e., a clear region) and superimposing it on the density plot:

st = ContourPlot[(x^2)/2 + (y^2)/3, {x, -1.2, 1.2}, {y, 0, 1.5}, 
    ContourStyle -> White, Contours -> {1/2, 1}, ContourShading -> {White, None}]
Show[dp, st]

More data for the original ListDensityPlot would fill out the horse-collar region.

Addendum

To demonstrate, let us linearly extrapolate the data in the question by

if = Interpolation[{{-1.2, 0, 22}, {1.2, 0, 16}, {0, 1.5, 34}}, InterpolationOrder -> All];
tt = DensityPlot[if[x, y], {x, -1.2, 1.2}, {y, 0, 1.5}];
Show[tt, st]

Of course, if Interpolation is to be used as an intermediate step, then the code in the question, appropriately modified, produces essentially the same plot.

DensityPlot[if[x, y], {x, -1.2, 1.2}, {y, 0, 1.5}, 
    RegionFunction -> Function[{x, y}, 0.5 < (x^2)/2 + (y^2)/3 < 1]]

Answer 2

You can wrap ListDensityPlot output with Texture and use it as the PlotStyle setting in RegionPlot:

llp = ListDensityPlot[{{-1.2, 0, 22}, {1.2, 0, 16}, {0, 1.5, 34}}, ImageSize -> Medium];

texture = Texture[Show[llp, PlotRangePadding -> 0, Frame -> False]];

rp = RegionPlot[0.5 < (x^2)/2 + (y^2)/3 < 1, {x, -1.2, 1.2}, {y, 0, 1.5}, 
   PlotStyle -> texture, BoundaryStyle -> None, ImageSize -> Medium];

Row[{llp, rp}, Spacer[10]]