Put random points into a specific 2D subregion

Question

I want to cover a particular subregion of the x-y plane with a random distribution.

As this particular region is not a circle I encounter some problems declaring the permitted zone where to put points.

The points must NOT be drawn in the blue zone. As in the graph

And I use the following code to drawing the graph:

mu = 0.000954;
h = x^2 - y^2 + 2 (1 - mu)/Norm[x + mu] + 2 mu/Norm[1 - x - mu];
S = 
  RegionPlot[h < 3.07, {x, -2, 2}, {y, -2, 2}, 
    Mesh -> None, PlotPoints -> 200, Axes -> True, Frame -> False]

Where h is the function.

P.s.: the drawing process is a bit long if you use more then 200 points, so keep this not so high.

Answer 1

How about

f[x_, y_] = h - 3.07;

Then, drawing candidates

dat = {RandomVariate[UniformDistribution[{-2, 2}], np], 
       RandomVariate[UniformDistribution[{-2, 2}], np]} // Transpose;

and selecting

Show[Select[dat, f @@ # > 0 &] // ListPlot[#, AspectRatio -> 1] &, S]

the corresponding data can be exported as

Export["test.dat",dat]

$cat test.dat

Answer 2

This is now easily done with the current region functionality:

With[{mu = 0.000954, val = 3.07},
     reg = ImplicitRegion[x^2 - y^2 + 2 (1 - mu)/Norm[x + mu] + 2 mu/Norm[1 - x - mu] <
                          val, {{x, -2, 2}, {y, -2, 2}}]];

(* discretize complement *)
cr = BoundaryDiscretizeRegion[RegionDifference[Rectangle[{-2, -2}, {2, 2}], reg]];

RegionPlot[reg, Epilog -> {AbsolutePointSize[1], Point[RandomPoint[cr, 5000]]}]