Testing Membership 

Test points for membership in a mesh.

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X mesh2d = BoundaryDiscretizeGraphics[Text[Style["\[Pi]", Bold]], _Text, MaxCellMeasure -> 0.1]; pts = {{2, 2}, {3, 3}};

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X RegionMember[mesh2d, pts]

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Create a RegionMemberFunction to test membership for a large set of points.

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X mf = RegionMember[mesh2d]; pts = RandomReal[{-3, 3}, {2000, 2}]; Show[mesh2d, Graphics[{PointSize[Small], Red, Point[Select[pts, mf]], Black, Point[Select[pts, ! mf[#] &]]}]]

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The same can be done in 3D.

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X i = ImplicitRegion[ x^4 - 5 x^2 + y^4 - 5 y^2 + z^4 - 5 z^2 + 11.8 <= 0, {x, y, z}]; mesh3d = BoundaryDiscretizeRegion[i, RegionBounds[i]]; mf = RegionMember[mesh3d]; pts = RandomVariate[UniformDistribution[RegionBounds[mesh3d]], 2000]; Show[HighlightMesh[mesh3d, Style[2, Opacity[0.2]]], Graphics3D[{Red, Point[Select[pts, mf]], Black, Point[Select[pts, ! mf[#] &]]}]]

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