Region Membership 

Test points for membership.

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X pts = {{0.5, 0.5}, {1, 1}};

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X RegionMember[Disk[{0, 0}, 1], pts]

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Compute region membership conditions in any dimension.

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X RegionMember[ ParametricRegion[{{s^2, s^3}, -1 <= s <= 1}, {s}], {x, y}]

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X RegionMember[ Simplex[{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}], {w, x, y, z}]

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Visualize point membership in red.

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X \[ScriptCapitalR] = ImplicitRegion[x^2 - 2 y^2 <= 1, {{x, -3, 3}, {y, -4, 4}}];

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X pts = RandomVariate[ UniformDistribution[RegionBounds[\[ScriptCapitalR]]], 10^4];

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X membership = RegionMember[\[ScriptCapitalR], pts];

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X Graphics[{AbsolutePointSize[ 1], {membership /. {False -> Gray, True -> Red}, Point /@ pts}\[Transpose]}]

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Create a RegionMember function to test membership for a large set of points.

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X \[ScriptCapitalR] = Cylinder[{{0, 1/2, 1}, {1.1, 1/2, 1}}, 1/2];

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X mf = RegionMember[\[ScriptCapitalR]]

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In[11]:=

X pts = RandomReal[{0, 1}, {2500, 3}];

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X Graphics3D[{EdgeForm[White], Yellow, {Opacity[0.2], \[ScriptCapitalR]}, {Red, Ball[Select[pts, mf], 0.01]}, Lighting -> "Neutral", {GrayLevel@0.3, Point[Select[pts, Not[mf[#]] &]]}}, Boxed -> False]

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