3
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According to the documentation RegionMember should work with regions that are true for ConstantRegionQ. However, the function does not work with simple polyhedra.

Consider:

ConstantRegionQ@RandomPolyhedron[5]
(*True*)
RegionMember@RandomPolyhedron[5]
(* {} *)

I am using version 12.0. Is this a bug?

And since RegionMember is not working, is there a way to test whether a point lies inside or outside a user defined polyhedra?

An example for a user defined polyhedron (here i have triangulated the individual faces because the vertices of a face does not have to be coplanar):

data = {{{0.9000000000, 9.803074361, -0.2788974201}, {0.6750000000, 
10.19278579, -0.2270388454}, {0.4500000000, 
9.803074361, -0.2935999100}}, {{0.6750000000, 
10.19278579, -0.2270388454}, {0.2250000000, 
10.19278579, -0.2379129589}, {0.4500000000, 
9.803074361, -0.2935999100}}, {{0.2250000000, 
10.19278579, -0.2379129589}, {0, 
9.803074361, -0.3066046725}, {0.4500000000, 
9.803074361, -0.2935999100}}, {{0, 
9.803074361, -0.3066046725}, {0.2250000000, 
9.413362929, -0.3638887807}, {0.4500000000, 
9.803074361, -0.2935999100}}, {{0.2250000000, 
9.413362929, -0.3638887807}, {0.6750000000, 
9.413362929, -0.3472567826}, {0.4500000000, 
9.803074361, -0.2935999100}}, {{0.6750000000, 
9.413362929, -0.3472567826}, {0.9000000000, 
9.803074361, -0.2788974201}, {0.4500000000, 
9.803074361, -0.2935999100}}, {{0.9000000000, 
9.803074361, -1.278897420}, {0.6750000000, 
10.19278579, -1.227038845}, {0.4500000000, 
9.803074361, -1.293599910}}, {{0.6750000000, 
10.19278579, -1.227038845}, {0.2250000000, 
10.19278579, -1.237912959}, {0.4500000000, 
9.803074361, -1.293599910}}, {{0.2250000000, 
10.19278579, -1.237912959}, {0, 
9.803074361, -1.306604673}, {0.4500000000, 
9.803074361, -1.293599910}}, {{0, 
9.803074361, -1.306604673}, {0.2250000000, 
9.413362929, -1.363888781}, {0.4500000000, 
9.803074361, -1.293599910}}, {{0.2250000000, 
9.413362929, -1.363888781}, {0.6750000000, 
9.413362929, -1.347256783}, {0.4500000000, 
9.803074361, -1.293599910}}, {{0.6750000000, 
9.413362929, -1.347256783}, {0.9000000000, 
9.803074361, -1.278897420}, {0.4500000000, 
9.803074361, -1.293599910}}, {{0.9000000000, 
9.803074361, -0.2788974201}, {0.9000000000, 
9.803074361, -1.278897420}, {0.7875000000, 
9.997930077, -0.7529681327}}, {{0.9000000000, 
9.803074361, -1.278897420}, {0.6750000000, 
10.19278579, -1.227038845}, {0.7875000000, 
9.997930077, -0.7529681327}}, {{0.6750000000, 
10.19278579, -1.227038845}, {0.6750000000, 
10.19278579, -0.2270388454}, {0.7875000000, 
9.997930077, -0.7529681327}}, {{0.6750000000, 
10.19278579, -0.2270388454}, {0.9000000000, 
9.803074361, -0.2788974201}, {0.7875000000, 
9.997930077, -0.7529681327}}, {{0.6750000000, 
10.19278579, -0.2270388454}, {0.6750000000, 
10.19278579, -1.227038845}, {0.4500000000, 
10.19278579, -0.7324759022}}, {{0.6750000000, 
10.19278579, -1.227038845}, {0.2250000000, 
10.19278579, -1.237912959}, {0.4500000000, 
10.19278579, -0.7324759022}}, {{0.2250000000, 
10.19278579, -1.237912959}, {0.2250000000, 
10.19278579, -0.2379129589}, {0.4500000000, 
10.19278579, -0.7324759022}}, {{0.2250000000, 
10.19278579, -0.2379129589}, {0.6750000000, 
10.19278579, -0.2270388454}, {0.4500000000, 
10.19278579, -0.7324759022}}, {{0.2250000000, 
10.19278579, -0.2379129589}, {0.2250000000, 
10.19278579, -1.237912959}, {0.1125000000, 
9.997930077, -0.7722588157}}, {{0.2250000000, 
10.19278579, -1.237912959}, {0, 
9.803074361, -1.306604673}, {0.1125000000, 
9.997930077, -0.7722588157}}, {{0, 9.803074361, -1.306604673}, {0, 
9.803074361, -0.3066046725}, {0.1125000000, 
9.997930077, -0.7722588157}}, {{0, 
9.803074361, -0.3066046725}, {0.2250000000, 
10.19278579, -0.2379129589}, {0.1125000000, 
9.997930077, -0.7722588157}}, {{0, 9.803074361, -0.3066046725}, {0,
9.803074361, -1.306604673}, {0.1125000000, 
9.608218645, -0.8352467266}}, {{0, 
9.803074361, -1.306604673}, {0.2250000000, 
9.413362929, -1.363888781}, {0.1125000000, 
9.608218645, -0.8352467266}}, {{0.2250000000, 
9.413362929, -1.363888781}, {0.2250000000, 
9.413362929, -0.3638887807}, {0.1125000000, 
9.608218645, -0.8352467266}}, {{0.2250000000, 
9.413362929, -0.3638887807}, {0, 
9.803074361, -0.3066046725}, {0.1125000000, 
9.608218645, -0.8352467266}}, {{0.2250000000, 
9.413362929, -0.3638887807}, {0.2250000000, 
9.413362929, -1.363888781}, {0.4500000000, 
9.413362929, -0.8555727816}}, {{0.2250000000, 
9.413362929, -1.363888781}, {0.6750000000, 
9.413362929, -1.347256783}, {0.4500000000, 
9.413362929, -0.8555727816}}, {{0.6750000000, 
9.413362929, -1.347256783}, {0.6750000000, 
9.413362929, -0.3472567826}, {0.4500000000, 
9.413362929, -0.8555727816}}, {{0.6750000000, 
9.413362929, -0.3472567826}, {0.2250000000, 
9.413362929, -0.3638887807}, {0.4500000000, 
9.413362929, -0.8555727816}}, {{0.6750000000, 
9.413362929, -0.3472567826}, {0.6750000000, 
9.413362929, -1.347256783}, {0.7875000000, 
9.608218645, -0.8130771013}}, {{0.6750000000, 
9.413362929, -1.347256783}, {0.9000000000, 
9.803074361, -1.278897420}, {0.7875000000, 
9.608218645, -0.8130771013}}, {{0.9000000000, 
9.803074361, -1.278897420}, {0.9000000000, 
9.803074361, -0.2788974201}, {0.7875000000, 
9.608218645, -0.8130771013}}, {{0.9000000000, 
9.803074361, -0.2788974201}, {0.6750000000, 
9.413362929, -0.3472567826}, {0.7875000000, 
9.608218645, -0.8130771013}}};

Polyhedron@data

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4
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You could also try:

RegionMember[
 BoundaryDiscretizeGraphics@CanonicalizePolyhedron[Polyhedron[data]]]
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4
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Polyhedrons are a very new data type and it is not unusual that not all possible functions have been overloaded for them.

You can extract the vertex coordinates of a Polyhedron R with R[[1]]; combined with ConvexHullMesh, this allows you to convert the Polyhedron to a MeshRegion and to apply `RandomPoint:

R = RandomPolyhedron[5];
S = ConvexHullMesh[R[[1]]];
RandomMember[S]

With the user-defined "polyhedron", you can do, e.g., this:

R = BoundaryDiscretizeGraphics@Graphics3D[
   Polygon[{{{0.9000000000, 
       9.803074361, -0.2788974201}, {0.6750000000, 
       10.19278579, -0.2270388454}, {0.4500000000, 
       9.803074361, -0.2935999100}}, {{0.6750000000, 
       10.19278579, -0.2270388454}, {0.2250000000, 
       10.19278579, -0.2379129589}, {0.4500000000, 
       9.803074361, -0.2935999100}}, {{0.2250000000, 
       10.19278579, -0.2379129589}, {0, 
       9.803074361, -0.3066046725}, {0.4500000000, 
       9.803074361, -0.2935999100}}, {{0, 
       9.803074361, -0.3066046725}, {0.2250000000, 
       9.413362929, -0.3638887807}, {0.4500000000, 
       9.803074361, -0.2935999100}}, {{0.2250000000, 
       9.413362929, -0.3638887807}, {0.6750000000, 
       9.413362929, -0.3472567826}, {0.4500000000, 
       9.803074361, -0.2935999100}}, {{0.6750000000, 
       9.413362929, -0.3472567826}, {0.9000000000, 
       9.803074361, -0.2788974201}, {0.4500000000, 
       9.803074361, -0.2935999100}}, {{0.9000000000, 
       9.803074361, -1.278897420}, {0.6750000000, 
       10.19278579, -1.227038845}, {0.4500000000, 
       9.803074361, -1.293599910}}, {{0.6750000000, 
       10.19278579, -1.227038845}, {0.2250000000, 
       10.19278579, -1.237912959}, {0.4500000000, 
       9.803074361, -1.293599910}}, {{0.2250000000, 
       10.19278579, -1.237912959}, {0, 
       9.803074361, -1.306604673}, {0.4500000000, 
       9.803074361, -1.293599910}}, {{0, 
       9.803074361, -1.306604673}, {0.2250000000, 
       9.413362929, -1.363888781}, {0.4500000000, 
       9.803074361, -1.293599910}}, {{0.2250000000, 
       9.413362929, -1.363888781}, {0.6750000000, 
       9.413362929, -1.347256783}, {0.4500000000, 
       9.803074361, -1.293599910}}, {{0.6750000000, 
       9.413362929, -1.347256783}, {0.9000000000, 
       9.803074361, -1.278897420}, {0.4500000000, 
       9.803074361, -1.293599910}}, {{0.9000000000, 
       9.803074361, -0.2788974201}, {0.9000000000, 
       9.803074361, -1.278897420}, {0.7875000000, 
       9.997930077, -0.7529681327}}, {{0.9000000000, 
       9.803074361, -1.278897420}, {0.6750000000, 
       10.19278579, -1.227038845}, {0.7875000000, 
       9.997930077, -0.7529681327}}, {{0.6750000000, 
       10.19278579, -1.227038845}, {0.6750000000, 
       10.19278579, -0.2270388454}, {0.7875000000, 
       9.997930077, -0.7529681327}}, {{0.6750000000, 
       10.19278579, -0.2270388454}, {0.9000000000, 
       9.803074361, -0.2788974201}, {0.7875000000, 
       9.997930077, -0.7529681327}}, {{0.6750000000, 
       10.19278579, -0.2270388454}, {0.6750000000, 
       10.19278579, -1.227038845}, {0.4500000000, 
       10.19278579, -0.7324759022}}, {{0.6750000000, 
       10.19278579, -1.227038845}, {0.2250000000, 
       10.19278579, -1.237912959}, {0.4500000000, 
       10.19278579, -0.7324759022}}, {{0.2250000000, 
       10.19278579, -1.237912959}, {0.2250000000, 
       10.19278579, -0.2379129589}, {0.4500000000, 
       10.19278579, -0.7324759022}}, {{0.2250000000, 
       10.19278579, -0.2379129589}, {0.6750000000, 
       10.19278579, -0.2270388454}, {0.4500000000, 
       10.19278579, -0.7324759022}}, {{0.2250000000, 
       10.19278579, -0.2379129589}, {0.2250000000, 
       10.19278579, -1.237912959}, {0.1125000000, 
       9.997930077, -0.7722588157}}, {{0.2250000000, 
       10.19278579, -1.237912959}, {0, 
       9.803074361, -1.306604673}, {0.1125000000, 
       9.997930077, -0.7722588157}}, {{0, 
       9.803074361, -1.306604673}, {0, 
       9.803074361, -0.3066046725}, {0.1125000000, 
       9.997930077, -0.7722588157}}, {{0, 
       9.803074361, -0.3066046725}, {0.2250000000, 
       10.19278579, -0.2379129589}, {0.1125000000, 
       9.997930077, -0.7722588157}}, {{0, 
       9.803074361, -0.3066046725}, {0, 
       9.803074361, -1.306604673}, {0.1125000000, 
       9.608218645, -0.8352467266}}, {{0, 
       9.803074361, -1.306604673}, {0.2250000000, 
       9.413362929, -1.363888781}, {0.1125000000, 
       9.608218645, -0.8352467266}}, {{0.2250000000, 
       9.413362929, -1.363888781}, {0.2250000000, 
       9.413362929, -0.3638887807}, {0.1125000000, 
       9.608218645, -0.8352467266}}, {{0.2250000000, 
       9.413362929, -0.3638887807}, {0, 
       9.803074361, -0.3066046725}, {0.1125000000, 
       9.608218645, -0.8352467266}}, {{0.2250000000, 
       9.413362929, -0.3638887807}, {0.2250000000, 
       9.413362929, -1.363888781}, {0.4500000000, 
       9.413362929, -0.8555727816}}, {{0.2250000000, 
       9.413362929, -1.363888781}, {0.6750000000, 
       9.413362929, -1.347256783}, {0.4500000000, 
       9.413362929, -0.8555727816}}, {{0.6750000000, 
       9.413362929, -1.347256783}, {0.6750000000, 
       9.413362929, -0.3472567826}, {0.4500000000, 
       9.413362929, -0.8555727816}}, {{0.6750000000, 
       9.413362929, -0.3472567826}, {0.2250000000, 
       9.413362929, -0.3638887807}, {0.4500000000, 
       9.413362929, -0.8555727816}}, {{0.6750000000, 
       9.413362929, -0.3472567826}, {0.6750000000, 
       9.413362929, -1.347256783}, {0.7875000000, 
       9.608218645, -0.8130771013}}, {{0.6750000000, 
       9.413362929, -1.347256783}, {0.9000000000, 
       9.803074361, -1.278897420}, {0.7875000000, 
       9.608218645, -0.8130771013}}, {{0.9000000000, 
       9.803074361, -1.278897420}, {0.9000000000, 
       9.803074361, -0.2788974201}, {0.7875000000, 
       9.608218645, -0.8130771013}}, {{0.9000000000, 
       9.803074361, -0.2788974201}, {0.6750000000, 
       9.413362929, -0.3472567826}, {0.7875000000, 
       9.608218645, -0.8130771013}}}]
   ];
RandomMember[R]

PS.: I have no clue at all what this new datatype is supposed to ship what is not already provided MeshRegions.

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  • $\begingroup$ thanks. what about the case if my polyhedron have faces wherein the vertices are not coplanar. What I have done is that I have triangulated those faces and can display a polyhedron successfully. Do you think the approach that you have mentioned above will work for such a polyhedron? $\endgroup$ – Ali Hashmi Jul 10 '19 at 4:32
  • $\begingroup$ kindly check the example that I posted in the question $\endgroup$ – Ali Hashmi Jul 10 '19 at 4:43
  • $\begingroup$ thanks. I was working on a solution myself as well. We both reached the same answer :) I will accept your answer however $\endgroup$ – Ali Hashmi Jul 10 '19 at 5:47
3
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here is a little fix that should work for both cases:

datamodified = data /. {0.4500000000, 9.803074361, -0.2935999100} -> {0.4500000000, 
9.803074361, -0.3935999100} (* this will make vertices of one face non-coplanar *)

poly = Polyhedron[datamodified];

points = DeleteDuplicates@Flatten[poly[[1]], 1];

assoc = AssociationThread[points -> Range[Length@points]];

order = Map[Lookup[assoc, #] &, poly[[1]]];

reg = DiscretizeGraphics@Region@Polyhedron[points, order];

enter image description here

RegionMember@reg

enter image description here

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