RegionMember does not work with RandomPolyhedron or Polyhedron object

Question

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

halmir · Accepted Answer · 2019-07-10 14:01:56Z

4

You could also try:

RegionMember[
 BoundaryDiscretizeGraphics@CanonicalizePolyhedron[Polyhedron[data]]]

answered Jul 10, 2019 at 14:01

halmir

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Henrik Schumacher · Accepted Answer · 2019-07-10 17:49:15Z

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.

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? — Ali Hashmi, Commented Jul 10, 2019 at 4:32
thanks. I was working on a solution myself as well. We both reached the same answer :) I will accept your answer however — Ali Hashmi, Commented Jul 10, 2019 at 5:47

Ali Hashmi · Accepted Answer · 2019-07-10 05:42:05Z

3

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];

RegionMember@reg

answered Jul 10, 2019 at 5:42

Ali Hashmi

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