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I would need to identify the types of regular polygons forming the surface of a convex hull of 3D points. If I e.g. take the following example of a regular polyhedron

ConvexHullMesh[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]]]

The convex hull routine returns a triangulated mesh surface. Is there any simple way to convince Mathematica to return the surface as polyhedrons (in this case pentagons) instead of a triangulation.

To illustrate the issue further, e.g if one applies

MeshCells[ConvexHullMesh[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]]], 2]

Mathematica only returns triangles.

If one applies

ConvexHullMesh[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]]] // FullForm

There is the option "CoplanarityTolerance". But I do not know how to use it.

Any ideas?

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  • $\begingroup$ Where do you see the CoplanarityTolerance option? $\endgroup$ – Szabolcs Oct 14 '15 at 11:15
  • $\begingroup$ Just look at the FullForm of the last ConvexHullMesh statement above. There you see the option Rule["CoplanarityTolerance", Automatic]. If one tries to use this option with a numerical parameter instead of Automatic an error is returned that this option is unknown... $\endgroup$ – Rainer Oct 14 '15 at 13:46
  • $\begingroup$ Ah, that's inside Method, and it's for BoundaryMeshRegion, not for ConvexHull or BoundaryMesh ... $\endgroup$ – Szabolcs Oct 14 '15 at 15:46
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The procedure groups triangles based on the same unit normal vector, then uses the vertices in each group to form a new polygon. The vertices are sorted in such a way that their polygon is not self-intersecting.

This method doesn't allow for coplanar tolerance. Triangles in the same group have the same unit normal vector determined to within the second argument of Round (10^-5 here).

The sorting function sort is modified from #48091, which is a 2D method. sort uses the XY-projection of the points, unless they're colinear in X or Y.

sort[pts_] := Module[
  {p, subspaceselector},
  p = coord[[#]] & /@ pts;
  subspaceselector = Which[
    p[[1, 1]] == p[[2, 1]] == p[[3, 1]], Rest,
    p[[1, 2]] == p[[2, 2]] == p[[3, 2]], Drop[#, {2}] &,
    True, Most
  ];
  SortBy[pts, N[ArcTan @@ subspaceselector[coord[[#]] - Mean[p]]] &]
];
unitnormal[verts_] := Round[
  Normalize[Cross[verts[[2]] - verts[[1]], verts[[3]] - verts[[1]]]],
  10^-5
];
convexhull = ConvexHullMesh[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]]];
coord = MeshCoordinates[convexhull];
trivertices = Level[MeshCells[convexhull, 2], {-2}];
polysets = GatherBy[
  trivertices,
  unitnormal[Function[i, coord[[i]]] /@ #] &
];
polyvertices = Map[sort][Union @@ # & /@ polysets];
MeshRegion[coord, Polygon /@ polyvertices]

New convex hull

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  • $\begingroup$ It doesn't quite work well for "Cube" instead of "Dodecahedron". The face vertices are not well ordered. $\endgroup$ – Szabolcs Oct 14 '15 at 13:08
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    $\begingroup$ In principle, two distinct faces of a polyhedron can have the same normal. But that will never happen if the polyhedron is a convex hull. So the idea should work well. $\endgroup$ – Szabolcs Oct 14 '15 at 13:11
  • $\begingroup$ Yes, I do rely on that assumption. $\endgroup$ – Taiki Oct 14 '15 at 13:12
  • $\begingroup$ I'm coding proper sort now... $\endgroup$ – Taiki Oct 14 '15 at 13:13
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    $\begingroup$ @Taiki. NICE work thanks for putting this online.... $\endgroup$ – Rainer Oct 14 '15 at 16:03
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Here is a solution that uses undocumented functionality to generate an appropriate MeshRegion[] object:

Graphics`Mesh`MeshInit[];
FirstCase[ConvexHull3D[N[PolyhedronData["Dodecahedron", "VertexCoordinates"]],
                       FlatFaces -> False], 
          GraphicsComplex[pts_, stuff_] :>
          MeshRegion[pts, Cases[stuff, _Polygon, ∞]], ∞]

dodecahedron as a MeshRegion[]

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