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computational geometry - How do I get ConvexHullMesh to return polygons instead of triangle as surface mesh?


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?



Answer



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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