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GeoVoronoi (and GeoDelaunay)

Posted 3 months ago
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An example of a GeoVoronoi

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Posted 3 months ago

That's because Voronoi already naturally wraps around, similar to how a planar Delaunay is bounded, but a planar Voronoi covers all of the plane.

POSTED BY: J. M.
Answer

Ah, that makes a lot of sense. You are basically getting only the "outer" hull (a bit like the upper/lower hull) instead of the full convex hull. I'll make that change if I submit these to the function repository.

Edit: I wonder why GeoVoronoi seems to work without that extra vertex.

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Posted 3 months ago

I agree; in a sense, that's what made the undocumented functions a little more convenient to use, since they were (apparently) orientation-preserving. In the planar case, you can do a reconstitution with ConvexHullMesh[], but it is murkier in the spherical case.

POSTED BY: J. M.
Answer

Thanks. The issue with MeshConnectivityGraph is that it destroys the ordering information. That is, it can tell you which faces are adjacent to which vertices, but not the order in which they are around it, and so it is difficult to use them to construct a polygon. (This is the same as the issue with AdjacentMeshCells).

Thanks for the paper as well. My original goal was to implement geo power diagrams (also known as Laguerre Voronoi diagrams), and I was referencing this paper which happens to be by the same author. The paper you sent, however, looks like it has some other information about computing the power diagram using only 3D convex hulls, which would be great, so I'll have to look into it.

Posted 3 months ago

This is very nice. I note that the spherical Voronoi implementation in my posts was written quite a while ago, and using undocumented properties like "VertexFaceConnectivity" was the easiest method. Nowadays, I would say it's more advisable to use MeshConnectivityGraph[]:

With[{mcg = MeshConnectivityGraph[chm, {2, 0}, 0]},
     Table[EdgeList[mcg, DirectedEdge[_, {0, k}]][[All, 1]][[All, 2]],
           {k, MeshCellCount[chm, 0]}]]

but of course you should do your own comparisons against your DualPolyhedron[] approach.

A good survey paper to read about all these things is this paper by Sugihara, which has also nice pointers to other literature.

POSTED BY: J. M.
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