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3D version of the built-in VoronoiDiagram

Posted 1 year ago
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Here I will show how to create 3D version of VoronoiDiagram in Wolfram Language.

Note that there's currently no way to represent a collection of 3D Voronoi mesh cells in a MeshRegion or BoundaryMeshRegion.

Here's a routine that takes the dual of the DelaunayMesh and returns an Association where the keys are the points and the values are their respective Voronoi cells.

pad[?_][{min_, max_}] := {min, max} + ?(max-min){-1, 1}

VoronoiCells[pts_] /; MatrixQ[pts, NumericQ] && 2 <= Last[Dimensions[pts]] <= 3 := 
  Block[{bds, dm, conn, adj, lc, pc, cpts, hpts, hns, hp, vcells},
    bds = pad[.1] /@ MinMax /@ Transpose[pts];
    dm = DelaunayMesh[pts];

    conn = dm["ConnectivityMatrix"[0, 1]];
    adj = conn . Transpose[conn];

    lc = conn["MatrixColumns"];
    pc = adj["MatrixColumns"];
    cpts = MeshCoordinates[dm];

    vcells = Table[
      hpts = PropertyValue[{dm, {1, lc[[i]]}}, MeshCellCentroid];
      hns = Transpose[Transpose[cpts[[DeleteCases[pc[[i]], i]]]] - cpts[[i]]];
      hp = MapThread[HalfSpace, {hns, hpts}];
      BoundaryDiscretizeGraphics[#, PlotRange -> bds]& /@ hp,
      {i, MeshCellCount[dm, 0]}
    ];

    AssociationThread[cpts, RegionIntersection @@@ vcells]
  ]

Example:

SeedRandom[10000];
pts = RandomReal[1, {10, 3}];

vc = VoronoiCells[pts]

enter image description here

Show[MapIndexed[
  BoundaryMeshRegion[#, MeshCellStyle -> {1 -> {Black, Thick}, 2 -> {ColorData[112][First[#2]]}}] &, 
  Values[vc]
]]

enter image description here

Show[
  MapIndexed[
    BoundaryMeshRegion[#, MeshCellStyle -> {1 -> Black, 2 -> {Opacity[0.5], ColorData[112][First[#2]]}}] &, 
    Values[vc]
  ], 
  Graphics3D[{PointSize[Large], Point[pts]}], 
  Method -> {"RelieveDPZFighting" -> True}
]

enter image description here

Note that this works in 2D too:

SeedRandom[10000];
pts = RandomReal[1, {10, 2}];

vc = VoronoiCells[pts];

Show[MapIndexed[
  BoundaryMeshRegion[#, MeshCellStyle -> {1 -> {Black, Thick}, 2 -> {ColorData[112][First[#2]]}}] &, 
  Values[vc]
], Epilog -> {PointSize[Large], Point[pts]}]

enter image description here

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11 Replies

Thanks for this function! will be very useful actually!

The code can be slightly simplified: MinMax also has padding built in, second argument:

MinMax[Â…,paddingspec]
Posted 1 year ago

Aha, thanks!

Posted 1 year ago

Some time ago, I submitted this Wolfram Demonstration: Three-Dimensional Voronoi Mesh but, since I did not use the Regions functionality, I had to limit it to 2D cross sections of the 3D mesh. Hope they includeChip's excellent function in V12?

A most excellent function and one that is needed. Thanks ! the function should be scaled up and made part of version 12.0 !

enter image description here - Congratulations! This post is now a Staff Pick as distinguished by a badge on your profile! Thank you, keep it coming, and consider contributing your work to the The Notebook Archive!

Hi. This is very interesting. Is it possible to link the points with the face centres of their cells? It will be nice to visualise this.

Great post. Thanks Fotos Stylianou

Very nice. Have you thought about submitting this to the function repository?

Also, do you know if this has a relationship to power diagrams?

He's had to think about contributing it to the WFR; I put in a request a couple of days ago.

I hope that such an important function can go beyond the function repository and can be incorporated in the next Mathematica release.

I have been using your method with more points (like 1000) and it has some issues due to probably an internal bug in BoundaryDiscretizeGraphics. You may want to check my post here https://mathematica.stackexchange.com/questions/219100/weird-behaviour-with-boundarydiscretizegraphics

I spent nearly a week programming this in Mathematica (v.5, as I recall) around 2001 for half of Figure 4.10 in my book Pattern classification (2nd ed). At that time, there was no Opacity[] functionality either. I believe mine is the first book to include such a figure. I am so glad Wolfram has included this functionality, and thank Chip Hurst for his contributions to it. It looks great!

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