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# 3D version of the built-in VoronoiDiagram

Posted 5 years ago
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Posted 2 years ago
 Chip and me were discussing his function a few days ago, when I mentioned to him that this can be used to implement Lloyd relaxation in 3D. In the spirit of this 2D Lloyd implementation and this spherical Lloyd implementation, I present the following: VoronoiCells[pts_List, ibds : (_?MatrixQ | Automatic) : Automatic] /; MatrixQ[pts, InternalRealValuedNumericQ] && 2 <= Last[Dimensions[pts]] <= 3 := Module[{bds, dm, conn, adj, lc, pc, cpts, hpts, hns, hp, vcells}, If[ibds === Automatic, bds = CoordinateBounds[pts, Scaled[0.1]], bds = ibds]; 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]] BlockRandom[SeedRandom["voronoi"]; pts = RandomReal[{-1, 1}, {32, 3}]]; lloyd = With[{maxit = 50,(*maximum iterations*) tol = 0.005 (*distance tolerance*)}, FixedPointList[Function[pts, RegionCentroid /@ Values[VoronoiCells[pts, {{-1, 1}, {-1, 1}, {-1, 1}}]]], pts, maxit, SameTest -> (Max[MapThread[EuclideanDistance, {#1, #2}]] < tol &)]]; frames = Function[pt, Show[MapIndexed[BoundaryMeshRegion[#, MeshCellStyle -> {1 -> Black, 2 -> {Opacity[0.5], ColorData[112] @@ #2}}] &, Values[VoronoiCells[pt, {{-1, 1}, {-1, 1}, {-1, 1}}]]], Graphics3D[{PointSize[Large], Point[pt]}], Axes -> True, Boxed -> True, Method -> {"RelieveDPZFighting" -> True}]] /@ lloyd; ListAnimate[frames] 
Posted 3 years ago
 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!
Posted 3 years ago
 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
Posted 3 years ago
 Very nice. Have you thought about submitting this to the function repository?Also, do you know if this has a relationship to power diagrams?
Posted 3 years ago
 He's had to think about contributing it to the WFR; I put in a request a couple of days ago.
Posted 3 years ago
 I hope that such an important function can go beyond the function repository and can be incorporated in the next Mathematica release.
Posted 3 years ago
 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
Posted 5 years ago
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Posted 5 years ago
 A most excellent function and one that is needed. Thanks ! the function should be scaled up and made part of version 12.0 !
Posted 5 years 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?
Posted 5 years ago
 Thanks for this function! will be very useful actually! The code can be slightly simplified: MinMax also has padding built in, second argument: MinMax[,paddingspec] `
Posted 5 years ago
 Aha, thanks!