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As a simple example of what I want to do, let's say I have a Voronoi mesh in a $10\times10$ area with $10$ random "seed points":

SeedRandom[1];
randMesh=VoronoiMesh[RandomReal[{1, 10}, {10, 2}], {{1, 10}, {1, 10}}]

enter image description here

And I am interested in all integer points within this area. i.e. the $100$ points determined by

points = Flatten[Table[{x, y}, {x, 10}, {y, 10}], 1]

What is an efficient way to group the points by the mesh section they reside in?

One way I have thought of doing this is is just going point-by-point and checking each region, such as something like

{5, 5} \[Element] # & /@ MeshPrimitives[randMesh, 2]

{False, False, True, False, False, False, False, False, False, False}

but applied to each point, and then grouping the points based on which mesh primitive they belong in. But this is not very efficient, as in what I am actually doing I have a $512\times512$ area with about $1000$ seed points for the mesh. Is there a better way to approach this?

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2 Answers 2

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This is possible already in version 12, but just undocumented.

grouped =  GatherBy[points, Region`Mesh`MeshNearestCellIndex[randMesh]];

Apparently, this can find only top-dimensional cells. This is a good example where the syntax of undocumented code was changed in the final version.

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    $\begingroup$ Ugh I feel like I always get hit by the undocumented things. Thanks for the simple fix $\endgroup$
    – BioPhysicist
    Mar 20, 2020 at 21:49
  • $\begingroup$ You're welcome. $\endgroup$
    – Henrik Schumacher
    Mar 20, 2020 at 21:50
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In version 12.1 you can use NearestMeshCells with GatherBy as follows:

grouped = GatherBy[points, NearestMeshCells[{randMesh, 2}, #] &];

Show[randMesh, 
 ListPlot[grouped, BaseStyle -> PointSize[Large], 
  PlotLegends -> ("group-" <> ToString[#] & /@ Range[Length @ grouped])]]

enter image description here

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  • $\begingroup$ What if I have 12.0? :) $\endgroup$
    – BioPhysicist
    Mar 20, 2020 at 21:40
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    $\begingroup$ @AaronStevens it’s worth the upgrade! Especially for when undocumented functions go documented like in this example! $\endgroup$
    – CA Trevillian
    Mar 20, 2020 at 22:39
  • $\begingroup$ kglr, the link and linked term are missing a necessary s, I could not edit as it is less than 6 characters $\endgroup$
    – CA Trevillian
    Mar 21, 2020 at 3:07
  • $\begingroup$ Thank you @CATrevillian. I fixed it. $\endgroup$
    – kglr
    Mar 21, 2020 at 3:13
  • $\begingroup$ Is it possible to generate a more efficient version of the function NearestMeshCells[{randMesh, 2}, #] & with NearestMeshCells[{randMesh, 2}]? Typically, this is possible with Nearest-related constructs. $\endgroup$
    – Henrik Schumacher
    Mar 21, 2020 at 8:21

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