Generate a interpolation function from a mesh with assigned value?

Questions
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1

Hi all,
I was trying to use Mathematica for preprocessing to generate a Voronoi mesh with assigned values. Here is my code to generate the mesh:

noPoly = 20; boxSize = 400; pts = 
 RandomReal[{-boxSize, boxSize}, {noPoly, 2}]; mesh = 
 VoronoiMesh[pts, {{-boxSize, boxSize}, {-boxSize, boxSize}}]

You will get something like:

7998Capture.PNG
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Now this is what I want to do:

  1. For each “grain” in the diagram, I want to assign a constant value for the whole grain. i.e., define a function f(x,y) on the mesh such that it is constant in the same grain but maybe different for different grains.

  2. Then I want to generate a list of x,y,f(x,y) for x,y are uniform grid points, like (1,0), (2,0), (1,1), (1,2)… If I can do this, I know how to generate text file for another software to read and generate this function.

  3. is easy if I could do 1), i.e. get the function f(x,y). I have check the Mathematica documentation but have no idea how to do this. It seems PropertyValue might help but how? How do I even get those value defined on a mesh?

Thanks in advance!

2

It seems no one have a solution yet.

If it is not possible to assign value to a mesh, is there other ways to generate a labelled Voronoi diagram? If I could label them, maybe I can substitute the labels with my values?

3

Can be done with Nearest. Here is that example.

SeedRandom[123456]
noPoly = 20; boxSize = 400; pts = 
 RandomReal[{-boxSize, boxSize}, {noPoly, 2}]; mesh = 
 VoronoiMesh[pts, {{-boxSize, boxSize}, {-boxSize, boxSize}}]

.png.png
.png.png360×359 6.58 KB

We create the function using a random value for each Voronoi point.

vals = RandomReal[1, noPoly];
nf = Nearest[pts -> "Index"];
fun[x_] := vals[[nf[x][[1]]]]

We’ll check it by making a large array of these values and using ArrayPlot to illustrate in color. I do some rearrangement to account for the (in my opinion, counterintuitive) ordering used by ArrayPlot.

tt = Table[fun[{x, y}], {x, -400, 400}, {y, -400, 400}];
ArrayPlot[Transpose[Map[Reverse, tt]], ColorFunction -> "Rainbow"]

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9135.png360×359 19.5 KB

4

Hi Daniel,

Thanks for the workable solution.

I noticed the mesh is not being used in your solution. Does that mean there is no simple ways to extend this question to some arbitrary mesh or diagrams?

5

If given an arbitrary mesh, something similar can be done as follows.

(1) Extract polygons.

(2) Define function values based on polygon membership.

It won’t be as fast though.

6

Thank you!