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Additively Weighted Voronoi Tessellation

Rodrigo Murta, Looqbox
Posted 12 years ago
Anyone know if there is any package where I can create Additively Weighted Voronoi Tessellation?
I found this code for R here. I know I can run R in MMA, but it's not what I need.
Here is how I create the Non Weighted Tessellation.
pts=MapIndexed[Flatten[{##}]&,RandomReal[{0,1},{20,2}]];
g1=ListDensityPlot[pts
    ,ColorFunction->Hue
    ,InterpolationOrder->0
    ,ImageSize->500
    ,PlotRange->{{0,1},{0,1}}
    ,Mesh->All
]
POSTED BY: Rodrigo Murta
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It may be that I misunderstood what is an additively weighted Voronoi tesselation as the concept was new to me, but would this method be useful to you?  (I.e. does this simply amount to having a different norm?)

Why do you want to avoid using R?  Is it because you're not familiar enough with R and it seems like too much effort to work it out (a reasonable decision) or do you have some technical reason?
POSTED BY: Szabolcs Horvát
Szabolcs nice to see you here.
The method that you suggested is interesing, but I need one form that I can latter work with individual faces as polygons, nos as image.
About R, yes, I don't have much affinity with it, and I would like to use the Weighted Voronoi Tessellation as part of a bigger MMA pack for Voronoi Tree. The only part that is missing is this.
tks
POSTED BY: Rodrigo Murta
Hello Rodrigo,
I am not sure if this is useful to you.

The ComputationalGeometry package has references to Nearest. However, it is not clear whether Nearest is being used to compute the triangulation and its dual.

I wasn't successful, but I tried to hack the package and use an alternative DistanceFunction for Nearest.  This might be a possible approach for you?

This is something that I had hoped to get back to someday. If you solve this, could you repost here?

Craig
POSTED BY: W. Craig Carter
@Craig,

The reason why a basic approach like this can't work is that the boundaries between the regions in an additively weighte Voronoi tesselatin are not straight lines.  The function from ComputationalGeometry describes the result using the assumption that the boundaries are straight.
POSTED BY: Szabolcs Horvát
Hello Szabolcs,
Yes, I can see that from your example---but I am still not sure what the definition of "additively weighted voronoi" is.

I was thinking that one could compute delaunay the triangulation using a different distance metric and then compute its dual. The dual in this case wouldn't satisify the voronoi definition (set  of "closest" points to a set of given points) as in the example.
POSTED BY: W. Craig Carter
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