does anyone have a definition for weighted voronoi diagram (this is like a voronoi diagram, only the points have different weights = different degrees of influence) ?
very interesting find indeed. if i can't get hold of a script until later tonight, i'll try and make one myself using the cones technique, although i need the resulting curves and i don't really know how to get them (except going in rhino and using the make2D command, which isn't really a good solution because i need to use those curves as input further in the definition...
i did give it a few tries.
for weighted voronoi the right way might be using spheres of different radii instead of cones at different heights.
here's the def using spheres and weights (i tried getting rid of those radial lines, but it wasn't an easy task so i forgot about it.
note that cells included whole in other cells won't work (because of the union procedure).
oh, and it's slow..
i REALLY hope someone would upload a proper def for this (meaning fast and that results in cells = closed curves for each zone)
I don't understand how intersecting spheres should give you the proper solution...
Daniel's idea is the one if your'e looking for a simple (though painfully slow) definition.
What's more, there are various types of weighted Voronoi diagrams. Among others there are the multiplicatively weighted and the additively weigthed diagram concepts. There's a good short review by Adam Dobrin here, which covers the definitions, as well as some of the properties and uses of different variations of the Voronoi diagram.
Interestingly, the intersecting cones method can be used to draw both MW and AW diagrams:
MW - cones with different base radius, same height:
AW - cones scaled accordingly to assigned weights:
i'm not sure what you are trying to say, but i am guessing that you accuse me of not mentioning where i got the inspiration for this (small) def.
if this is the case, you probably didn't read one post above:
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i might have found the solution on this blog http://madeincalifornia.blogspot.com/2009/03/simple-voronoi-01.html
There's a quick way to approximate additively weighted Voronoi diagrams:
- draw circles positioned at the generator points, with radii sizes corresponding to point weights
- sample (divide) the circles
- create an ordinary Voronoi diagram using the division points as generator points
- merge cells around base generator points
It's a very simple and fast method which came to my mind after reading Adam Dobrin's paper. I've attached a simple definition - works fine as long as generator points do not dominate each other: