voronoi diagrams

A voronoi diagram is a way of decomposition or subdivision of space based on an initial set of objects or points (for more information http://en.wikipedia.org/wiki/Voronoi). Voronoi diagrams have countless applications from statistics to biology and urban planning.

[construction method of a 2d voronoi cell]

[several examples of 2d voronoi]

Voronoi diagrams can also be useful in an architectural context, for several reasons:
a. Their structural properties, both in 2d and 3d.
b. As a way to subdivide/organize space, based on proximity/closest neighbor.
c. The fact that they can describe many natural formations, like soap bubbles, sponges or bone cells, which can inform architecture with new ways to organize and structure space.

This project is an attempt to explore the possibilities that voronoi diagrams can provide to architectural design, and to find ways in which they can be used in order to create an architectural process.

In that context, the two main problems that this project tries to study are:
a. a method to construct the voronoi diagrams in such a way that they can be incorporated inside standard 3d packages like rhino or 3dstudio max.
b. a way to define the initial set of points (that are necessary in order for a voronoi diagram to be created).

Concerning the first question, there are already several applications that can be found online. For example a plug in the constructs voronoi diagrams in rhino is already developed and can be found here. (however at the time that this project was developed the plugin was failing to calculate correctly complex 3d voronoi diagrams. A more reliable application is qhull. However custom scripts have to be created in order for the results of qhull to be imported into a 3d software.

[applying different smoothing algorithms to the edges of the voronoi cells]

[using lines as the generators of the diagrams]

-Aranda, B., Lasch, C. Tooling New York: Princeton Architectural Press, 2006.
-Bollobas, B., Riordan, O. Percolation Cambridge: Cambridge University Press, 2006.
-Kim, M.S., Shimada, K., (eds.) Geometric Modeling and Processing – GMP 2006 Berlin: Springer, 2006.
-Klein, R. Concrete and Abstract Voronoi Diagrams Berlin: Springer-Verlag, 1987.
-Okabe, A., Boots, B., Sugihara, K., Chiu, S.N. Spatial Tessellations, Concepts and Applications of Voronoi Diagrams West Sussex: John Wiley & Sons Ltd, 2000 (f.e. 1992).

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gabe f said...


I realise you probably just reproduced the images from one of the books, and you may not know how to create such models...But once you have modelled your voronoi structure - which I have - do you know how to get the vertices to be all nice and rounded/smooth? Im new to rhino...


dimitris gourdoukis said...

all the voronoi diagrams were produced in rhino using qhull. The "smoothing" however was done in 3dstudio max. It will be hard to do something similar in rhino. Maybe you can do it if you build your voronoi diagram with polygons and then you use t-splines.

Goran Jankovic said...

Hi, i tried to follow you trough different blog posts to see if i can understand how to create 5-6 voronoi cells (the one shown in this post in three step evolution is the one i need). I have to use it for my exam, and i'm thrilled that i found your blog, but i just can't get it.
Is it possible to model 1st step in rhino as well as the other two? And then i hve to export it for 3dmax for smoothing? Or just first is done with qhull and rhino?!
You say, rhino can't build me Vcells, so you used qhull plugn. I tried to download it, and sincerly i don't have a clue how to install and use it. Is there on the web some kind of tutorial how to do exactly the same kinda cells you shown here? Please help, :) i have so much time that i can barly learn all this stuff to be able to apply it.
p.s.- did you have to use grasshopper for those (i hope not) :)