Voronoi Diagrams and their applications

What is a Voronoi Diagram?

In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points (called seeds, sites, or generators) is specified beforehand, and for each seed there is a corresponding region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells. The Voronoi diagram of a set of points is dual to its Delaunay triangulation.

Other names that this diagram is known are Voronoi tessellation, Voronoi decomposition, Voronoi partition and Dirichlet tessellation.

The formal definition of a Voronoi Diagram

Matematically speaking, we can define a Voronoi Diagram as the following: Let X be a metrix space with distance function d. Let k be a set of indices and let (P_{k})_{k\epsilon K} be a tuple (an ordered collection) of nenempty subsets (the sites) in the space X. The Voronoi cell, also called Voronoi region, R_{k}, associated with the site P_{k} is the set of all points in X whose distance to P_{k} is not greater than their distance to the other sites P_{j}, where j is any index different form k. In other words, if d\left (x,A\right )=inf \left \{d\left (x,A\right )\left | a\epsilon A)\right \} denotes the distance between the point x and the subset A, then:

R_{k}=\left \{ x\epsilon X\left | d\left ( x,P_{x} \right ) \leq d\left ( x,P_{j} \right )for\, all\, j\neq k \right \}

So, the Voronoi diagram is simple the tuple of cells  \left ( R_{k} \right )_{k\epsilon K}. In principle some of the sites can intersect and even coincide but usually they ase assumed to the disjoint.

An application through Parametric Engineering

Thanks to parametric software like Sverchok (for Blender), Grasshopper (for Rhino) and many other applications (all of them work through nodes), we can design Voronoi cells in a plane, copy then into a parallel plane and loft the cells.

Here you are an example made by Grasshopper (Click on it to get bigger):

To do that is easy if whe have parametrised the cells of the Voronoi Diagram, I mean, we can construct lines of the borders that configure the cells, copy them into another plane that is parallel to the original plane and lastly we can loft (or extrude or whatever it is called depending on the software you are using) and connect all the cells.

After that, as usually in all the parametric softwares, to make the figure as something we can manipulate in a 3D software, we have to bake the final node. And export it in a suitable file to append (or merge or link…) into a scene ready for render a final image of the object. You have an example from Blender below:

And another render image without the bottom face:

The result in a wall

We can see many instances that people have applied this concept in art, façades, etc. on the Net.

One of the sites I have recently visited inspired me to try  the previous example and look for more information about Voronoi Diagram, is the C_Wall, from Matsys Design. They designed a cellular structure combining honeycomb and voronoi geometries that produces some interesting structural, thermal and visual performances into the building.

I show you some images of the project made by MATSYS:

The processus to make a fabrication layout is the following:

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