proprieties/unity-delaunay

A Delaunay/Voronoi library for Unity, and a simple destruction effect

Voronoi destruction effect

I was interested in implementing some algorithms for generating Delaunay triangulations and Voronoi diagrams, and since it’s basically obligatory for anyone making Voronoi diagram libraries to make a destruction effect, this is mine!

The code is written in C#, and made for the Unity engine. If you want just the Delaunay/Voronoi parts, they’re in the Assets/Plugins/Delaunay folder. They’re not quite as polished as I would like, but I would say that they’re suitable for production. If you use it and have any problems, let me know! In fact, if you use it at all, I’d really appreciate hearing about it!

Demo

Higher res version

Implementation details

I’m generating the Delaunay triangulation using the incremental algorithm (Bowyer-Watson), and generating the voronoi diagram from the delaunay triangulation. You can also clip the voronoi diagram to any convex shape (it uses Sutherland-Hodgman clipping, adapted to voronoi diagrams).

The fact that you can clip to any convex shape is neat, because it means you can “recursively” generate voronoi diagrams, so you can shatter already generated voronoi shards.

All the “generator” classes (the delaunay generator, the voronoi generator, and the voronoi clipper) are designed in such a way as to not generate any garbage if they are reused (though I’m currently not doing that in the example code). If you were to do this in a real game, it would be suitable to run the generators on background threads, though they’re certainly performant enough to run in real time in a single-threaded environment.

I was going to develop this a bit further, but the limiting factor in this kind of destruction effect is the physics. If you generate these kinds of shards and let them pile up, the physics engine basically implodes. Given that that was the limitation, I sort-of lost interest in the project. As such, the non-Delaunay/Voronoi parts of the code are super-unoptimized and kind of a mess :)

Bowyer-Watson

It was an interesting experience to implement Bowyer-Watson, because I realized that virtually every implementation I could find online was wrong. The algorithm generates a Delaunay triangulation by first creating one big “container” triangle, and then iteratively adding sites to it splitting the big triangle into smaller triangles. However, the algorithm is only accurate if two of the vertices of the containing triangle are outside every circumcircle formed by any three points in the set points (the third vertex must be a “corner vertex” of sorts). Many implementations solve this by placing the vertices of the corners some large distance away, or by using some formula that guarantees that the triangle completely surrounds all the points (such as here or here).

This is a mistake. It’s not enough that the “super” triangle surrounds all the points, you have to guarantee that the two vertices are outside every possible circumcircle of any three points, and these circumcircles can very easily become huge. In fact, if three points are collinear, the “circumcircle” is essentially an infinite half-plane. Even if no three points are collinear, if they’re even close to collinear, the radius of the circle is massive. If you don’t account for this, the delaunay triangulation will not be correct. One easy way to spot incorrect implementations is that the the convex hull will not be part of the triangulation, and the convex hull is always part of the Delaunay triangulation. Virtually every implemention of this algorithm i could find online made this mistake, and you can often see it in the visualizations (here, for instance).

To do it right, you have to treat the vertices of the surrounding triangle as “symbolic” points. One way of thinking about them is that they’re sort of “infinitely” far away, but it’s more correct to say that they’re points with special rules for how they’re compared to the rest of the points. Computational Geometry: Algorithms and Applications describes this approach, but even they are wrong in one small detail: the test on the bottom on page 204 is wrong.

This experience has lead me to belive that using the incremental approach to generate Delaunay triangulations and Voronoi diagrams is a bad way to do it. It seems simpler to implement than other versions, but this subtlety with the containing triangle the fact that you have to construct an intricate tree structure makes the algorithm more complex and easier to get wrong than something like Quickhull or Fortune’s sweep-line algorithm. It’s also usually a bit slower. Still, it was a fun thing to implement, and having powered through it, you end up with a pretty stable and robust Delaunay/Voronoi calculator