unity-delaunay VS delaunator

I wrote a moderately popular Delaunay/Voronoi library for Unity a few years back [1] with a neat little demo video [2]. I implemented the incremental Bowyer-Watson algorithm for generating the triangulations, and then took the dual to generate the Voronoi tesselation (I also added a "clipper" that clips the voronoi diagram to a convex outline, which was fun, I haven't seen that anywhere else before and had to figure out how to do it myself).

Bowyer-Watson (which is described in this article) seems very simple to implement when you start: just start with a "big triangle" and then add points iteratively to it, and perform the flips you need to do. To do it performant, you have to build up a tree structure as you go, but it's not very tricky.

However: almost every description and implementation of Bowyer-Watson I could find was wrong. There's an incredibly subtle and hard to deal with issue with the algorithm, which is the initial "big triangle". Most people who implement it (and indeed I did the same initally) just make the triangle big enough to contain all the points, but that's enough: it needs to be big enough to contain all the points in all the circumcircles of the triangles. These circumcircles can get ENORMOUS: in the limit of three points on a line, it's an infinite half-plane. Even if they aren't literally collinear, just close, the triangle becomes way to huge to deal with.

The answer is that you have to put the points "at infinity", which is a very weird concept. Basically, you have to have special rules for these points when doing comparisons, it's really tricky and very hard to get right.

If I were doing this again, I wouldn't use Bowyer-Watson, this subtlety is too tricky and hard to get right. Fortune's sweep-line is more complex on the surface, but that's the one I would go with. Or the 3D convex hull technique (which I also wrote a library for, by the way [3]).

[1]: https://github.com/OskarSigvardsson/unity-delaunay

[2]: https://www.youtube.com/watch?v=f3T5jtsokz8

[3]: https://github.com/OskarSigvardsson/unity-quickhull

Fortunately for me, the internet came in clutch with the Oskar Sigvardsson Delaunay Unity Library with the vast majority of the work done for me!

Here's a link for the destruction use i mentioned: https://github.com/OskarSigvardsson/unity-delaunay

Well, the easiest option would probably be to use a different Delaunay/Voronoi library. Delaunator (C# bindings here) should be a pretty good option: it's fast, and there's a good explanation of how to use the data structures it generates. There is some weirdness with how points at the edge of the domain work that you may have to work around, but that will probably be easier than trying to modify another algorithm or write your own. I know I've struggled to implement the algorithms for calculating Voronoi diagrams/Delaunay triangulations, even though I feel like I have a decent high-level understanding of how Fortune's algorithm is supposed to work.

Great article!

How does this algorithm compare to others, e.g. sweep algorithms like the one used by delaunator[0]?

An obvious difference is sweep algorithms sort the points in some way before adding them, is that a key to efficiency gains?

0: https://github.com/mapbox/delaunator/blob/main/README.md#pap...