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Bowyer–Watson algorithm might be the most famous method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. Its Wikipedia page and most textbooks on unstructured meshing 1 introduced it as the same way in the original article of Watson's 2. However, when I looked into Bowyer's original article 3, it seems to me his version is different from Watson's version. Are they really equivalent? If so, how can they be converted into each other?

Bowyer's version:

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Watson's version:

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1 Farrashkhalvat, M., and J. P. Miles. Basic Structured Grid Generation: With an introduction to unstructured grid generation. Elsevier, 2003.

2 Watson, David F. (1981). "Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes". Comput. J. 24 (2): 167–172. doi:10.1093/comjnl/24.2.167

3 Bowyer, Adrian (1981). "Computing Dirichlet tessellations". Comput. J. 24 (2): 162–166. doi:10.1093/comjnl/24.2.162

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Bowyer's version looks like it's expressed in terms of the vertices of a Voronoi diagram. Voronoi diagrams and Delaunay triangulations are "dual" to each other, meaning that each vertex of the Voronoi diagram corresponds to one triangle of the Delaunay triangulation. In fact, the Voronoi vertex is at the circumcenter of the corresponding Delaunay triangle.

So, when Bowyer speaks of removing any "vertex that is nearer to the new point than to its forming points", that's the same thing as removing triangles (simplices) that have the new point within their circumcircle (circumsphere).

In both articles, the process of finding new vertices/simplices is, uhh, not exceptionally cleary explained. But if you squint a bit, they're both talking about filling in the "hole" created by deleting the old vertices/simplices, by combining the new point with each of the edges around the "hole".

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  • $\begingroup$ Thanks for the answer! Does it mean that they are equivalent in the sense of their general idea instead of the detailed algorithm? $\endgroup$
    – 8cold8hot
    Mar 29, 2022 at 8:20
  • $\begingroup$ I would say that they are talking about the same idea using different language and terminology. $\endgroup$
    – Nathan Reed
    Mar 29, 2022 at 17:02

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