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Alexandre,

I am interested in using (and possibly implementing) software for (constained) 3d tetrahedralization and so I was concerned when I saw your comment:

"3D delaunay (I assume you want tetrahedra, not that you're trying to fit a surface) is very very very very ill conditioned and almost impossible to get right robustly. Actually, the problem is less with the algorithm than with the typical data which is fed to it. – Alexandre C. Feb 4 '11 at 14:20"

Can you possibly elaborate on why you feel that 3d delaunay is so ill-conditioned? In particular, could you point me to a reference so that I can read on it myself?

Thanks,

Dan L.

sitiposit
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  • Your question is posted as a regular StackOverflow question, but it seems that you want to ask a specific person. Can you point to the context of that question? Is that a discussion in another question? – lrineau Jan 14 '15 at 10:43
  • The context to this question is entirely contained in the three quoted lines above. Although I addressed this question to the author of the comment, I meant to direct it to anyone who can help justify the claim. – sitiposit Jan 21 '15 at 14:08
  • See https://stackoverflow.com/a/45240506/4994003 for a robust spherical Delaunay triangulation. – ree2k Jan 08 '18 at 21:39

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The problem is that in the input points, there can easily be five points that are almost co-spherical, and for those points, the insphere predicate of the Delaunay triangulation is difficult to evaluate.

Actually, not so difficult. The following article Efficient Exact Geometric Predicates for Delaunay Triangulations explains how it is implemented in the 3D Delaunay triangulation of CGAL: the article explains how the predicates can be computed exactly, with filters that ensure the efficiency in non-degenerated cases.

lrineau
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