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I'd like to have random tessellations of regions in a hyperbolic space.

In the Euclidean plane I get good results by scattering random points and performing a periodic Delaunay triangulation using CGAL.

For the hyperbolic case, though, there is nothing yet available in the library, even tough work on the implementation of non-Euclidean triangulations and meshes in CGAL was ongoing already in 2011, and essentially ready by 2014.

A purportedly "easy" recipe for implementing the hyperbolic triangulation has been long available (arxiv.org:0903.3287), but I don't think it's trivial to implement it reliably.

Is there any other implementation of hyperbolic Delaunay triangulations, preferably with periodic boundary conditions?

stafusa
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    For CGAL I see this branch https://github.com/CGAL/cgal-public-dev/tree/Periodic_4_hyperbolic_triangulation_2-IIordanov , don't know how far along it is exactly (or if the name of the branch means that it contains the feature you want), but the latest commits are rather recent. – Marc Glisse May 19 '17 at 05:51
  • Thanks Marc. As Monique points out in her answer, that code is for periodic, not random triangulations, but I was not aware of it: +1 (had I reputation for that). – stafusa May 19 '17 at 07:24
  • I'd really like to know why the question was downvoted. Being a niche question shouldn't be a reason, the information I provide makes clear it's well researched, and if there's something vague about it (which I think there isn't), asking for clarification (even if in addition to downvoting) would be far more constructive. – stafusa May 19 '17 at 07:27
  • I guess that it is not such a niche. The literature shows that this tool is used by many physicists. That's also why we are working on the code, I would not invest time in software if I did not have a good hope that it would be used at some point. – Monique Teillaud May 19 '17 at 07:43
  • What do you mean by "that code is ... not random triangulations"? If you insert random points, doesn't it provide you with what you need? – Monique Teillaud May 19 '17 at 07:45
  • @MoniqueTeillaud, from the branch name and the comment in your answer, "is computing periodic triangulations" I had understood it provides regular tessellations in the Poincaré disk. I see now, from your other comment, that I'm mistaken, and the "periodic" in the name refers instead to periodic boundary conditions. (BTW, I also don't think it's niche, I only mentioned that because it's a possible downvote reason.) – stafusa May 19 '17 at 10:08

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The code that Marc mentions is computing periodic triangulations (along translations corresponding to the hyperbolic octagon), following the paper soon to be presented at SoCG'17 (see https://hal.inria.fr/hal-01411415 for a preliminary version).

We also have code that computes Delaunay triangulations in the hyperbolic plane, as presented in our JoCG paper (see http://jocg.org/index.php/jocg/article/view/141). The branch is currently private in github, but we will make it public soon. Some parts need polishing, though, and the documentation is not yet written.

Monique Teillaud
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  • Thanks a lot Monique. I just joined your CGAL mailing-list -- when the branch is made public, will that be announced there? – stafusa May 19 '17 at 07:29
  • no, there is no announcement for branches, there are only announcements for official releases. Anyway, the branch was just made public, see https://github.com/CGAL/cgal-public-dev/tree/Hyperbolic_triangulation_2-MBogdanov. However, if you need periodic boundary conditions, the branch mentioned by Marc may be the one that you need. It depends on which boundary conditions you need. As I mentioned, we provide periodicity conditions for the regular hyperbolic octagon. – Monique Teillaud May 19 '17 at 07:36
  • Thanks for making it public and for the clarifications! – stafusa May 19 '17 at 10:13