2

*Note this question applies to all languages

I am using triangulation in MATLAB for a Monte Carlo simulation of a physical surface. The triangulation represents a tethered sphere network and I have a certain constraint for the triangulation. In this simulation, I need the length of the tethers, represented by the edges, to be within a certain range of length. Note that is not the typical constraint used for constrained triangulation. How can I triangulate a surface, such that the the edge have lengths between a minimum and maximum length?

If there is an easier way to do this in another language, I am also willing to consider that.

TanMath
  • 598
  • 1
  • 9
  • 27
  • Not sure why you set montecarlo tag, nothing related to montecarlo in question. Do you need to triangulate sphere only in a such way, that triangle to be of certain size? – Severin Pappadeux Feb 11 '18 at 21:44
  • @SeverinPappadeux all I care is that the edge lengths are under a certain constant.... By that nature I guess the size of the triangle is also constrained... Note that the triangle needs to be changed during each iteration of the Monte Carlo simulation, but this edge length constraint is still present – TanMath Feb 11 '18 at 21:47
  • For planar sampling of points above certain distance there is so-called Poisson Disk algorithm (https://www.jasondavies.com/poisson-disc/). You want something like this but on the surface. Quick search produced link https://99db2f6c-a-62cb3a1a-s-sites.googlegroups.com/site/yandongming/publication/2012_SIGA_Brief_AMPS.pdf, check, for example, Fig 4, is it what you want? – Severin Pappadeux Feb 11 '18 at 22:15
  • @SeverinPappadeux I might not be understanding the paper but it seems to focus on angle constraint and also has a minimum length... I need a maximum length. – TanMath Feb 11 '18 at 22:35
  • Yes, Poisson disk cares about minimum length only, but maximum length usually is taken care by having maximal sampling (basically, you cannot add more points to the plane), and usually it is within 20% of radius of the disk. I'm beyond my knowledge here, never dealt with surface Poisson disk sampling, and would venture this is too much of a niche to get good answer on SO. You might think about asking authors of the paper I mentioned for advice and (maybe) code - maximal sampling might solve your max.length problem for all practical cases – Severin Pappadeux Feb 11 '18 at 22:48
  • @SeverinPappadeux would this work even if my lower limit is simply zero? What is the distribution used for the random points generated with Poisson disk sampling? – TanMath Feb 11 '18 at 23:53
  • well, in case of plane sampling lowering low limit to zero makes it usual 2D uniform-in-the-square sampling, no need of fancy algorithm. I guess the same will work for surface/sphere sampling - you end up with uniform-on-the-sphere sampling (http://mathworld.wolfram.com/SpherePointPicking.html) – Severin Pappadeux Feb 12 '18 at 00:15
  • You might find the answers [here](https://stackoverflow.com/q/42806059/3372061) useful, specifically the GA. – Dev-iL Feb 12 '18 at 10:41
  • I looked again at Yan&Wonka paper, Fig.4 and table. If we consider dart throwing to be just sphere points picking, then their algorithm behavior is a LOT more uniform than that. Did you look at the table below Fig.4? If I read it right, it is pretty much uniformly-sized triangles, not sure if you looking for a better option. – Severin Pappadeux Feb 14 '18 at 15:58
  • @SeverinPappadeux The triangulation does not have to necessarily be uniform. In fact, the simulations start out with a sphere that has a triangulation that is quite random. However, this constraint is still placed. – TanMath Feb 15 '18 at 20:29
  • ok, I'm out of my depth, beyond poisson disks on a surface nothing comes to my mind. Good luck! – Severin Pappadeux Feb 16 '18 at 03:46

0 Answers0