CGAL users discussion list

[Timothée Nicolas <>]

Dear all,

I would like to clarify one or two points. I am learning to mesh 3D domain with Mesh_3 from a surface .OFF file, and I would like to have more understanding on how to use Mesh_criteria_3 to get the most regular mesh, in terms of Radius Ratios defined as the ratio between the radii of the inscribed and circumscribed spheres (normalized by 3 so that the regular tetrahedron has a radius ratio of 1.0). The problem is that Mesh_criteria_3 provides a criterion which is the radius to edge ratio, that is, the ratio between the radius circumscribed sphere and the shortest edge. It is well known that this measure eliminates all bad tetrahedron except for slivers. Then, my first question is

1) What criterion to use in order to decrease the number of slivers ? I thought at first that facet_angle would be the job, but it seems to act only on the surface mesh. I picked up CGAL because I would like to obtain the nice results of the paper by Pierre Alliez et al, (Alliez, P., Cohen-Steiner, D., Yvinec, M., & Desbrun, M. (2005). Variational tetrahedral meshing. ACM Transactions on Graphics (TOG), 24(3), 617-625.), which is about removing slivers. See in particular Figure 10.

I have an other question

2) How precise are the algorithms supposed to be ?  I have produced a mesh where I specified a maximum radius to edge ratio of 1.05 (the minimum for the regular tet is ~0.6124), and in the result I still have some tets with a larger radius to edge ratio (though I can see that the parameter does ave some effect). I have attached a figure where the problem is apparent. There is a clear drop around 1.05, but still dozens of tets with a ratio larger than 1.05, up to 1.4 !

Thx

Timothee

Attachment: RadiusEdgeRatios.png
Description: PNG image

Attachment: Tetrahedron.png
Description: PNG image