make_mesh_3 with arbitrary sizing fields

[larsmunaf]

Dear all,

when replicating the Sizing Field as an Analytical Function example from the 3D Mesh Generation package with a spherical sizing field, I get the expected result:

Now, when changing the sizing field to some different function, e.g., a torus SDF, I get a result that is only somehow expected (by me). The resulting mesh resembles the torus shape, but the shape of the tetrahedra appears to change not as smooth as in the previous example.

Next, when changing the sizing field to a more complex function that encodes some notion of density around a given point cloud, the resulting mesh does not resemble the given sizing field at all:

For all examples above, I have used the same parameters

Mesh_criteria criteria(params::facet_angle(30).facet_size(0.1).facet_distance(0.025).
                                 cell_radius_edge_ratio(2).cell_size(size));

Obviously, make_mesh_3 outputs meshes based exactly on the criteria I provide, but:

• Is there some documentation or intuition on which criteria/parameters need to be tweaked when make_mesh_3 produces meshes that obviously need improvement (as in the second example) or do not correspond to the given sizing field at all (as in the third example)?
• Are there any requirements to the sizing field that is passed to make_mesh_3()?

Regards!

[sloriot]

You should provide a minimal example for the third case so that we can tell whether something is missing or if it's a bug in CGAL.

[MaelRL]

Internally, the cell size criterion compares the size of a tetrahedron (i.e., the radius of the circumscribing ball) to the size prescribed by a sizing field at the circumcenter of the tetrahedron. In other words, it is not a continuous criterion that would be checking if the tetrahedron is more or less of the correct size with respect to any point within the cell.

In your last example, everything is more or less yellow, so it is likely that your circumcenter falls onto a yellow zone, prescribes a large sizing, and thus no tetrahedron gets refined.

[larsmunaf]

This explanation helps a lot. I ended up removing the smoothing kernel from the size field so that it returns the actual distance to the nearest point, which fixed my issue. Thanks for your help!