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[Marc Alexa <>]

Hi Laurent,

thanks for the clarification about the weights. I’m not using sharp features so I should be fine. 

So what is the definition of restricted Delaunay? That the circumcenter needs to be part of the domain? That would explain why sliver on the boundary are not part of the triangulation.

Thanks!

-Marc

On 11. Jan 2019, at 11:06, Laurent Rineau (CGAL/GeometryFactory) <> wrote:

On Saturday, January 5, 2019 1:53:50 PM CET Marc Alexa wrote:

Dear all,

I am generating a tetrahedral mesh using the suite of algorithms in Mesh_3.
I want to use the resulting mesh in my code, which is based on
Triangulation_3. The vertex and cell base classes are incompatible. My
preferred way for doing this would be writing the mesh to a file.
Unfortunately, the formats of the stream operators are different. Is there
an easy way to “reduce” what is being written by Mesh_3 to what is being
expected by Triangulation_3?

On a side note: I understand that if I am not using sliver exudation, the
triangulation generated by Mesh_3 will be Delaunay. This means I could
export the points only (and then reconstruct the tetrahedra). Then I only
need information on which tetrahedra are in the interior of the meshed
surface.

Hi Marc,

Note that the trianguation is not a Delaunay triangulation, but a *weighted* 
Delaunay triangulation (call a regular triangulation). If your mesh domain 
does not have 1D featured curves, then all the weights will be 0, and then 
only the triangulation is equivalent to a Delaunay triangulation. (And as soon 
as you use the sliver exuder, then the non-zero weights reappear.)

I tried to circumvent this by using a simple convex surface,
namely a sphere. Yet even for the sphere some tetrahedra are excluded,
despite clearly lying in the interior. I do understand that the excluded
tetrahedra are slivers, but wouldn’t it make sense that a tetrahedral mesh
with the boundary vertices on the sphere contained everything inside the
sphere?

I agree that could be surprising, but that is not the definition of the 
Delaunay triangulation restricted to the mesh domain.

-- 
Laurent Rineau, PhD
R&D Engineer at GeometryFactory           http://www.geometryfactory.com/
Release Manager of the CGAL Project       http://www.cgal.org/




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