CGAL users discussion list

["Laurent Rineau (CGAL/GeometryFactory)" <>]

On Friday, January 11, 2019 5:14:00 PM CET Marc Alexa wrote:
> 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.

Yes, you are right. For each simplex of the Delaunay triangulation (with 
weights or not), then the simplex is in the triangulation restricted to a 
sub-space if and only if its dual intersects the sub-space.

For a tetrahedron, it is in the triangulation restricted to the domain if and 
only if its circumcenter is in the domain.

For a triangle, it is in the triangulation restricted to the boundary of the 
domain if and only if its dual segment (or dual ray, for triangles in convex 
position) intersects the boundary of the domain.

There are illustrations in 2D and 3D on Google Image. For example:

https://www.semanticscholar.org/paper/Conforming-restricted-Delaunay-mesh-generation-for-Engwirda/3607169f6d38b2085a582b3b2597515e86d6719d/figure/0
https://www-sop.inria.fr/geometrica/software/cgalmesh/restrictedDT.html

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