CGAL users discussion list

[Olivier Devillers <>]

 a écrit :
> I am interested in knowing how general the Triangulation_3 class is.  It 
> requires that the triangulation be a triangulation of a 4-sphere;  the basic 
> idea is that the triangulation is a finite triangulation of some convex body 
> together with a point at infinity.
>
> In some cases it would be possible to extend a tetrahedral mesh of a 
> nonconvex body to a mesh of the convex hull of the mesh vertices, but I 
> believe it is not always possible to do this without adding new vertices.
If the boundary of your object is non convex, you may link all facets of 
the boundary to the point at infinity
and then overload the is_valid test of a facet such that it does not 
check convexity.

>   For example, consider a tetrahedral mesh with a hole in the shape of a 
> Schonhardt polyhedron.
>   
If you have holes, then you can add one "virtual point"per hole, such a 
point play the same role as point at infinity.
You need to be able to distinguish between "usual tetrahedra" and 
"virtual tetrahedra" to overload the is_valid test

a virtual tetrahedron is a tetrahedron with a virtual vertex (or the 
vertex at infinity).
vertices can have an extra field "virtual"  or be stored in a map
(in current implementation, the only virtual vertex = vertex at infinity 
is stored in the triangulation object to be compared to any vertex)