Perturbations for Delaunay and weighted Delaunay 3D triangulations

The Delaunay triangulation and the weighted Delaunay triangulation are not uniquely defined when the input set is degenerate. We present a new symbolic perturbation that allows to always define these triangulations in a unique way, as soon as the points are not all coplanar. No flat tetrahedron exists in the defined triangulation. The perturbation scheme is easy to code; It is implemented in cgal, and guarantees that both vertex insertion and vertex removal are fully robust.