5.7 Delaunay Slivers

A sliver has been introduced in Section 3.1 as an element with a very specific bad shape in the context of mesh generation. They are sometimes also called flat tetrahedra. Here, in the context of a Delaunay Triangulation and for the following chapter it is worthwhile to distinguish three sliver versions (Fig. 5.10).

• A sliver tetrahedron which does not satisfy the Delaunay criterion.
• A sliver tetrahedron which satisfies the Delaunay criterion in a strict sense. Its vertices are not part of a set of cospherical points and the Delaunay Triangulation around the sliver is unique.
• A sliver tetrahedron which satisfies the Delaunay criterion, but which is composed of vertices out of a set of cospherical points. The Delaunay Triangulation is not unique.

Figure 5.10: (a) Non-Delaunay sliver with circumsphere and two adjacent tetrahedra in the back (b) Strict sense Delaunay sliver with an empty circumsphere (c) Delaunay sliver with a cospherical point set.

The first type is of less importance and will clearly be absent in a Delaunay Triangulation. The second type, a Delaunay sliver, unfortunately exists in a Delaunay Triangulation for a given point set $P$ in three dimensions. The various optimality properties of a Delaunay Triangulation were given in Section 5.1. The minimum dihedral angle is not guaranteed to be optimal in three dimensions. A different non-Delaunay tetrahedralization might exist which avoids slivers with small dihedral angles but which is in other sense less optimal. In theory manipulation of $P$ by constructing a Steiner Triangulation with Steiner points at circumcenters does not help to eliminate Delaunay slivers. The question arises whether a different type of Steiner point insertion is capable to manipulate $P$ such that a Delaunay Triangulation results which does not contain Delaunay slivers. Alternatively, it would be of interest to examine how pronounced the occurrence of Delaunay slivers is in a practical mesh and how often they really survive the insertion of Steiner points at circumcenters in practice. Also, there is a chance that the third type sliver which is less critical exists much more often than a strict sense Delaunay sliver. Local transformations can be applied while maintaining the Delaunay property. The Delaunay Triangulation is not unique. The mesh examples in Section 3.2.3 have shown two different Delaunay Triangulations of an identical ortho-product point set where only one tetrahedralization contained sliver elements. A similar distinction of types can be made for the twisted prism. Its relation to slivers will be important in the next chapter (Section 6.4.3, Fig. 6.29).