ABSTRACT
Three-dimensional triangulations are sometimes called tetrahedralizations. Delaunay tetrahedralizations are not quite as effective as planar Delaunay triangulations at producing elements of good quality, but they are nearly as popular in the mesh generation literature as their two-dimensional cousins. Many properties of Delaunay triangulations in the plane generalize to higher dimensions, but many of the optimality properties do not. Notably, Delaunay tetrahedralizations do not maximize the minimum angle (whether plane angle or dihedral angle). Figure 4.1 depicts a three-dimensional counterexample. The hexahedron at the top is the convex hull of its five vertices. The Delaunay triangulation of those vertices, to the left, includes a thin tetrahedron known as a sliver or kite, whose vertices are nearly coplanar and whose dihedral angles can be arbitrarily close to 0◦ and 180◦. A triangulation of the same vertices that is not Delaunay, at lower right, has better quality.
