First we measure shape quality by the minimum solid angle. Other shape metrics are equally effective, but unlike 2D the ratio of the shortest to longest edges on the elements is a poor metric because of the degenerate "sliver" tetrahedra shown on the bottom. The vertices of this pathological element are nearly co-planar and the slightest movement of any vertex can cause the sign of the volume of the element to flip.

We include a volume term which encourages elements to have roughly the same volume as the ideal volume.