Tetrahedralization of Simple and Non-Simple Polyhedra

It is known that not all simple polyhedra can be tetrahedralized, i.e., decomposed into a set of tetrahedra without adding new vertices (tetrahedralization). We investigate several classes of simple and non-simple polyhedra that admit such decompositions. In particular, we show that certain classes of rectilinear (isothetic) simple polyhedra can always be tetrahedralized in O(n2) time where n is the number of vertices in the polyhedron. We also show that polyhedral slabs (even with holes) as well as subdivision slabs can always be tetrahedralized in O(n log n) time. Furthermore, for simple polyhedral slabs O(n) time suffices. We show that polyhedra that are the union of three convex polyhedra can always be tetrahedralized in O(n2) time. Finally we show that if a polyhedron is the union of four or more convex polyhedra it does not necessarily admit a tetrahedralization.