Efficiently Hex-Meshing Things with Topology

topological quadrilateral mesh Q of a connected surface in ℝ^3 can be extended to a topological hexahedral mesh of the interior domain if and only if Q has an even number of quadrilaterals and no odd cycle in Q bounds a surface inside . Moreover, if such a mesh exists, the required number of hexahedra is within a constant factor of the minimum number of tetrahedra in a triangulation of that respects Q . Finally, if Q is given as a polyhedron in ℝ^3 with quadrilateral facets, a topological hexahedral mesh of the polyhedron can be constructed in polynomial time if such a mesh exists. All our results extend to domains with disconnected boundaries. Our results naturally generalize results of Thurston, Mitchell, and Eppstein for genus-zero and bipartite meshes, for which the odd-cycle criterion is trivial.