Filling Polyhedral Molds

In the manufacturing industry, nding an orientation for a mold that eliminates surface defects and insures a complete ll after termination of the gravity casting process, is an important and diicult problem. We study the problem of determining a favorable position of a mold (modeled as a polyhedron), such that when it is lled, no air bubbles and ensuing surface defects arise. Given a polyhedron in a xed orientation, we present a linear time algorithm that determines whether the mold can be lled from that orientation without forming air bubbles. We also present an algorithm that determines the most favorable orientation for a polyhedral mold in O(n 2) time. A reduction from a well-known problem indicates that improving the O(n 2) bound is unlikely for general polyhedral molds. We relate llability to some well known classes of polyhedra. For some of these classes of objects, an optimal direction of llability can be determined in linear time. Finally, for molds that satisfy a local regularity condition, we give an improved algorithm that runs in time O(nk log 2 n loglog(n=k)), where k is the number of local maxima.