Filling polyhedral molds
In the manufacturing industry, finding an orientation for a mold that eliminates surface defects and insures a complete fill after termination of the injection process is an important problem. We study the problem of determining a favorable position of a mold (modeled as a polyhedron), such that when it is filled, no air bubbles and ensuing surface defects arise. Given a polyhedron in a fixed orientation, we present a linear time algorithm that determines whether the mold can be filled from that orientation without forming air bubbles. We also present an algorithm that determines the most favorable orientation for a polyhedral mold in O(n2) time. A reduction from a well-known problem indicates that improving the O(n2) bound is unlikely for general polyhedral molds. But we give an improved algorithm for molds that satisfy a local regularity condition that runs in time O(nk log2n log log(n/k)), where k is the number of local maxima. Finally, we relate fillability to certain known classes of polyhedral.