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 polyhedra.

[1]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[2]  K. C. Hui,et al.  Mould design with sweep operations - a heuristic search approach , 1992, Comput. Aided Des..

[3]  Nimrod Megiddo,et al.  Linear-time algorithms for linear programming in R3 and related problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[4]  Avraham A. Melkman,et al.  On-Line Construction of the Convex Hull of a Simple Polyline , 1987, Inf. Process. Lett..

[5]  David Avis,et al.  A Linear Algorithm for Finding the Convex Hull of a Simple Polygon , 1979, Inf. Process. Lett..

[6]  David P. Dobkin,et al.  The Complexity of Linear Programming , 1980, Theor. Comput. Sci..

[7]  Micha Sharir,et al.  Fat Triangles Determine Linearly Many Holes , 1994, SIAM J. Comput..

[8]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[9]  Bernard Chazelle,et al.  Triangulating a simple polygon in linear time , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[10]  Eric L. Buckleitner,et al.  Molds for Reaction Injection, Structural Foam and Expandable Styrene Molding , 1987 .

[11]  Godfried T. Toussaint,et al.  Movable Separability of Sets , 1985 .