Mold Accessibility via Gauss Map Analysis

Abstract

In manufacturing processes like injection molding or die casting, a two-piece mold is required to be separable, that is, be able to have both pieces of the mold removed in opposite directions while interfering neither with the mold nor with each other. The fundamental problem is to find a viewing (i.e., separating) direction, from which a valid partition line (i.e., the contact curves of the two mold pieces) exists. While previous research work on this problem exists for polyhedral models, verifying and finding such a partition line for general freeform shapes, represented by NURBS surfaces, is still an open question. This paper shows that such a valid partition exists for a compact surface of genus g, if and only if there is a viewing direction from which the silhouette consists of exactly g+1 nonsingular disjoint loops. Hence, the two-piece mold separability problem is essentially reduced to the topological analysis of silhouettes. In addition, we deal with removing almost vertical surface regions from the mold so that the form can more easily be extracted from the mold. It follows that the aspect graph, which gives all topologically distinct silhouettes, allows one to determine the existence of a valid partition as well as to find such a partition when it exists. In this paper, we present an aspect graph computation technique for compact free-form objects represented as NURBS surfaces. All the vision event curves (parabolic curves, flecnodal curves, and bitangency curves) relevant to mold separability are computed by symbolic techniques based on the NURBS representation, combined with numerical processing. An image dilation technique is then used for robust aspect graph cell decomposition on the sphere of viewing directions. Thus, an exact solution to the two-piece mold separability problem is given for such models.