Finding Undercut-Free Parting Directions for Polygons with Curved Edges

Abstract

We consider the problem of whether a given geometry can be molded in a two-part, rigid, reusable mold with opposite removal directions. We describe an efficient algorithm for solving the opposite direction moldability problem for a 2D “polygon” bounded by edges that may be either straight or curved. We introduce a structure, the normal graph of the polygon, that represents the range of normals of the polygon’s edges, along with their connectivity. We prove that the normal graph captures the directions of all lines corresponding to feasible parting directions. Rather than building the full normal graph, which could take time O(nlogn) for a polygon bounded by n possibly curved edges, we build a summary structure in O(n) time and space, from which we can determine all feasible parting directions in time O(n).