Determining Moldability and Parting Directions for Polygons with Curved Edges

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(n logn) 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). 1 Background and Previous Work In molding or casting manufacturing processes, material is reshaped in a hollow mold. A simple reusable mold consists of two rigid halves that are removed in opposite directions; the orientation of the removal directions is called the parting direction. In order for a part geometry to be de-moldable, it must be oriented relative to the parting direction so that the two mold halves can be removed from the part via translation along the parting direction without colliding with the part. Surfaces where collisions occur, preventing extraction of the part, are called undercuts. They occur where the mold extends into the area between the part and the parting surface, relative to the parting direction (Figure 1). Forming undercuts requires additional mold inserts which increase the cost of the mold, so we would like to avoid Corresponding author. them if possible. Finding a feasible two part molding orientation (one without undercuts) for an arbitrary geometry is subject to geometric accessibility constraints; not all geometries admit such an orientation. Many manufacturing researchers who have implemented algorithms for finding a feasible parting direction for a two part mold for a given 3D geometry (possibly defined by curved faces) only look at a limited number of potential parting directions, such as the three principle axes [1, 2], bounding box axes [3], or use a heuristic search approach [4,5]. If a valid parting direction exists but is not among the directions tested, these algorithms will not find it. Multi-piece and/or sacrificial mold design, where constraints on the number of mold pieces and/or demoldability are relaxed, has been studied by [6], [7] and [8], but these molds are more expensive than two part molds, and sacrificial molds are not as suited to mass production. In the computational geometry literature, algorithms for finding if any feasible two part mold orientation exists have been presented, but only for faceted geometry. Rappaport and Rosenbloom describe an O(n) time algorithm for determining if a 2D polygon can be made by a 2-part mold with arbitrary (not necessarily opposite) removal directions for the mold halves, and an O(n logn) time algorithm for a variation on the opposite removal direction problem, again for a 2D polygon [9]. Ahn et al. describe an algorithm for opposite direction mold removal parting line direction determination for a 3D faceted polyhedron, returning all combinatorially distinct feasible directions in time O(n4) [10]. Although this algorithm could of course be applied to find feasible mold removal directions for a tessellated approximation of a curved geometry, the running time would increase prohibitively the more accurately the approximation fit the original geometric surface. 1 Copyright c © 2004 by ASME parting direction upper mold half lower mold half parting surface part (a) Mold for a simple 2D part parting direction