Computing a flattest, undercut-free parting line for a convex polyhedron, with application to mold design

AbstractA parting line for a polyhedron is a closed curve on its surface, which identifies the two halves of the polyhedron for which mold-boxes must be made. A parting line is undercut-free if the two halves that it generates do not contain facets that obstruct the de-molding of the polyhedron. Computing an undercut-free parting line that is as “flat” as possible is an important problem in mold design. In this paper, algorithms are presented to compute such a parting line for a convex polyhedron, based on different flatness criteria

A Technique for Adding Range Restrictions to Generalized Searching Problems

In a generalized searching problem, a set S of n colored geometric objects has to be stored in a data structure, such that for any given query object q, the distinct colors of the objects of S intersected by q can be reported efficiently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (resp. fat triangles) with a fat triangle (resp. point). For both problems, a data structure is obtained having size O(n 1+ffl ) and query time O((log n) 2 + C). Here, C denotes the number of colors reported by the query, and ffl is an arbitrarily small positive constant. Keywords: Computational geometry, data structures, intersection searching, range restriction. 1 Introduction Geometric searching problems arise in a large variety of application areas, such as computer graphics, robotics, VLSI layout design, and databases. In such a problem, a set S of n geometric objects h..

Computing a Flattest, Undercut-Free Parting Line for a Convex Polyhedron, With Application to Mold Design

A parting line for a polyhedron is a closed curve on its surface, which identifies the two halves of the polyhedron for which mold-boxes must be made. A parting line is undercut-free if the two halves that it generates do not contain facets that obstruct the de-molding of the polyhedron. Computing an undercut-free parting line that is as "flat" as possible is an important problem in mold design. In this paper, algorithms are presented to compute such a parting line for a convex polyhedron, based on different flatness criteria. Keywords: Casting/molding, computational geometry, optimization. Additional keywords: Arrangements, shortest paths, visibility, point-set width. 1 Introduction We consider a geometric problem arising in the design of molds for casting and injection molding. Consider the construction of a sand mold for casting a polyhedral solid. First a prototype, P, of the polyhedron is made. Two halves of P are then identified and a separate mold-box is made for each. This i..

The Rectangle Enclosure and Point-Dominance Problems Revisited

We consider the problem of reporting the pairwise enclosures in a set of n axesparallel rectangles in IR 2 , which is equivalent to reporting dominance pairs in a set of n points in IR 4 . Over a decade ago, Lee and Preparata 6 gave an O(n log 2 n + k)--time and O(n)--space algorithm for these problems, where k is the number of reported pairs. Since that time, the question of whether there is a faster algorithm has remained an intriguing open problem. In this paper, we give an algorithmwhich uses O(n) space and runs in O(n log n log log n+ k log log n) time. Thus, although our result is not a strict improvement over the Lee-- Preparata algorithm for the full range of k, it is, nevertheless, the first result since Ref. (6) to make any progress on this long--standing open problem. Our algorithm is based on the divide--and--conquer paradigm. The heart of the algorithm is the solution to a red--blue dominance reporting problem (the "merge" step). We give a novel solution for this..

Efficient Algorithms for Counting and Reporting Pairwise Intersections between Convex Polygons

Let S be a set of convex polygons in the plane with a total of n vertices, where a polygon consists of the boundary as well as the interior. Efficient algorithms are presented for the problem of reporting output-sensitively (resp. counting) the I pairs of polygons that intersect. The algorithm for the reporting (resp. counting) problem runs in time O(n 4=3+ffl +I) (resp. O(n 4=3+ffl )), where ffl ? 0 is an arbitrarily small constant. This result is based on an interesting characterization of the intersection of two convex polygons in terms of the intersection of certain trapezoids from their trapezoidal decomposition. Also given is an alternative solution to the reporting problem, which runs in O(n 4=3 log O(1) n + I) time, and is based on characterizing the intersection of two convex polygons via the intersection of their upper and lower chains and their leftmost vertices. The problems are interesting and challenging because the output size, I , can be much smaller..