A parting line for a polyhedron is a closed curve on its surface, which identiies 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 \\at" as possible is an important problem in mold design. ...

A number of different polyhedral decomposition problems have previously been studied, most notably the problem of triangulating a simple polygon. We are concerned with the polyhedron triangulation problem: decomposing a three-dimensional polyhedron into a set of nonoverlapping tetrahedra whose vertices must be vertices of the polyhedron. It has previously been shown that some polyhedra cannot b...

We define a notion of local overlaps in polyhedron unfoldings. We use this concept to construct convex polyhedra for which certain classes of edge unfoldings contain overlaps, thereby negatively resolving some open conjectures. In particular, we construct a convex polyhedron for which every shortest path unfolding contains an overlap. We also present a convex polyhedron for which every steepest...

We prove that there is a polyhedron with genus 6 whose faces are orthogonal polygons (equivalently, rectangles) and yet the angles between some faces are not multiples of 90, so the polyhedron itself is not orthogonal. On the other hand, we prove that any such polyhedron must have genus at least 3. These results improve the bounds of Donoso and O’Rourke [4] that there are nonorthogonal polyhedr...

We use the terminology polyhedron for a closed polyhedral surface which is permitted to touch itself but not self-intersect (and so a doubly covered polygon is a polyhedron). A flat folding of a polyhedron is a folding by creases into a multilayered planar shape ([7], [8]). A. Cauchy [4] in 1813 proved that any convex polyhedron is rigid: precisely, if two convex polyhedra P, P ′ are combinator...

The polyhedron model has proved to be a useful tool in studying methods for the automatic parallelization of loop nests. Most of the mathematical tools developed for the polyhedron model require the coefficients of variables to be constants. This restriction has turned out to be a severe limitation for several recent developments in the polyhedron model. We show how the polyhedron model can be ...

Given a graph G = (V, E) and a weight function on the edges w : E 7→ R, we consider the polyhedron P (G, w) of negative-weight flows on G, and get a complete characterization of the vertices and extreme directions of P (G, w). Based on this characterization, and using a construction developed in [11], we show that, unless P = NP , there is no output polynomial-time algorithm to generate all the...

We present a simple algorithm to compute a convex decomposition of a non-convex, non-manifold polyhedron of arbitrary genus (handles). The algorithm takes a non-convex polyhedron with n edges and r notches (features causing non-convexity in the polyhedra) and produces a worst-case optimal O(r2 ) number of convex polyhedra Si, with U;S; = S, in O(nr2 ) time and O(nr) space. Recenlly, Chazelle an...

This paper describes the results of our experiments with gluing together partial hyperbolic paraboloids, or hypars. We make a paper model of each hypar by folding a polygonal piece of paper along concentric polygons in an alternating fashion. Gluing several hypars together along edges, we obtain a beautiful collection of closed, curved surfaces which we call hyparhedra. Our main examples are gi...