Given a convex polyhedron with n vertices and F faces, what is the fewest number of pieces, each of which unfolds to a simple polygon, into which it may be cut by slices along edges? Shephard’s conjecture says that this number is always 1, but it’s still open. The fewest nets problem asks to provide upper bounds for the number of pieces in terms of n and/or F . We improve the previous best know...

Based on an extension of the Hermite-Biehler Theorem to the quasipolynomial stability problem, this paper studies the problem of stabilizing a second-order plant with dead time via a PID controller. The region in PID parameters space for the closed-loop stability is given. For a feasible proportional gain ( p k ), the region of all the admissible integral gains ( i k ) and derivative gains ( d ...

We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q “lives on a cone” to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new gen...

An effective strategy for accelerating the calculation of convex hulls for point sets is to filter the input points by discarding interior points. In this paper, we present such a straightforward and efficient preprocessing approach by exploiting the GPU. The basic idea behind our approach is to discard the points that locate inside a convex polygon formed by 16 extreme points. Due to the fact ...

In this paper an application of constraint logic programming (CLP) to the resolution of nesting problems is presented. Nesting problems are a special case of the cutting and packing problems, in which the pieces generally have non-convex shapes. Due to their combinatorial optimization nature, nesting problems have traditionally been tackled by heuristics and in the recent past by meta-heuristic...

A technique to triangulate planar convex polygons presenting T{vertices is described. Simple strip or fan tessellation of a polygon with T{vertices can result in zero{area triangles and compromise the rendering process. Our technique splits such a polygon into one triangle strip and, at most, one triangle fan. The utility is particularly useful in multiresolution or adaptive representation of p...

A random polygon is the convex hull of uniformly distributed random points in a convex body K ⊂ R. General upper bounds are established for the variance of the area of a random polygon and also for the variance of its number of vertices. The upper bounds have the same order of magnitude as the known lower bounds on variance for these functionals. The results imply a strong law of large numbers ...

After McCallum and Avis [4] showed that the convex hull of a simple polygon P with n vertices can be constructed in O(n) time, several authors [1,2,3] devised simplified algorithms for this problem. Graham and Yao [2] presented a particularly simple and elegant one. After finding two points of the convex hull, their algorithm generated all other hull vertices using only one stack for intermedia...

We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): “Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?” We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem re...

The associahedron is a convex polytope whose vertices correspond to triangulations of a convex polygon. We define two signed or hyperoctahedral analogues of the associahedron, one of which is shown to be a simple convex polytope, and the other a regular CW-sphere.