A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel planes, then the band is nested. We prove that a...

In this paper we discuss an algorithm to perform the conversion from the interior to the boundary of d dimensional polyhedra where both the d polyhedron and its d boundary faces are represented as BSP trees This approach allows also to compute the BSP tree of the set intersection between any hyperplane in E and the BSP representation of a d polyhedron If such section hyperplane is the a ne supp...

We present an algorithm to build covering polyhedra for digital 3D objects, by iteratively filling local concavities. The resulting covering polyhedron is convex and is a good approximation of the convex hull of the object. The algorithm uses 3 x 3 x 3 operators and requires a few minutes for a 128 x 128 x 128 image, when implemented on a sequential computer. Once the covering polyhedron has be...

In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron—in particular the edge lengths and dihedral angles of the polyhedron. Cauchy’s rigidity theorem of 1813 states that the dihedral angles are uniquely determined. Finding them is a significant algorithmic problem which we ex...

Can you have a non-convex polyhedron with a bigger volume than a convex polyhedron with the isometric surface? Imagine you start blowing air into a polyhedron with a bendable but non-stretchable surface. What can be said about the limiting shape? These are the key questions I will consider. I will start the talk by discussing the history of the problem, presenting examples, and surveying earlie...

We relate an axiomatic generalization of matroids, called a jump system, to poly-hedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the ...

A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel planes, then the band is nested. We prove that a...

the complex of ce(iii), [ ce (phensc) (h 0) ] (c10 ) , was synthesized using the phensc, a hexadentate ligand,2,9-diformyl-1,lo- phenanthroline bis (semicarbazone), and characterized by x-ray diffraction. the cerium atom has an unusual coordination number of ten involving six donor atoms of the planar ligand in the equatorial plane and four oxygen atoms from four axial water molecules. the comp...

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...

In this paper we study the problem of nding the shortest path between two points lying on the surface of a (possibly nonconvex) polyhedron which is constrained so that a given point on the surface of the polyhedron must be seen from some point in the path. Our algorithm runs in time O(n 2 log n) for the convex polyhedron and O(n 3 log n) for nonconvex case. The method used is based on the subdi...