The iterated di erence of polyhedra V P n P n Pk has been proposed independently in and as a su cient condition for V to be exactly computable by a two layered neural network An algorithm checking whether V IR is an iterated di erence of polyhedra is proposed in However this algorithm is not practically usable because it has a high computational complexity and it was only conjectured to stop wi...

We generalize Cauchy’s celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space E to the context of circle polyhedra in the 2-sphere S. We prove that any two convex and bounded non-unitary c-polyhedra with Möbiuscongruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings as wel...

Every convex polyhedron in the Euclidean space Rd admits both H-representation and V-representation. When working with convex polyhedra, in particular largescale ones in high dimensions, it is useful to have a canonical representation that is minimal and unique up to some elementary operations. Such a representation allows one to compare two H-polyhedra or two V-polyhedra efficiently. In this p...

Linear invariants are essential in many optimization and veri cation tasks. The domain of convex polyhedra (sets of linear inequalities) has the potential to infer all linear relationships. Yet, it is rarely applied to larger problems due to the join operation whose most precise result is given by the convex hull of two polyhedra which, in turn, may be of exponential size. Recently, Sankaranara...

A stabbing line for a set of convex polyhedra is extremal if it passes through four edges and is tangent to the polyhedra containing those edges. In this paper we present three constructions of convex polyhedra with many extremal lines. The rst construction shows (n 2) extremal stabbing lines constrained to meet two skew lines. The second shows (n 4) extremal lines which are tangent to two poly...

Triangulated polyhedra are simpler to process than arbitrary polyhedra for many graphics operations. Algorithms that compute the boundary representation of a constructive solid geometry (csg) model, however, may perform poorly if the model involves triangulated polyhedral primitives. A new csg algorithm speciically tailored to triangulated primitives is presented. The key features of this algor...

A chiral polyhedron has a geometric symmetry group with two orbits on the flags, such that adjacent flags are in distinct orbits. Part I of the paper described the discrete chiral polyhedra in ordinary Euclidean space E3 with finite skew faces and finite skew vertex-figures; they occur in infinite families and are of types {4, 6}, {6, 4} and {6, 6}. Part II completes the enumeration of all disc...

An isohedron is a 3-dimensional polyhedron all faces of which are equivalent under symmetries of the polyhedron. Many well known polyhedra are isohedra; among them are the Platonic solids, the polars of Archimedean polyhedra, and a variety of polyhedra important in crystallography. Less well known are isohedra with nonconvex faces. We establish that such polyhedra must be starshaped and hence o...

It will be proved that the simple 3-polyhedra with f + 1 facets are obtained from all simple 3-polyhedra with f facets by 2-face splits. The numbers of combinatorial types of simple 3-polyhedra with up to 15 facets are stated with respect to their automorphism group order.

The reverse split rank of an integral polyhedron P is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is P. Already in R3 there exist polyhedra with infinite reverse split rank. We give a geometric characterization of the integral polyhedra in Rn with infinite reverse split rank.