Three-dimensional (3D) displays having regular-polyhedron structures are proposed and their imaging characteristics are analyzed. Four types of conceptual regular-polyhedron 3D displays, i.e., hexahedron, octahedron, dodecahedron, and icosahedrons, are considered. In principle, regular-polyhedron 3D display can present omnidirectional full parallax 3D images. Design conditions of structural fac...

Note: c is the objective function, A a matrix and b, c the column vectors. Simplex method finds a solution to such a problem. • The set of feasible points of our LP is a polyhedron P := {x|Ax ≤ b} If P is non-empty then we have a convex polyhedron, where vertices are defined by d constraints, there where they are tight: aix = bi and ai are the rows of matrix A. Recall that: • A polyhedron is a ...

A well-studied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with co...

We report a small advance on a question raised by Robertson, Schweitzer, andWagon in [RSW02]. They constructed a genus-13 polyhedron built from bricks without corners, and asked whether every genus-0 such polyhedron must have a corner. A brick is a parallelopiped, and a corner is a brick of degree three or less in the brick graph. We describe a genus-3 polyhedron built from bricks with no corne...

It is shown how to transform the General Routing Problem (GRP) into a variant of the Graphical Travelling Salesman Problem (GTSP). This transformation yields a projective characterisation of the GRP polyhedron. Using this characterisation, it is shown how to convert facets of the GTSP polyhedron into valid inequalities for the GRP polyhedron. The resulting classes of inequalities subsume severa...

A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obtained. A characterization of spherical polyhedron surfaces in terms of the tri...

where V , E, and F are, respectively, the number of vertices, edges, and faces of p. The formula was first stated in print by Euler in 1758 [11]. The proof given here is based on Poincaré’s linear algebraic proof, stated in [17] (with a corrected proof in [18]), as adapted by Imre Lakatos in the latter’s Proofs and Refutations [15]. As is well known, Euler’s formula is not true for all polyhedr...

Degeneracy of a polyhedron is a source of difficulties for both theory and computation in mathematical programming. This phenomenon may be avoided by considering slight perturbations of degenerate polyhedra, that is, by approximating the degenerate polyhedron by a nondegenerate one. This method was applied to a well-known proof of finiteness of the simplex algorithm [5, 7]. Another application ...

A well-studied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that \open" polyhedra with co...

Split cuts form a well-known class of valid inequalities for mixed-integer programming problems (MIP). Cook et al. (1990) showed that the split closure of a rational polyhedron P is again a polyhedron. In this paper, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We also show how this result can be used to prove that some generaliz...