AD-polyhedron is a polyhedron P such that if x, y are in P then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, Dpolyhedra are a unifying genera...

Convex polyhedra, commonly employed for the analysis and verification of both hardware and software, may be defined either by a finite set of linear inequality constraints or by finite sets of generating points and rays of the polyhedron. Although most implementations of the polyhedral operations assume that the polyhedra are topologically closed (i.e., all the constraints defining them are non...

Whiteley, W., Weavings, sections and projections of spherical polyhedra, Discrete Applied Mathematics 32 (1991) 275-294. In this paper we give simultaneous answers to three questions: (a) When does a plane picture of lines, weaving over and under in the plane, lift and separate into a configuration of disjoint lines in 3-space? (b) When is a configuration of lines in the plane the cross-section...

We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have a disconnected interior: the triangles are connected at vertices, but not necessarily joined along edges.

We construct a polyhedron that is topologically convex (i.e., has the graph of a convex polyhedron) yet has no vertex unfolding: no matter how we cut along the edges and keep faces attached at vertices to form a connected (hinged) surface, the surface necessarily unfolds with overlap.

We prove that any polyhedron of genus zero or genus one built out of rectangular faces must be an orthogonal polyhedron, but that there are nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This leads to a resolution of a question posed by Biedl, Lubiw, and Sun [BLS99].

Recent progress is described on the unsolved problem of unfolding the surface of an orthogonal polyhedron to a single non-overlapping planar piece by cutting edges of the polyhedron. Although this is in general not possible, partitioning the faces into the natural vertex-grid may render it always achievable. Advances that have been made on various weakenings of this central problem are summariz...

The problem of determining whether a polyhedron has a constrained Delaunay tetrahedralization is NP-complete. However, if no five vertices of the polyhedron lie on a common sphere, the problem has a polynomial-time solution. Constrained Delaunay tetrahedralization has the unusual status (for a small-dimensional problem) of being NP-hard only for degenerate inputs.

We present a recursive algorithm for finding the minimum norm point in an unbounded, wnvex and pointed polyhedron defined by finite sets of points and rays. The algorithm can start at an arbitrary point of the polyhedron and does not require to solve systems of linear equations.

We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov’s gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n) time for polygons whose gluings are specified by n labels.