Related papers: Unfolding Orthogrids with Constant Refinement
An unzipping of a polyhedron P is a cut-path through its vertices that unfolds P to a non-overlapping shape in the plane. It is an open problem to decide if every convex P has an unzipping. Here we show that there are nearly flat convex…
We prove that, for any two polyhedral manifolds $\mathcal P, \mathcal Q$, there is a polyhedral manifold $\mathcal I$ such that $\mathcal P, \mathcal I$ share a common unfolding and $\mathcal I,\mathcal Q$ share a common unfolding. In other…
If all tiles in a tiling are congruent, the tiling is called monohedral. Tiling by convex polygons is called edge-to-edge if any two convex polygons are either disjoint or share one vertex or one entire edge in common. In this paper, we…
While orthogonal drawings have a long history, smooth orthogonal drawings have been introduced only recently. So far, only planar drawings or drawings with an arbitrary number of crossings per edge have been studied. Recently, a lot of…
The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed…
We introduce a new family of orthogonal polynomials on the disk that has emerged in the context of wave propagation in layered media. Unlike known examples, the polynomials are orthogonal with respect to a measure all of whose even moments…
The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and…
We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged…
We generalize regular subdivisions (polyhedral complexes resulting from the projection of the lower faces of a polyhedron) introducing the class of recursively-regular subdivisions. Informally speaking, a recursively-regular subdivision is…
We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular there exists no combinatorial obstruction to a positive resolution of Durer's unfoldability problem, which answers a question…
In this article we propose an original classification method for unobscured imaging systems unfolded in two dimensions. This classification is based on a study of off-axis properties, and relies on topology and algorithm of real algebraic…
Two tetrahedra are called orthologic if the lines through vertices of one and perpendicular to corresponding faces of the other are intersecting. This is equivalent to the orthogonality of non-corresponding edges. We prove that the…
In this work, we show the geometric properties of a family of polyhedra obtained by folding a regular tetrahedron along regular triangular grids. Each polyhedron is identified by a pair of nonnegative integers. The polyhedron can be cut…
We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again…
The aim of this paper is to study finite orthogonal polynomials on a cone of revolution and its surface. We define two classes of finite orthogonal polynomials on the solid cone and derive their corresponding differential equations and…
The notion of meromorphic convexity is defined and studied on complex manifolds. Using this notion, in analogy with Stein manifolds, a new class of complex manifolds, called {\calligra M }-manifolds, is introduced. This is a class of…
Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction…
We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.
Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and…
We study $C^1$-regular surfaces in $R^3$ that admit tilings by a finite number of rigid motion congruence classes of tiles. We construct examples with various topologies and present a framework for a systematic study, mainly concentrating…