Related papers: Straight Skeletons of Three-Dimensional Polyhedra

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In…

This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical,…

This paper investigated the problem of embedding a simple Hamiltonian Cycle with n vertices on n points inside a simple polygon. This problem seeks to embed a straight-line cycle (without bends), which does not intersect either itself or…

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we…

Object skeletons are useful for object representation and object detection. They are complementary to the object contour, and provide extra information, such as how object scale (thickness) varies among object parts. But object skeleton…

A novel self-assembly strategy for polypeptide nanostructure design was presented in [Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments, Nature Chemical Biology 9 (2013) 362--366]. The first mathematical…

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In…

In this paper, we present a novel algorithm that integrates deep learning with the polycube method (DL-Polycube) to generate high-quality hexahedral (hex) meshes, which are then used to construct volumetric splines for isogeometric…

Standard sweep algorithms require an order of discrete points in Euclidean space, and rely on the property that, at a given point, all points in the halfspace below come earlier in this order. We are motivated by the problem of…

M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for…

It is conjectured that all decomposable (i.e. interior can be triangulated without adding new vertices) polyhedra with vertices in convex position are infinitesimally rigid and only recently has it been shown that this is indeed true under…

Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear…

A convex polyhedron $P$ is $k$-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of $P$ are $k$-gon for some fixed value of $k$. Since 1968, it is an open problem to construct all…

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time…

We study the simplex method over polyhedra satisfying certain "discrete curvature" lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint…

The study of bodies of constant width is a classical subject in convex geometry, with the 3-dimensional Meissner bodies being canonical examples. This paper presents a novel geometric construction of a body of constant width in $\mathbb…

We construct compact polyhedra with $m$-gonal faces whose links are generalized 3-gons. It gives examples of cocompact hyperbolic bildings of type $P(m,3)$. For $m=3$ we get compact spaces covered by Euclidean buildings of type $A_2$.

The geodesic complexity of a length space $X$ quantifies the required number of case distinctions to continuously choose a shortest path connecting any given start and end point. We prove a local lower bound for the geodesic complexity of…

The surface reconstruction problem from sets of planar parallel slices representing cross sections through 3D objects is presented. The final result of surface reconstruction is always based on the correct estimation of the structure of the…

A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as…