Related papers: Locked and Unlocked Polygonal Chains in 3D
In this paper, we study the following problem of reconstructing a simple polygon: Given a cyclically ordered vertex sequence of an unknown simple polygon P of n vertices and, for each vertex v of P, the sequence of angles defined by all the…
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…
We investigate by exact optimization method properties of two- and three-dimensional systems of elastic lines in presence of splayed columnar disorder. The ground state of many lines is separable both in 2d and 3d leading to a random walk…
An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron…
The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards…
Geometric embedding of graphs in a point set in the plane is a well known problem. In this paper, the complexity of a variant of this problem, where the point set is bounded by a simple polygon, is considered. Given a point set in the plane…
Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a Schnyder wood that has proven powerful…
The turning distance is a well-studied metric for measuring the similarity between two polygons. This metric is constructed by taking an $L^p$ distance between step functions which track each shape's tangent angle of a path tracing its…
The present paper suggests a new approach for geometric representation of 3D spatial models and provides a new compression algorithm for 3D meshes, which is based on mathematical theory of convex geometry. In our approach we represent a 3D…
In this exploration paper, we design algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. Such a contraction is a kind of shrinking or reducing process. In our algorithms, we need to…
Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expressed as the projection of a much simpler set in higher…
In an exploration paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, we designed algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete…
We are interested in easy geometric transformations which regularize n-polygons in the non-euclidean plane. A transformation is called easy if it can be easily implemented into an algorithm. This article is motivated by preceding work on…
We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard…
Polyominoes are a subset of polygons which can be constructed from integer-length squares fused at their edges. A system of polygons P is interlocked if no subset of the polygons in P can be removed arbitrarily far away from the rest. It is…
We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and…
We present a computational approach for unfolding 3D shapes isometrically into the plane as a single patch without overlapping triangles. This is a hard, sometimes impossible, problem, which existing methods are forced to soften by allowing…
We recall some abstract connectivity concepts, and apply them to special chains in partially ordered sets, called veins, that are defined as order-convex chains that are contained in every maximal chain they meet. Veins enable us to define…
We consider closed chains of circles $C_1,C_2,\ldots,C_n,C_{n+1}=C_1$ such that two neighbouring circles $C_i,C_{i+1}$ intersect or touch each other with $A_i$ being a common point. We formulate conditions such that a polygon with vertices…
Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar…