Simple Polygon Research Articles

Modeling crack discontinuities without element‐partitioning in the extended finite element method

SummaryIn this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.

On the Beer Index of Convexity and Its Variants

Let S be a subset of mathbb {R}^d with finite positive Lebesgue measure. The Beer index of convexity{text {b}}(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio{text {c}}(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of convexity. We show that every set Ssubseteq mathbb {R}^2 with simply connected components satisfies {text {b}}(S)leqslant alpha {text {c}}(S) for an absolute constant alpha , provided {text {b}}(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of {text {b}}(S). For 1leqslant kleqslant d, the k-index of convexity{text {b}}_k(S) of a set Ssubseteq mathbb {R}^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every dgeqslant 2 there is a constant beta (d)>0 such that every set Ssubseteq mathbb {R}^d satisfies {text {b}}_d(S)leqslant beta {text {c}}(S), provided {text {b}}_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma (d)>0 such that for every varepsilon in (0,1) there is a set Ssubseteq mathbb {R}^d of Lebesgue measure 1 satisfying {text {c}}(S)leqslant varepsilon and {text {b}}_d(S)geqslant gamma frac{varepsilon }{log _2{1/varepsilon }}geqslant gamma frac{{text {c}}(S)}{log _2{1/{text {c}}(S)}}.

Approximation algorithms for guarding holey polygons

Guarding edges of polygons is a version of art gallery problem.The goal is finding the minimum number of guards to cover the edges of a polygon. This problem is NP-hard, and to our knowledge there are approximation algorithms just for simple polygons. In this paper we present two approximation algorithms for guarding polygons with holes. Keywords: guarding, approximation algorithm, vertex guard, edge guard

An efficient incremental algorithm for generating the characteristic shape of a dynamic set of points in the plane

ABSTRACTSeveral algorithms have been proposed to generate a polygonal ‘footprint’ to characterize the shape of a set of points in the plane. One widely used type of footprint is the χ-shape. Based on the Delaunay triangulation (DT), χ-shapes guaranteed to be simple (Jordan) polygons. This paper presents for the first time an incremental χ-shape algorithm, capable of processing point data streams. Our incremental χ-shape algorithm allows both insertion and deletion operations, and can handle streaming individual points and multiple point sets. The experimental results demonstrated that the incremental algorithm is significantly more efficient than the existing, batch χ-shape algorithm for processing a wide variety of point data streams.

Polynomially solvable cases of the bipartite traveling salesman problem

Given two sets, R and B, consisting of n cities each, in the bipartite traveling salesman problem one looks for the shortest way of visiting alternately the cities of R and B, returning to the city of origin. This problem is known to be NP-hard for arbitrary sets R and B. In this paper we provide an O(n6) algorithm to solve the bipartite traveling salesman problem if the quadrangle property holds. In particular, this algorithm can be applied to solve in O(n6) time the bipartite traveling salesman problem in the following cases: S=R∪B is a convex point set in the plane, S=R∪B is the set of vertices of a simple polygon and V=R∪B is the set of vertices of a circular graph. For this last case, we also describe another algorithm which runs in O(n2) time.

Polygon guarding with orientation

The art gallery problem is a classical sensor placement problem that asks for the minimum number of guards required to see every point in an environment. The standard formulation does not take into account self-occlusions caused by a person or an object within the environment. Obtaining good views of an object from all orientations despite self-occlusions is an important requirement for surveillance and visual inspection applications. We study the art gallery problem under a constraint, termed △-guarding, that ensures that all sides of any convex object are always visible in spite of self-occlusion.Our contributions in this paper are three-fold: We first prove that Ω(n) guards are always necessary for △-guarding the interior of a simple polygon having n vertices. Second, we present a O(log⁡copt) factor approximation algorithm for △-guarding polygons with or without holes, when the guards are restricted to vertices of the polygon. Here, copt is the optimal number of guards. Third, we study the problem of △-guarding a set of line segments connecting points on the boundary of the polygon. This is motivated by applications where an object or person of interest can only move along certain paths in the polygon. We present a constant factor approximation algorithm for this problem – one of the few such results for art gallery problems.

A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon

Let P be a closed simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack et al. (Discrete Comput Geom 4(1): 611–626, 1989) showed an $$O(n\log n)$$ O ( n log n ) -time algorithm that computes the geodesic center of P. Since then, a longstanding question has been whether this running time can be improved. In this paper we affirmatively answer this question and present a deterministic linear-time algorithm to solve this problem.

Контроль правильности установления границ пространственных объектов площадного типа

Abstract: Conceptual approaches to the verification of spatial object delineation forming the territory coverage are considered. The necessity of integrated spatial object geometry control based on requirements for object geometry correctness is proved. Verification control includes: polygon self-crossing, metric proximity of polygons, polygon crossings and spacing, tame embedding of simple polygons inside of composed ones. Techniques for revealing wrong situations in object geometry and their elimination are described. Both techniques are based on the building of the effective object topological structure including elements of arcs, nodes, polygons, and relations between them. Examples of pseudo-situations in object geometry before and after their elimination using a special-purpose graphics editor. Software used for verification purposes is outlined.

A Simple Polygon Search Algorithm

This paper considers simple polygon search problem. How many searchers find a mobile intruder that is arbitrarily faster than the searcher within polygon art gallery? This paper uses the visibility graph that is connected with edges for mutually visible vertices. Given visibility graph, we select vertex  that is conjunction  in   for   ≤ . We decide 1-searchable if  ≤ ≤  and 2-searchable if ≥  . We also present searcher's shortest path. This algorithm is verified by varies 1 or 2-searchable polygons. ▸

CudaChain: an alternative algorithm for finding 2D convex hulls on the GPU.

This paper presents an alternative GPU-accelerated convex hull algorithm and a novel Sorting-basedPreprocessingApproach (SPA) for planar point sets. The proposed convex hull algorithm termed as CudaChain consists of two stages: (1) two rounds of preprocessing performed on the GPU and (2) the finalization of calculating the expected convex hull on the CPU. Those interior points locating inside a quadrilateral formed by four extreme points are first discarded, and then the remaining points are distributed into several (typically four) sub regions. For each subset of points, they are first sorted in parallel; then the second round of discarding is performed using SPA; and finally a simple chain is formed for the current remaining points. A simple polygon can be easily generated by directly connecting all the chains in sub regions. The expected convex hull of the input points can be finally obtained by calculating the convex hull of the simple polygon. The library Thrust is utilized to realize the parallel sorting, reduction, and partitioning for better efficiency and simplicity. Experimental results show that: (1) SPA can very effectively detect and discard the interior points; and (2) CudaChain achieves 5×–6× speedups over the famous Qhull implementation for 20M points.Electronic supplementary materialThe online version of this article (doi:10.1186/s40064-016-2284-4) contains supplementary material, which is available to authorized users.

Plane geodesic spanning trees, Hamiltonian cycles, and perfect matchings in a simple polygon

Let S be a finite set of points in the interior of a simple polygon P. A geodesic graph, GP(S,E), is a graph with vertex set S and edge set E such that each edge (a,b)∈E is the shortest geodesic path between a and b inside P. GP is said to be plane if the edges in E do not cross. If the points in S are colored, then GP is said to be properly colored provided that, for each edge (a,b)∈E, a and b have different colors. In this paper we consider the problem of computing (properly colored) plane geodesic perfect matchings, Hamiltonian cycles, and spanning trees of maximum degree three.

Approximating the Maximum Overlap of Polygons under Translation

Let $$P$$P and $$Q$$Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present a $$(1-\varepsilon )$$(1-ź)-approximation algorithm, for finding the translation of $$Q$$Q, which maximizes its area of overlap with $$P$$P. Our algorithm runs in $$O\left( {c n}\right) $$Ocn time, where c is a constant that depends only on k and $$\varepsilon $$ź. This suggests that for polygons that are close to being convex, the problem can be solved (approximately), in near linear time.

An Empirical Study on Randomized Optimal Area Polygonization of Planar Point Sets

While random polygon generation from a set of planar points has been widely investigated in the literature, very few works address the construction of a simple polygon with minimum area (MINAP) or maximum area (MAXAP) from a set of fixed planar points. Currently, no deterministic algorithms are available to compute MINAP/MAXAP, as the problems have been shown to be NP-complete. In this article, we present a greedy heuristic for computing the approximate MINAP of any given planar point set using the technique of randomized incremental construction. For a given point set of n points, the proposed algorithm takes O ( n 2 log n ) time and O ( n ) space. It is rather simplistic in nature, hence very easy for implementation and maintenance. We report on various experimental studies on the behavior of a randomized heuristic on different point set instances. Test data have been taken from the SPAETH cluster data base and TSPLIB library. Experimental results indicate that the proposed algorithm outperforms its counterparts for generating a tighter upper bound on the optimal minimum area polygon for large-sized point sets.

Area-Preserving Simplification and Schematization of Polygonal Subdivisions

In this article, we study automated simplification and schematization of territorial outlines. We present a quadratic-time simplification algorithm based on an operation called edge-move . We prove that the number of edges of any nonconvex simple polygon can be reduced with this operation. Moreover, edge-moves preserve area and topology and do not introduce new orientations. The latter property in particular makes the algorithm highly suitable for schematization in which all resulting lines are required to be parallel to one of a given set of lines (orientations). To obtain such a result, we need only to preprocess the input to use only lines that are parallel to one of the given set. We present an algorithm to enforce such orientation restrictions, again without changing area or topology. Experiments show that our algorithms obtain results of high visual quality.

CONVEX TRAVERSAL IN POLYGON WITH SEGMENT STABBING

Given a set S of disjoint simple polygons on a bounded 2D-Plane, we studied the problem of finding a set of polygon P, such that stabbing S by segment stabbing region in optimal way. We present a novel approach for solving the convex traversing method in PTIME. Convex traversing is a mechanism of layering continues convex hull extracted from points P. Whenever adding a point in the convex traversing plane, it is necessary to compute the whole convexation again, which contradict the better runtime. We introduce the new notion of dynamic algorithm to reduce the runtime of this problem into LOG-SPACE bounded. Hence output produced at every dynamic modification results only a LOG-SPACE bounded reconstruction.

Nonobtuse Triangulations of PSLGs

We show that any planar straight line graph (PSLG) with $n$ vertices has a conforming triangulation by $O(n^{2.5})$ nonobtuse triangles (all angles $\leq 90^\circ$), answering the question of whether any polynomial bound exists. A nonobtuse triangulation is Delaunay, so this result also improves a previous $O(n^3)$ bound of Eldesbrunner and Tan for conforming Delaunay triangulations of PSLGs. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only $O(n^2)$ triangles are needed, improving an $O(n^4)$ bound of Bern and Eppstein. We also show that for any $\epsilon >0$, every PSLG has a conforming triangulation with $O(n^2/\epsilon^2)$ elements and with all angles bounded above by $90^\circ + \epsilon$. This improves a result of S. Mitchell when $\epsilon = 3 \pi /8 = 67.5^\circ $ and Tan when $\epsilon = 7\pi/30 =42^\circ$.

두 가시성 다각형의 교집합과 합집합을 구하는 효율적인 RMESH 알고리즘

We can consider the following problems for two given points p and q in a simple polygon P. (1) Compute the set of points of P which are visible from both p and q. (2) Compute the set of points of P which are visible from either p or q. They are corresponding to the problems which are to compute the intersection and the union of two visibility polygons. In this paper, we consider algorithms for solving these problems on a reconfigurable mesh(in short, RMESH). The algorithm in [1] can compute the intersection of two general polygons in constant time on an RMESH with size O(), where n is the total number of vertices of two polygons. In this paper, we construct the planar subdivision graph in constant time on an RMESH with size O() using the properties of the visibility polygon for preprocessing. Then we present O() time algorithms for computing the union as well as the intersection of two visibility polygons, which improve the processor-time product from O() to O().

Subdividing barycentric coordinates

Barycentric coordinates are commonly used to represent a point inside a polygon as an affine combination of the polygon's vertices and to interpolate data given at these vertices. While unique for triangles, various generalizations to arbitrary simple polygons exist, each satisfying a different set of properties. Some of these generalized barycentric coordinates do not have a closed form and can only be approximated by piecewise linear functions. In this paper we show that subdivision can be used to refine these piecewise linear functions without losing the key barycentric properties. For a wide range of subdivision schemes, this generates a sequence of piecewise linear coordinates which converges to non-negative and C1 continuous coordinates in the limit. The power of the described approach comes from the possibility of evaluating the C1 limit coordinates and their derivatives directly. We support our theoretical results with several examples, where we use Loop or Catmull–Clark subdivision to generate C1 coordinates, which inherit the favourable shape properties of harmonic coordinates or the small support of local barycentric coordinates.

Shortest path in a polygon using sublinear space

We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon $P$ with $n$ vertices in a read only memory, and additional working memory of size $m$, the new algorithm computes the shortest path (in $P$) in $O( n^2 / m )$ expected time, assuming $m = O(n/\log^2 n)$. This requires several new tools, which we believe to be of independent interest. Specifically, we show that violator space problems, an abstraction of low dimensional linear-programming (and LP-type problems), can be solved using constant space and expected linear time, by modifying Seidel's linear programming algorithm and using pseudo-random sequences.

An Approach for Indoor Path Computation among Obstacles that Considers User Dimension

People often transport objects within indoor environments, who need enough space for the motion. In such cases, the accessibility of indoor spaces relies on the dimensions, which includes a person and her/his operated objects. This paper proposes a new approach to avoid obstacles and compute indoor paths with respect to the user dimension. The approach excludes inaccessible spaces for a user in five steps: (1) compute the minimum distance between obstacles and find the inaccessible gaps; (2) group obstacles according to the inaccessible gaps; (3) identify groups of obstacles that influence the path between two locations; (4) compute boundaries for the selected groups; and (5) build a network in the accessible area around the obstacles in the room. Compared to the Minkowski sum method for outlining inaccessible spaces, the proposed approach generates simpler polygons for groups of obstacles that do not contain inner rings. The creation of a navigation network becomes easier based on these simple polygons. By using this approach, we can create user- and task-specific networks in advance. Alternatively, the accessible path can be generated on the fly before the user enters a room.