Polygonal Holes Research Articles

Deterministic Rendezvous of Asynchronous Bounded-Memory Agents in Polygonal Terrains

Two mobile agents, modeled as points starting at different locations of an unknown terrain, have to meet. The terrain is a polygon with polygonal holes. We consider two versions of this rendezvous problem: exact RV, when the points representing the agents have to coincide at some time, and ?-RV, when these points have to get at distance less than ? in the terrain. In any terrain, each agent chooses its trajectory, but the movements of the agent on this trajectory are controlled by an adversary that may, e.g., speed up or slow down the agent. Agents have bounded memory: their computational power is that of finite state machines. Our aim is to compare the feasibility of exact and of ?-RV when agents are anonymous vs. when they are labeled. We show classes of polygonal terrains which distinguish all the studied scenarios from the point of view of feasibility of rendezvous. The features which influence the feasibility of rendezvous include symmetries present in the terrains, boundedness of their diameter, and the number of vertices of polygons in the terrains.

Hole filling on surfaces by discrete variational splines

This paper concerns about a method to fill polygonal holes in a given surface by using smoothing variational splines. We develop two different approaches of this technique: discontinuous filling and regular filling. In the discontinuous case, we fill the holes with spline functions in a finite element space that minimizes an energy functional. Such fillings are chosen to be smooth and as close as possible to the original surface in the neighborhoods of the holes. In this approach the global reconstructed surface will not be continuous. On the contrary, in the regular case, we do not only fill the holes, but we also replace the known surface with another very similar in such a way that the global reconstructed surface will have the desired regularity. We give convergence results and we illustrate the developed theory with several graphical and numerical examples.

A novel hybrid finite element analysis of two polygonal holes in an infinite elastic plate

2D plates containing interacting double diamond holes, double square holes and double rectangular holes are considered. A novel finite element method together with the Hellinger–Reissner variational principle is used to formulate a super n-sided polygonal hybrid element that contains part of an inner conner of a polygonal hole. The generalized stress intensity factors (GSIFs) at the angular corner tip are systematically calculated for various geometry and spacing. The numerical results show that the present method yields satisfactory results with fewer elements. Compared with the conventional finite element methods, the present method is more suitable for dealing with complicated problems of stress singularity in elasticity.

On the stress concentration around a hole in an infinite plate subject to a uniform load at infinity

In this paper stress concentration around a hole in an infinite plate that is subjected to a uniform load at infinity is considered. The stress is calculated by using a modified Muskhelishvili complex variable method. The method is illustrated by several examples of stress distribution around polygonal holes of a complex geometry utilizing the Schwartz–Chistoffel mapping function.

Searching for mobile intruders in circular corridors by two 1-searchers

We consider the problem of searching for a mobile intruder in a circular corridor by two mobile searchers, who hold one flashlight. A circular corridor is a polygon with one polygonal hole such that its outer and inner boundaries are mutually weakly visible. Both 1-searchers always direct their flashlights at the inner boundary. The objective is to decide whether there exists a search schedule for two 1-searchers to detect the intruder, no matter how fast he moves, and if so, generate a search schedule. We give a characterization of the circular corridors, which are searchable by two 1-searchers. Based on our characterization, an O ( n log n ) time algorithm is then presented to determine the searchability of a circular corridor, where n denotes the total number of vertices of the outer and inner boundaries. Moreover, a search schedule can be reported in time linear in its size, if it exists.

Boron fullerenes with 32-56 atoms: irregular cage configurations and electronic properties

Unbiased search by first-principles simulated annealing revealed irregular cage configurations for medium-sized B n clusters (namely, boron fullerene) with n=32-56, which are more stable than the previously proposed symmetric cages. The stability of these irregular cages can be understood by the three-center bonds as well as the polygonal holes on the cage surface. The delocalized distribution of molecular orbitals as well as the negative nucleus-independent chemical shifts (NICS) values for each boron cage indicates strong aromaticity.

Radar Backscatter from Conducting Polyhedral Spheres

Electromagnetic backscatter characteristics of conducting polyhedral spheres, constructed of 12 pentagons and a large number of hexagons, were investigated. The polyhedral structures were classified by their number of vertices and by the width of the conducting edges in their frames. Polyhedral spheres with 60, 80, 180, 240, 320, 500, 540, 960, and 1500 vertices were evaluated for their ability to scatter electromagnetic waves back to a source. The edge widths were varied between the limits of (1) nearly zero, to produce a wire-frame sphere; and (2) full surface closure of the polygon holes in the sphere, to form solid conducting structures. Using Method-of-Moments simulations with the WIPL-D code, the monostatic scattering properties of these spheres were studied over a wide range of structural and frequency parameters. The spherical wire frames had up to a 10 dB larger radar cross section than the completely solid spheres. This result provides models for new light-weight radar cross section targets used as satellite-calibration spheres with enhanced radar cross section for easier detection, low mass for easier launch, and low drag cross section for long orbit life. With selected edge widths, the monostatic radar cross section becomes vanishingly small at specific anti-backscatter frequencies, for conducting spherical shells with a regular pattern of polygon holes. These shells can be placed around solid spheres to nearly eliminate radar reflection from the composite object at specific frequencies. The polyhedral conducting-mesh (PCM) spheres with vanishing radar cross section at a desired frequency have applications for isotropic scattering sources that reflect no energy back to the electromagnetic source. the anti-backscatter polyhedral conducting-mesh spheres may provide insight for future stealth applications with spherical targets.

Optimal arrangement for through-holes in a three-dimensional thin-film battery

In 3D thin-film batteries (3D-TFBs) and in trenched capacitors, the surface area is increased by forming holes in a substrate such as a silicon wafer. In this paper we define area gain (AG) as a ratio of the total new surface area (including the surface areas of all the holes) to the original surface area of the substrate (“footprint”). We analyze the AG for different configurations of convex polygonal holes within the substrate. In most cases, the AG can be computed based on the P/ S ratio, where P is the sum of all hole perimeters and S is the surface area of one side of the substrate. Assuming that the diameter of each hole is not less than D and that the distance between any two holes is not less than s, the P/ S ratio never exceeds 4/( D + 2 s). Tesselation of the surface into a regular grid of polygonal cells of diameter D + s with regular polygonal holes of the same form (of diameter D) results in P/ S = 4 D/( D + s) 2. We propose two alternative tessellations (one with regular square holes and another with regular triangular holes) which have P/ S ratios slightly better than 4 D/( D + s) 2 and which satisfy the assumptions of the hole diameter and the wall width. It is found that a smoothed triangular tessellation provides the largest AG.

Mechanical Circle-Squaring

Click to increase image sizeClick to decrease image size This article is part of the following collections: Mathematics of Pi Additional informationNotes on contributorsBarry CoxBarry Cox (barryc@uow.edu.au; www.uow.edu.au/~barryc) is a lecturer in applied mathematics at the University of Wollongong (Australia) and holds a postdoctoral fellowship from the Australian Research Council. The primary focus of his research is modelling nanoscaled devices. He became interested in the problem of drilling polygonal holes through the Macalester Problem of the Week, to which he has subscribed for many years. Barry's personal interests include recreational mathematics, public speaking, tennis, and chess.Stan WagonStan Wagon (wagon@macalester.edu; www.stanwagon.com) is professor of mathematics at Macalester College in St. Paul, Minnesota. He has written extensively about mathematics (this is his 100th paper), and enjoys seeing how the power of Mathematica can be used to bring abstract constructions to life. The relation between squares and circles has interested him ever since he heard that a square wheel will roll smoothly on a road of linked catenaries, and built a 25-foot roadbed on which to ride the full-sized, square-wheel bicycle he built. Other interests include competitive snow sculpture, wilderness skiing, mountain climbing, and nordic ski racing. He has completed 100-mile races both on foot and on skis.

Filling polygonal holes with minimal energy surfaces on Powell–Sabin type triangulations

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function f in a part of a polygonal domain D ⊂ R 2 . We obtain the final reconstructed surface over the whole domain D . We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell–Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of f and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace f outside the hole by a smoothing bivariate spline s f , and then we fill the hole also with a Powell–Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.

Conduction Shape Factor Models for Hollow Cylinders with Nonuniform Gap Spacing

An analytical model is developed for the conduction shape factor for the two-dimensional annular region formed between concentricisothermal cylinders, where the inner and outer boundarieshavesimilar or different shapes. The model is based on the equivalent circular annulus with a modified independent parameter to accountforlocal variationsin the gap thickness. Validation of the model is performedusing existingnumerical data and analytical expressions from the literature for a variety of different geometries, including polygonal and rectangular cylinders with a circular hole and the circular cylinder with a polygonal hole. The average RMS difference between the available data and the model of all casesis less than 5 %. Nomenclature

Approximate convex decomposition of polygons

We propose a strategy to decompose a polygon, containing zero or more holes, into ''approximately convex'' pieces. For many applications, the approximately convex components of this decomposition p...

Approximate convex decomposition of polygons

We propose a strategy to decompose a polygon, containing zero or more holes, into “ approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller and can be computed more efficiently. Moreover, our approximate convex decomposition (ACD) provides a mechanism to focus on key structural features and ignore less significant artifacts such as wrinkles and surface texture. We propose a simple algorithm that computes an ACD of a polygon by iteratively removing ( resolving) the most significant non-convex feature ( notch). As a by product, it produces an elegant hierarchical representation that provides a series of ‘increasingly convex’ decompositions. A user specified tolerance determines the degree of concavity that will be allowed in the lowest level of the hierarchy. Our algorithm computes an ACD of a simple polygon with n vertices and r notches in O ( n r ) time. In contrast, exact convex decomposition is NP-hard or, if the polygon has no holes, takes O ( n r 2 ) time. Models and movies can be found on our web-pages at: http://parasol.tamu.edu/groups/amatogroup/.

A piecewise hole filling algorithm in reverse engineering

While scanning a complex part in reverse engineering, it is not possible to acquire all part of the scanned surface. Data are inevitably missing due to the complexity of the scanned part or imperfect scanning process. Missing scanned data cause holes in the created triangular mesh, so that a hole-free mesh model is prerequisite for fitting watertight surfaces. Although a number of hole filling algorithms have been investigated, they enable to fill holes only on the smooth regions of a model. They are not always robust in the regions of high curvature. This paper proposes a novel methodology that can automatically fill complex polygonal holes with a piecewise manner. It incrementally splits a complex hole into several simple holes with respect to the 3D shape of the hole boundary, and then it consecutively fills each divided simple hole with planar triangulation method until the entire complex hole is firmly closed. Finally smoothing and subdivision techniques are applied for enhancing the hole triangles. The newly created vertices and triangles are added to their respective lists and the topology information is updated. The method has proven to be robust and effective from the result of test with a variety of complex holes. Examples are given and discussed to validate the methodology.

Polygon trapezoidation by sets of open trapezoids

A new efficient algorithm is described for the simple trapezoidation of polygons based on a sweep-line paradigm. As the sweep-line glides over the plane, a set of so-called open trapezoids is generated and maintained. It is shown that a boundary case (more polygon vertices are located on the sweep-line) can be solved safely and does not slow down the algorithm. If desired, the polygon holes can be trapezoidated simultaneously. This proposed algorithm when compared with the fastest known algorithm developed by Seidel resulted in more efficiency for different classes of polygons.

On Approximate Solutions for Two-Dimensional Thermoelastic Problems with a Nearly Circular Hole

ABSTRACTThe general approximate solutions for the two-dimensional thermoelastic problems with a nearly circular hole are provided in this study. Based on Stroh formalism and the method of conformal mapping, the boundary perturbation analysis is applied to solve the problems of a hole with arbitrary shape. The radius of the hole considered here is represented as a sum of a reference constant and a perturbation magnitude that is expanded into a Fourier series. In order to illustrate the applicability and efficiency of the present approach, special examples associated with polygonal hole problems are solved explicitly and discussed in detail. Since the general solutions have not been found in the literature, comparison is made with some special cases for which the analytical solutions exist, which shows that our proposed method is effective and general.

General solutions for thermopiezoelectrics with various holes under thermal loading

The thermoelectroelastic problems for a hole of various shapes embedded in an infinite matrix are considered in this paper. Using Stroh’s formalism and the technique of conformal mapping, a unified solution is obtained in closed-form for an infinite thermopiezoelectric plate with various holes induced by thermal loads. The loads may be uniform remote heat flow, point heat source and temperature discontinuity. Further a detailed discussion on the critical points for the mapping function of a piezoelectric plate with polygonal holes is presented and the study shows that the transformation is nonsingle-valued. A simple approach is given to treat such a situation. As an application of the proposed solutions, a system of singular integral equations for the unknown temperature discontinuity and the dislocation of elastic displacement and electric potential defined on crack faces is developed and solved numerically. Numerical results are presented to illustrate the application of the proposed formulation.

The biaxial tension of an elastoplastic space with a prismatic inclusion

The problem on the elastoplastic state of a thick isotropic plate (the case of plane deformation) is solved. A larger prismatic inclusion is made a close fit in a polygonal hole in the plate. The plate is stretched at infinity by constant mutually perpendicular forces. The problem is solved by the small-parameter method and by the theory of ideal plasticity. The axisymmetric state of the plane with a circular hole stiffened by a round ring with a constant force applied to its inner contour is considered as a zero approximation. Some specific shapes of the hole and reinforcing elastic rigid ring are considered.

Filling polygonal holes using [formula omitted] cubic triangular spline patches

We use the method of energy minimization to fill polygonal holes by C 1 cubic triangular spline patches. We implement the method in MATLAB. Several numerical examples are shown.

MINIMUM NUMBER OF PIECES IN A CONVEX PARTITION OF A POLYGONAL DOMAIN

Let [Formula: see text] be a given nonempty family of directions in the plane. For a multiply connected polygonal domain P with polygonal holes, possibly degenerate, we determine the minimum number of convex polygons into which P is partitioned by linear cuts in the directions from [Formula: see text].