Showing papers on "Regular polygon published in 2005"

Discrete conformal mappings via circle patterns

Abstract: We introduce a novel method for the construction of discrete conformal mappings from (regions of) embedded meshes to the plane. Our approach is based on circle patterns, i.e., arrangements of circles---one for each face---with prescribed intersection angles. Given these angles the circle radii follow as the unique minimizer of a convex energy. The method has two principal advantages over earlier approaches based on discrete harmonic mappings: (1) it supports very flexible boundary conditions ranging from natural boundaries to control of the boundary shape via prescribed curvatures; (2) the solution is based on a convex energy as a function of logarithmic radius variables with simple explicit expressions for gradients and Hessians, greatly facilitating robust and efficient numerical treatment. We demonstrate the versatility and performance of our algorithm with a variety of examples.

Regular polygon detection

Abstract: This paper describes a new robust regular polygon detector. The regular polygon transform is posed as a mixture of regular polygons in a five dimensional space. Given the edge structure of an image, we derive the a posteriori probability for a mixture of regular polygons, and thus the probability density function for the appearance of a mixture of regular polygons. Likely regular polygons can be isolated quickly by discretising and collapsing the search space into three dimensions. The remaining dimensions may be efficiently recovered subsequently using maximum likelihood at the locations of the most likely polygons in the subspace. This leads to an efficient algorithm. Also the a posteriori formulation facilitates inclusion of additional a priori information leading to real-time application to road sign detection. The use of gradient information also reduces noise compared to existing approaches such as the generalised Hough transform. Results are presented for images with noise to show stability. The detector is also applied to two separate applications: real-time road sign detection for on-line driver assistance; and feature detection, recovering stable features in rectilinear environments.

Ehrhart polynomials and successive minima

Abstract: We investigate the Ehrhart polynomial for the class of 0-sym- metric convex lattice polytopes in Euclidean n-space R n . It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima are closely related by their geometric and arithmetic mean. We also show that the roots of lattice n-polytopes with or without interior lattice points differ essentially. Furthermore, we study the structure of the roots in the planar case. Here it turns out that their distribution reflects basic properties of lattice polygons.

LDPC codes from generalized polygons

Abstract: We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation.

Not necessarily closed convex polyhedra and the double description method

Abstract: Since the seminal work of Cousot and Halbwachs, the domain of convex polyhedra has been employed in several systems for the analysis and verification of hardware and software components. Although most implementations of the polyhedral operations assume that the polyhedra are topologically closed (i.e., all the constraints defining them are non-strict), several analyzers and verifiers need to compute on a domain of convex polyhedra that are not necessarily closed (NNC). The usual approach to implementing NNC polyhedra is to embed them into closed polyhedra in a higher dimensional vector space and reuse the tools and techniques already available for closed polyhedra. In this work we highlight and discuss the issues underlying such an embedding for those implementations that are based on the double description method, where a polyhedron may be described by a system of linear constraints or by a system of generating rays and points. Two major achievements are the definition of a theoretically clean, high-level user interface and the specification of an efficient procedure for removing redundancies from the descriptions of NNC polyhedra.

Rigidity and polynomial invariants of convex polytopes

Abstract: We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of the Robbins conjecture on the degree of generalized Heron polynomials.

Sublinear Geometric Algorithms

Abstract: We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in two-dimensional triangulations and Voronoi diagrams, and ray shooting in convex polyhedra, all of which run in expected time $O(\sqrt{n}\,)$, where $n$ is the size of the input We also provide sublinear solutions for the approximate evaluation of the volume of a convex polytope and the length of the shortest path between two points on the boundary

Ginzburg-Landau description of confinement and quantization effects in mesoscopic superconductors

Abstract: An approach to the Ginzburg–Landau problem for superconducting regular polygons is developed making use of an analytical gauge transformation for the vector potential A which gives An=0 for the normal component along the boundary line of different symmetric polygons. As a result the corresponding linearized Ginzburg–Landau equation reduces to an eigenvalue problem in the basis set of functions obeying Neumann boundary condition. Such basis sets are found analytically for several symmetric structures. The proposed approach allows for accurate calculations of the order parameter distributions at low calculational cost (small basis sets) for moderate applied magnetic fields. This is illustrated by considering the nucleation of superconductivity in squares, equilateral triangles and rectangles, where vortex patterns containing antivortices are obtained on the Tc–H phase boundary. The calculated phase boundaries are compared with the experimental Tc(H) curves measured for squares, triangles, disks, rectangles,...

Matheron's Conjecture for the Covariogram Problem

Abstract: The covariogram of a convex body provides the volumes of the intersections of with all its possible translates. Matheron conjectured in 1986 that this information determines among all convex bodies, up to certain known ambiguities. It is proved that this is the case if is not , or it is not strictly convex, or its boundary contains two arbitrarily small open portions 'on opposite sides'. Examples are also constructed that show that this conjecture is false in for any.

Smooth Two-Dimensional Interpolations: A Recipe for All Polygons

Abstract: This paper presents a method for constructing smooth and bounded interpolations on any polygon, whether convex or concave in shape, or even one containing holes or isolated points in its interior. The resulting two-dimensional function distributes the value at any given vertex or internal node over the remaining portion of the domain. The representation depends only on simple geometrical properties such as lengths and areas. Accordingly, it is invariant with respect to any chosen coordinate system. The resulting set of interpolations is smooth within the domain. Within a triangle, the behavior is akin to a linear color gradient. If necessary, linear boundary behavior can also be assured. A Java implementation is available online.

Hinged dissection of polyominoes and polyforms

Abstract: A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be rotated into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a polygon P that join corresponding edges of P. This construction uses kn pieces, where k is the number of vertices of P. When P is a regular polygon, we show how to reduce the number of pieces to ⌈k/2⌉ (n - 1). In particular, we consider polyominoes (made up of unit squares), polyiamonds (made up of equilateral triangles), and polyhexes (made up of regular hexagons). We also give a hinged dissection of all polyabolos (made up of right isosceles triangles), which do not fall under the general result mentioned above. Finally, we show that if P can be hinged into Q, then any edge-to-edge gluing of n congruent copies of P can be hinged into any edge-to-edge gluing of n congruent copies of Q.

Some functional inequalities and inclusion relationships associated with certain families of integral operators

Abstract: By making use of certain familiar integral operators, the authors introduce and investigate several new subclasses of starlike, convex, close-to-convex, and quasi-convex functions. Among other results presented here, the authors establish a number of inclusion relationships associated with some of these integral operators. Some of the results established in this paper would provide extensions of those given in earlier works.

A Rigidity Criterion for Non-Convex Polyhedra

Abstract: Let P be a (non-necessarily convex) embedded polyhedron in R3, with its vertices on the boundary of an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C1, . . ., Cn with disjoint interiors, whose vertices are the vertices of P. Suppose that there exists an ellipsoid which contains no vertex of P but intersects all the edges of the Ci. Then P is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra.

Signatures of spin-related phases in transport through regular polygons

Abstract: We address the subject of transport in one-dimensional ballistic polygon loops subject to Rashba spin-orbit coupling. We identify the role played by the polygon vertices in the accumulation of spin-related phases by studying interference effects as a function of the spin-orbit coupling strength. We find that the vertices act as strong spin-scattering centers that hinder the developing of Aharonov-Casher and Berry phases. In particular, we show that the oscillation frequency of the interference pattern can be doubled by modifying the shape of the loop from a square to a circle.

Hausdorff approximation of convex polygons

Abstract: We develop algorithms for the approximation of convex polygons with n vertices by convex polygons with fewer (k) vertices. The approximating polygons either contain or are contained in the approximated ones. The distance function between convex bodies which we use to measure the quality of the approximation is the Hausdorff metric. We consider two types of problems: min-#, where the goal is to minimize the number of vertices of the output polygon, for a given distance e, and min-e, where the goal is to minimize the error, for a given maximum number of vertices. For min-# problems, our algorithms are guaranteed to be within one vertex of the optimal, and run in O(n log n) and O(n) time, for inner and outer approximations, respectively. For min -e problems, the error achieved is within an arbitrary factor α > 1 from the best possible one, and our inner and outer approximation algorithms run in O(f(α, P) ċ n log n) and O (f (α, P) ċ n) time, respectively. Where the factor f (α, P) has reciprocal logarithmic growth as α decreases to 1, this factor depends on the shape of the approximated polygon P.

Cyclic polygons with given edge lengths: Existence and uniqueness

Abstract: Let a1, ..., a n be positive numbers satisfying the condition that each of the a i ’s is less than the sum of the rest of them; this condition is necessary for the a i ’s to be the edge lengths of a (closed) polygon. It is proved that then there exists a unique (up to an isometry) convex cyclic polygon with edge lengths a1, ..., a n . On the other hand, it is shown that, without the convexity condition, there is no uniqueness—even if the signs of all central angles and the winding number are fixed, in addition to the edge lengths.

On constructing the minimum orthogonal convex polygon for the fault-tolerant routing in 2-D faulty meshes

Abstract: The rectangular faulty block model is the most commonly used fault model for designing fault-tolerant, and deadlock-free routing algorithms in mesh-connected multicomputers. The convexity of a rectangle facilitates simple, efficient ways to route messages around fault regions using relatively few or no virtual channels to avoid deadlock. However, such a faulty block may include many nonfaulty nodes which are disabled, i.e., they are not involved in the routing process. Therefore, it is important to define a fault region that is convex, and at the same time, to include a minimum number of nonfaulty nodes. In this paper, we propose an optimal solution that can quickly construct a set of minimum faulty polygons, called orthogonal convex polygons, from a given set of faulty blocks in a 2-D mesh (or 2-D torus). The formation of orthogonal convex polygons is implemented using either a centralized, or distributed solution. Both solutions are based on the formation of faulty components, each of which consists of adjacent faulty nodes only, followed by the addition of a minimum number of nonfaulty nodes to make each component a convex polygon. Extensive simulation has been done to determine the number of nonfaulty nodes included in the polygon, and the result obtained is compared with the best existing known result. Results show that the proposed approach can not only find a set of minimum faulty polygons, but also does so quickly in terms of the number of rounds in the distributed solution.

Mean curvature Flow with convex Gauss image

Abstract: We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence results in both cases. We also study the asymptotic behavior of these solutions when $t\to\infty$.

A Linear Programming Solution for Exact Collision Detection

Abstract: This paper addresses the issue of real-time collision detection between pairs of convex polyhedral objects undergoing fast rotational and translational motions. Accurate contact information between objects in virtual reality based simulations such as product design, assembly analysis, performance testing and ergonomic analysis of products are critical factors to explore when desired realism is to be achieved. For this purpose, fast, accurate and robust collision detection algorithms are required. The method described in the text models the exact collision detection problem between convex objects as a linear program. One of the strengths of the proposed methodology is its capability of addressing high speed interframe collision. In addition to the interframe collision detection, experimental data demonstrate that mathematical programming approaches offer promising results in terms of speed and robustness as well. DOI: 10.1115/1.1846053

Allocating Vertex π-Guards in Simple Polygons via Pseudo-Triangulations

Abstract: We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices? Our results are as follows: (1) Any simple polygon with n vertices can be monitored by at most \lfloor n/2 \rfloor general vertex π-guards. This bound is tight up to an additive constant of 1. (2) Any simple polygon with n vertices, k of which are convex, can be monitored by at most \lfloor (2n – k)/3 \rfloor edge-aligned vertexπ-guards. This is the first non-trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.

An isoperimetric problem for point interactions

Abstract: We consider a Hamiltonian with N point interactions in , all with the same coupling constant, placed at vertices of an equilateral polygon . It is shown that the ground-state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.

The Small Octagon with Longest Perimeter

Abstract: The convex octagon with unit diameter and maximum perimeter is determined. This answers an open question dating from 1922. The proof uses geometric reasoning and an interval arithmetic based global optimization algorithm to solve a series of non-linear and non-convex programs involving trigonometric functions.

Approximation Algorithms for Cutting Out Polygons with Lines and Rays

Abstract: This paper studies the problem of cutting out a given polygon, drawn on a convex piece of paper, in the cheapest possible way. For the problems of cutting out convex polygons with line cuts and ray cuts, we present a 7.9-approximation algorithm and a 6-approximation algorithm, respectively. For the problem of cutting out ray-cuttable polygons, an O(log n)-approximation algorithm is given.

Generalizing Monotonicity: On Recognizing Special Classes of Polygons and Polyhedra

Abstract: A simple polyhedron is weakly-monotonic in direction provided that the intersection of the polyhedron and any plane with normal is simply-connected (i.e. empty, a point, a line-segment or a simple polygon). Furthermore, if the intersection is a convex set, then the polyhedron is said to be weakly-monotonic in the convex sense. Toussaint10 introduced these types of polyhedra as generalizations of the 2-dimensional notion of monotonicity. We study the following recognition problems: Given a simple n-vertex polyhedron in 3-dimensions, we present an O(n log n) time algorithm to determine if there exists a direction such that when sweeping over the polyhedron with a plane in direction , the cross-section (or intersection) is a convex set. If we allow multiple convex polygons in the cross-section as opposed to a single convex polygon, then we provide a linear-time recognition algorithm. For simply-connected cross-sections (i.e. the cross-section is empty, a point, a line-segment or a simple polygon), we derive an O(n2) time recognition algorithm to determine if a direction exists. We then study variations of monotonicity in 2-dimensions, i.e. for simple polygons. Given a simple n-vertex polygon P, we can determine whether or not a line l can be swept over P in a continuous manner but with varying direction, such that any position of l intersects P in at most two edges. We study two variants of the problem: one where the line is allowed to sweep over a portion of the polygon multiple times and one where it can sweep any portion of the polygon only once. Although the latter problem is slightly more complicated than the former since the line movements are restricted, our solutions to both problems run in O(n2) time.

Improved Signal To Noise Ratio And Computational Speed For Gradient-Based Detection Algorithms

Abstract: Image gradient-based feature detectors offer great advantages over their standard edge-only equivalents. In driver support systems research, the radial symmetry detection algorithm has given real-time results for speed sign recognition. The regular polygon detector is a scan line algorithm for these features facilitating recognition of other road signs such as stop and give way signs. Radial symmetry has also been applied to real-time face detection, and the polygon detector is showing promising results as a feature detector for SLAM. However, gradient-based feature detection is more sensitive to noise than standard edge-based algorithms. As the total gradient magnitude at a pixel decreases, the component of the gradient at that point that arises from image noise increases. When a pixel votes in its gradient direction out to an extended radius, its position is more likely to be inaccurate if the gradient magnitude is low. In this paper, we analyse the performance of the radial symmetry and regular polygon detector algorithms under changes to the threshold on gradient magnitude. We show that the number of pixels correctly voting on a circle is not greatly reduced by thresholds that decrease the total number of pixels that vote in the image to 20%. This greatly reduces the noise component in the image, with only slight impact on the signal. This improves the performance, particularly for the regular polygon detector where the voting mechanism is complex and constitutes a large amount of the processing per pixel. This facilitates a real-time implementation, which is presented here.

Improved flip chip mmic on board performance using periodic electromagnetic bandgap structures

Abstract: A hybrid assembly having improved cross talk characteristics includes a substrate having an upper surface. Conductive paths on the upper surface are provided for conducting high frequency signals. Regular polygons made of an electromagnetic band gap (EBG) material having slow wave characteristics are deposited on the upper surface and form a lattice for tessellating the upper surface. Each of the polygons has a periphery. The polygons are separated along their periphery from adjacent polygons by an interspace and are covered with an insulating material. Second polygons, also made of an electromagnetic band gap material, are deposited over the insulating material. Semiconductor structures are mounted over the second polygons. The semiconductor structures have a plurality of electrical contacts with the conductive paths. The regular polygons can be hexagons, triangles, octagons or any other combination that forms a lattice and can be printed onto the substrate.

On geometric dilation and halving chords

Abstract: Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and q (on edges or vertices) of G, is the ratio between the shortest path in G between p and q and their Euclidean distance. The supremum over all pairs of points of all these ratios is called the geometric dilation of G. Our research is motivated by the problem of designing graphs of low dilation. We provide a characterization of closed curves of constant halving distance (i.e., curves for which all chords dividing the curve length in half are of constant length) which are useful in this context. We then relate the halving distance of curves to other geometric quantities such as area and width. Among others, this enables us to derive a new upper bound on the geometric dilation of closed curves, as a function of D/w, where D and w are the diameter and width, respectively. We further give lower bounds on the geometric dilation of polygons with n sides as a function of n. Our bounds are tight for centrally symmetric convex polygons.