Showing papers on "Regular polygon published in 2013"

Visualizing Mutually Nondominating Solution Sets in Many-Objective Optimization

Abstract: As many-objective optimization algorithms mature, the problem owner is faced with visualizing and understanding a set of mutually nondominating solutions in a high dimensional space. We review existing methods and present new techniques to address this problem. We address a common problem with the well-known heatmap visualization, since the often arbitrary ordering of rows and columns renders the heatmap unclear, by using spectral seriation to rearrange the solutions and objectives and thus enhance the clarity of the heatmap. A multiobjective evolutionary optimizer is used to further enhance the simultaneous visualization of solutions in objective and parameter space. Two methods for visualizing multiobjective solutions in the plane are introduced. First, we use RadViz and exploit interpretations of barycentric coordinates for convex polygons and simplices to map a mutually nondominating set to the interior of a regular convex polygon in the plane, providing an intuitive representation of the solutions and objectives. Second, we introduce a new measure of the similarity of solutions - the dominance distance - which captures the order relations between solutions. This metric provides an embedding in Euclidean space, which is shown to yield coherent visualizations in two dimensions. The methods are illustrated on standard test problems and data from a benchmark many-objective problem.

Polyhedral particles for the discrete element method

Abstract: The geometry of convex polyhedra is described by a set of half spaces. This geometry representation is used in the discrete element method to model polyhedral particles. An algorithm for contact detection and the calculation of the interaction forces for these particles is presented. Finally the presented model is exemplified by simulating the particle flow through a hopper.

Distance Distributions in Regular Polygons

Abstract: This paper derives the exact cumulative density function (cdf) of the distance between a randomly located node and any arbitrary reference point inside a regular L-sided polygon. Using this result, we obtain the closed-form probability density function of the Euclidean distance between any arbitrary reference point and its nth neighbor node when N nodes are uniformly and independently distributed inside a regular L-sided polygon. First, we exploit the rotational symmetry of the regular polygons and quantify the effect of polygon sides and vertices on the distance distributions. Then, we propose an algorithm to determine the distance distributions, given any arbitrary location of the reference point inside the polygon. For the special case when the arbitrary reference point is located at the center of the polygon, our framework reproduces the existing result in the literature.

Liouville–Arnold integrability of the pentagram map on closed polygons

Abstract: The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras, and integrable systems. Integrability of the pentagram map was conjectured by Schwartz and proved by the present authors for a larger space of twisted polygons. In this article, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.

Integrability of the Pentagram Map

Abstract: The pentagram map was introduced by Schwartz in 1992 for convex planar polygons. Recently, Ovsienko, Schwartz, and Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, that is, for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, and we study the corresponding spectral curve and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever–Phong’s universal formula.

Approximate Guarding of Monotone and Rectilinear Polygons

Abstract: We show that vertex guarding a monotone polygon is NP-hard and construct a constant factor approximation algorithm for interior guarding monotone polygons. Using this algorithm we obtain an approximation algorithm for interior guarding rectilinear polygons that has an approximation factor independent of the number of vertices of the polygon. If the size of the smallest interior guard cover is OPT for a rectilinear polygon, our algorithm produces a guard set of size O(OPT 2).

A HIGH FREQUENCY hp BOUNDARY ELEMENT METHOD FOR SCATTERING BY CONVEX POLYGONS

Abstract: In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Global and fine approximation of convex functions

Abstract: Let U subset of R-d be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f:U -> R can be approximated by real analytic convex functions, uniformly on all of U. We also show that C-0-fine approximation of convex functions by smooth (or real analytic) convex functions on R-d is possible in general if and only if d = 1. Nevertheless, for d >= 2, we give a characterization of the class of convex functions on R-d which can be approximated by real analytic (or just smoother) convex functions in the C-0-fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every C-1 convex and proper function on U can be approximated by C-infinity convex functions in the C-1-fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

Interpolation error estimates for mean value coordinates over convex polygons

Abstract: In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach π.

Deformations of Polyhedra and Polygons by the Unitary Group

Abstract: We introduce the set of framed convex polyhedra with N faces as the symplectic quotient C^2N//SU(2) A framed polyhedron is then parametrized by N spinors living in C^2 satisfying suitable closure constraints and defines a usual convex polyhedron plus a phase for each face We show that there is an action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any polyhedron onto any other with the same total area This realizes the isomorphism of the space of framed polyhedra with the Grassmannian space U(N)/SU(2)*U(N-2) We show how to write averages and correlations of geometrical observables over the ensemble of polyhedra as polynomial integrals over U(N) and we use the Itzykson-Zuber formula from matrix models as the generating function for them In the quantum case, a canonical quantization of the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners The individual face areas are quantized as half-integers (spins) and the Hilbert spaces for fixed total area form irreducible representations of U(N) We define coherent intertwiner states peaked on classical framed polyhedra and transforming consistently under U(N) transformations The U(N) character formula can be considered as an extension of the Itzykson-Zuber to the quantum level and generates the traces of all polynomial observables over the Hilbert space of intertwiners We finally show that classical (convex) polygons can be described in a similar fashion trading U(N) for O(N) We conclude with a discussion of the possible (deformation) dynamics that one can define on the space of polygons or polyhedra This work is a priori useful in the context of discrete geometry but it should also be relevant to (loop) quantum gravity in 2+1 and 3+1 dimensions when the quantum geometry is defined in terms of gluing of (quantized) polygons and polyhedra

Deformations of polyhedra and polygons by the unitary group

Abstract: We introduce the set of framed (convex) polyhedra with N faces as the symplectic quotient C2N// SU (2). A framed polyhedron is then parametrized by N spinors living in C2 satisfying suitable closure constraints and defines a usual convex polyhedron plus extra U(1) phases attached to each face. We show that there is a natural action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any (framed) polyhedron onto any other with the same total (boundary) area. This identifies the space of framed polyhedra to the Grassmannian space U(N)/ (SU(2)×U(N−2)). We show how to write averages of geometrical observables (polynomials in the faces' area and the angles between them) over the ensemble of polyhedra (distributed uniformly with respect to the Haar measure on U(N)) as polynomial integrals over the unitary group and we provide a few methods to compute these integrals systematically. We also use the Itzykson-Zuber formula from matrix models as the generating function ...

Fractional vortex molecules and vortex polygons in a baby Skyrme model

Abstract: We construct a molecule of fractional vortices with fractional topological lump charges as a baby Skyrmion with the unit topological lump charge in the antiferromagnetic (or XY) baby Skyrme model, that is, an $O(3)$ sigma model with a four-derivative term and an antiferromagnetic or XY-type potential term quadratic in fields. We further construct configurations with topological lump charges $Q\ensuremath{\le}7$ and find that bound states of vortex molecules constitute regular polygons with $2Q$ vertices as vortices, where the rotational symmetry $SO(2)$ in real space is spontaneously broken into a discrete subgroup ${\mathbf{Z}}_{Q}$. We also find metastable and arrayed bound states of fractional vortices for $Q=5$, 6. On the other hand, we find for $Q=7$ that the regular polygon is metastable and the arrayed bound state is stable. We calculate binding energies of all configurations.

Analysis of the convergence rate for the cyclic projection algorithm applied to basic semi-algebraic convex sets

Abstract: In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semi-algebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semi-algebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the basic semi-algebraic convex sets.

Computing shortest paths among curved obstacles in the plane

Abstract: In this paper, we study the problem of finding Euclidean shortest paths among curved obstacles in the plane. We model curved obstacles as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge, and each curved edge is assumed to be of O(1) complexity. Given in the plane two points s and t and a set of h pairwise disjoint splinegons with a total of n vertices, we present an algorithm that can compute a shortest path from s to t avoiding the splinegons in O(n+hlogeh+k) time for any e>0, where k is a parameter sensitive to the input splinegons and k=O(h2). If all splinegons are convex, a common tangent of two splinegons is "free" if it does not intersect the interior of any splingegon; our techniques yield an output sensitive algorithm for computing all free common tangents of the h splinegons in O(n+hlogh+k) time and O(n) working space, where k is the number of all free common tangents.

Ball and spindle convexity with respect to a convex body

Abstract: Let \({C \subset \mathbb{R}^n}\) be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either \({\emptyset}\) , or \({\mathbb{R}^n}\) . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Caratheodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.

On topology preservation in triangular, square, and hexagonal grids

Abstract: There are three possible partitionings of the continuous plane into regular polygons that leads to triangular, square, and hexagonal grids. The topology of the square grid is fairly well-understood, but it cannot be said of the remaining two regular sampling schemes. This paper presents a general characterization of simple pixels and some simplified sufficient conditions for topology-preserving operators in all the three types of regular grids.

Random Distances Associated with Arbitrary Triangles: A Recursive Approach with an Arbitrary Reference Point

Abstract: In this work, we propose a decomposition and recursion approach in order to obtain the distance distributions associated with arbitrary triangles. The focus of this work is to derive the distance distributions from an arbitrary reference point to a random point within the triangle, where the reference point can be inside or outside of the triangle. Our approach is based on the distance distributions from a vertex of an arbitrary triangle to a random point inside. By decomposing the original triangle, using the probabilistic sum, and using the distance distributions from the vertex of the decomposed triangles, we obtain the desired distance distributions. We compare our analytical results with those of simulation, where a close match can be seen between them. Since any polygon can be decomposed into triangles, this approach also applies to the random distances from an arbitrary reference point to an arbitrary polygon, regardless convex or concave.

Periodic Trajectories in the Regular Pentagon, II

Abstract: In our recent paper [1], we studied periodic billiard trajectories in a regular pentagon and in the isosceles triangle with with the angles (π/5, π/5, 3π/5). We provided a full computation of the lengths of these trajectories, both geometric and combinatoric, and formulated some conjectures concerning symbolic periodic trajectories. The main goal of this article is to prove two of these conjectures. Technically, the study of billiard trajectories in a regular pentagon is essentially equivalent to the study of geodesics in the “double pentagon,” a translation surface obtained from two centrally symmetric copies of a regular pentagon by pairwise pasting the parallel sides. The result is a surface of genus 2 that has a flat structure inherited from the plane and a conical singularity. See [2, 3, 4, 5, 6, 7] for surveys of flat surfaces and rational polygonal billiards. Let us describe the relevant results from [1]. First of all, a periodic linear trajectory is always included into a parallel family of such trajectories, and when we talk about the period, length, symbolic orbit, etc., we always mean these parallel families. See Figure 1. Second, the double pentagon has an involution, the central symmetry that exchanges the two copies of the regular pentagon. This involution interchanges the linear trajectories that have the opposite directions. For this reason, we identify the opposite directions, so the set of directions is the real projective line RP. We identify this projective line with the circle at infinity of the hyperbolic plane in the Poincare disc model. It is clear from Figure 1 that the directions of the sides of the pentagons are periodic: every linear trajectory in this direction is closed. In fact, the socalled Veech dichotomy applies to the double pentagon: if there exists a periodic trajectory in some direction then all parallel trajectories are also periodic (and they form two strips, longer – shaded in Figure 1, and shorter – left unshaded,

Measuring regularity of convex polygons

Abstract: We propose several measures to evaluate to which extent the shape of a given convex polygon is close to be regular, focusing on a range of characteristics of regularity: optimal ratio area-perimeter, equality of angles and edge lengths, regular fitting, angular and areal symmetry. We prove that our measures satisfy a number of reasonable requirements that guarantee them to be well defined, and provide efficient algorithms for their computation. All these algorithms have been implemented, and we provide experimental results on all the proposed measures.

Cutting sequences, regular polygons, and the Veech group

Abstract: We describe the cutting sequences associated to geodesic flow on regular polygons, in terms of a combinatorial process called derivation. This work is an extension of some of the ideas and results in Smillie and Ulcigrai’s recent paper, where the analysis was made for the regular octagon. It turns out that the main structural properties of the octagon generalize in a natural way.

Affine unfoldings of convex polyhedra

Abstract: We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular there exists no combinatorial obstruction to a positive resolution of Durer's unfoldability problem, which answers a question of Croft, Falconer, and Guy. Among other techniques, the proof employs a topological characterization for embeddings among the planar immersions of the disk.

The Spectral Dirichlet--Neumann Map for Laplace's Equation in a Convex Polygon

Abstract: We provide a new approach to studying the Dirichlet--Neumann map for Laplace's equation on a convex polygon using Fokas' unified method for boundary value problems. By exploiting the complex analytic structure inherent in the unified method, we provide new proofs of classical results using mainly complex analytic techniques. The analysis takes place in a Banach space of complex valued, analytic functions and the methodology is based on classical results from complex analysis. Our approach gives way to new numerical treatments of the underlying boundary value problem and the associated Dirichlet--Neumann map. Using these new results we provide a family of well-posed weak problems associated with the Dirichlet--Neumann map and prove relevant coercivity estimates so that standard techniques can be applied.

Convex shape optimization for the least biharmonic Steklov eigenvalue

Abstract: The least Steklov eigenvalue d 1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L 2 -norm of harmonic functions. These applications suggest to address the problem of minimizing d 1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain of fixed measure minimizing d 1 is known, although the optimal shape is still unknown. We perform several numerical experiments which strongly suggest that the optimal planar shape is the regular pentagon. We prove the existence of a domain minimizing d 1 also among convex domains having fixed perimeter and present some numerical results supporting the conjecture that, among planar domains, the disk is the minimizer.

Convexity of level sets for elliptic problems in convex domains or convex rings: two counterexamples

Abstract: This paper deals with some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings. Constant boundary conditions are imposed on the single component of the boundary when the domain is convex, or on each of the two components of the boundary when the domain is a convex ring. A function is called quasiconcave if its superlevel sets, defined in a suitable way when the domain is a convex ring, are all convex. In this paper, we prove that the superlevel sets of the solutions do not always inherit the convexity or ring-convexity of the domain. Namely, we give two counterexamples to this quasiconcavity property: the first one for some two-dimensional convex domains and the second one for some convex rings in any dimension.

Map annotation method and device by means of conflict detection mechanism

Abstract: The invention discloses map annotation method and device by means of a conflict detection mechanism and aims to solve the problem that map annotation is low in efficiency in the prior art. The map annotation method includes: adopting a gift wrapping algorithm to calculate the minimum outer-packaging convex polygon of a map label to be annotated; calculating the minimum enclosing rectangle of each outer-packaging convex polygons; building a spatial index tree by utilizing the minimum outer-packaging rectangles of all elements of a map as root nodes and the minimum enclosing rectangles as leaf nodes; sequentially inquiring other enclosing rectangles intersected with each enclosing rectangle in the spatial index tree, annotating corresponding map labels in the enclosing rectangles not intersected with other enclosing rectangles; judging whether the outer-packaging convex polygons in the intersected outer-packaging rectangles are intersected or not when the enclosing rectangles are intersected with other enclosing rectangles; if yes, annotating the corresponding map labels in the outer-packaging convex polygons not intersected, and if not, annotating a map label corresponding to one of the outer-packaging convex polygons in the intersected outer-packaging convex polygons. By the map annotation method and device, map annotating efficiency is improved.

Convex Polygons are Self-Coverable

Abstract: We introduce a new notion for geometric families called self-coverability and show that homothets of convex polygons are self-coverable. As a corollary, we obtain several results about coloring point sets such that any member of the family with many points contains all colors. This is dual (and in some cases equivalent) to the much investigated cover-decomposability problem.

A Laplace Type Problem for Regular Lattices with Convex-Concave Cell and Obstacles Rhombus

Abstract: In this paper we consider two regular lattices with the cell represented in the figure 1, and we compute the probability that a segment of random position and of costant lenght intersects a side of lattice. In particular we obtain the probability determinated in the previous work, then the Laplace probability.