Showing papers on "Regular polygon published in 2009"

Stabilisation of infinitesimally rigid formations of multi-robot networks

Abstract: This article considers the design of a formation control for multivehicle systems that uses only local information. The control is derived from a potential function based on an undirected infinitesimally rigid graph that specifies the target formation. A potential function is obtained from the graph, from which a gradient control is derived. Under this controller the target formation becomes a manifold of equilibria for the multivehicle system. It is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold. A complete study of the stability of the regular polygon formation is presented and results for directed graphs are presented as well. Finally, the controller is validated experimentally.

Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space

Abstract: We consider minimal surfaces in three dimensional anti-de-Sitter space that end at the AdS boundary on a polygon given by a sequence of null segments. The problem can be reduced to a certain generalized Sinh-Gordon equation and to SU(2) Hitchin equations. We describe in detail the mathematical problem that needs to be solved. This problem is mathematically the same as the one studied by Gaiotto, Moore and Neitzke in the context of the moduli space of certain supersymmetric theories. Using their results we can find the explicit answer for the area of a surface that ends on an eight-sided polygon. Via the gauge/gravity duality this can also be interpreted as a certain eight-gluon scattering amplitude at strong coupling. In addition, we give fairly explicit solutions for regular polygons.

Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space

Abstract: We consider minimal surfaces in three dimensional anti-de-Sitter space that end at the AdS boundary on a polygon given by a sequence of null segments. The problem can be reduced to a certain generalized Sinh-Gordon equation and to SU(2) Hitchin equations. We describe in detail the mathematical problem that needs to be solved. This problem is mathematically the same as the one studied by Gaiotto, Moore and Neitzke in the context of the moduli space of certain supersymmetric theories. Using their results we can find the explicit answer for the area of a surface that ends on an eight-sided polygon. Via the gauge/gravity duality this can also be interpreted as a certain eight-gluon scattering amplitude at strong coupling. In addition, we give fairly explicit solutions for regular polygons.

Cutting circles and polygons from area-minimizing rectangles

Abstract: A set of circles, rectangles, and convex polygons are to be cut from rectangular design plates to be produced, or from a set of stocked rectangles of known geometric dimensions The objective is to minimize the area of the design rectangles The design plates are subject to lower and upper bounds of their widths and lengths The objects are free of any orientation restrictions If all nested objects fit into one design or stocked plate the problem is formulated and solved as a nonconvex nonlinear programming problem If the number of objects cannot be cut from a single plate, additional integer variables are needed to represent the allocation problem leading to a nonconvex mixed integer nonlinear optimization problem This is the first time that circles and arbitrary convex polygons are treated simultaneously in this context We present exact mathematical programming solutions to both the design and allocation problem For small number of objects to be cut we compute globally optimal solutions One key idea in the developed NLP and MINLP models is to use separating hyperplanes to ensure that rectangles and polygons do not overlap with each other or with the circles Another important idea used when dealing with several resource rectangles is to develop a model formulation which connects the binary variables only to the variables representing the center of the circles or the vertices of the polytopes but not to the non-overlap or shape constraints We support the solution process by symmetry breaking constraints In addition we compute lower bounds, which are constructed by a relaxed model in which each polygon is replaced by the largest circle fitting into that polygon We have successfully applied several solution techniques to solve this problem among them the Branch&Reduce Optimization Navigator (BARON) and the LindoGlobal solver called from GAMS, and, as described in Rebennack et al [21], a column enumeration approach in which the columns represent the assignments Good feasible solutions are computed within seconds or minutes usually during preprocessing In most cases they turn out to be globally optimal For up to 10 circles, we prove global optimality up to a gap of the order of 10-8 in short time Cases with a modest number of objects, for instance, 6 circles and 3 rectangles, are also solved in short time to global optimality For test instances involving non-rectangular polygons it is difficult to obtain small gaps In such cases we are content to obtain gaps of the order of 10%

Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

Abstract: We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time.

Contributing vertices-based Minkowski sum computation of convex polyhedra

Abstract: Minkowski sum is an important operation. It is used in many domains such as: computer-aided design, robotics, spatial planning, mathematical morphology, and image processing. We propose a novel algorithm, named the Contributing Vertices-based Minkowski Sum (CVMS) algorithm for the computation of the Minkowski sum of convex polyhedra. The CVMS algorithm allows to easily obtain all the facets of the Minkowski sum polyhedron only by examining the contributing vertices-a concept we introduce in this work, for each input facet. We exploit the concept of contributing vertices to propose the Enhanced and Simplified Slope Diagram-based Minkowski Sum (ESSDMS) algorithm, a slope diagram-based Minkowski sum algorithm sharing some common points with the approach proposed by Wu et al. [Wu Y, Shah J, Davidson J. Improvements to algorithms for computing the Minkowski sum of 3-polytopes. Comput Aided Des. 2003; 35(13): 1181-92]. The ESSDMS algorithm does not embed input polyhedra on the unit sphere and does not need to perform stereographic projections. Moreover, the use of contributing vertices brings up more simplifications and improves the overall performance. The implementations for the mentioned algorithms are straightforward, use exact number types, produce exact results, and are based on CGAL, the Computational Geometry Algorithms Library. More examples and results of the CVMS algorithm for several convex polyhedra can be found at http://liris.cnrs.fr/hichem.barki/mksum/CVMS-convex.

Optimizing Condition Numbers

Abstract: In this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite $n\times n$ matrices. We show that the condition number is a Clarke regular strongly pseudoconvex function. We prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program. This asymptotic analysis provides a valuable tool for designing an implementable algorithm for solving the problem of minimizing condition numbers.

3D Origami Design Based on Tucking Molecules

Abstract: This paper presents a new approach for folding a piece of paper into an arbitrary three-dimensional polyhedral surface based on ”tucking molecules.” The approach is to align the polygons of the three-dimensional surface onto a convex region of a plane and fill the blank area between the polygons with ”tucking molecules.” Tucking molecules are aligned between segments and vertices of the polygons, and ”paste” them together by being folded flat. We propose a novel method for generating crease pattern for tucking molecules. Necessary and sufficient conditions to enable construction of the required surface are also shown. Additionally we show that the necessary conditions can be represented in inequality conditions by using the ”tuck proxies.”

Decomposition of multiple coverings into many parts

Abstract: Let m(k) denote the smallest positive integer m such that any m-fold covering of the plane with axis-parallel unit squares splits into at least k coverings. J. Pach [J. Pach, Covering the plane with convex polygons, Discrete and Computational Geometry 1 (1986) 73-81] showed that m(k) exists and gave an exponential upper bound. We show that m(k)=O(k^2), and generalize this result to translates of any centrally symmetric convex polygon in the place of squares. From the other direction, we know only that m(k)>[email protected]?4k/[email protected]?-1.

Static analysis of memory manipulations by abstract interpretation -- Algorithmics of tropical polyhedra, and application to abstract interpretation

Abstract: In this thesis, we define a static analysis by abstract interpretation of memory manipulations. It is based on a new numerical abstract domain, which is able to infer program invariants involving the operators min and max. This domain relies on tropical polyhedra, which are the analogues of convex polyhedra in tropical algebra. Tropical algebra refers to the set IR U {-oo} endowed with max as addition and + as multiplication. This abstract domain is provided with sound abstract primitives, which allow to automatically compute over-approximations of semantics of programs by means of tropical polyhedra. Thanks to them, we develop and implement a sound static analysis inferring min- and max-invariants over the program variables, the length of the strings, and the size of the arrays in memory. In order to improve the scalability of the abstract domain, we also study the algorithmics of tropical polyhedra. In particular, a tropical polyhedron can be represented in two different ways, either internally, in terms of extreme points and rays, or externally, in terms of tropically affine inequalities. Passing from the external description of a polyhedron to its internal description, or inversely, is a fundamental computational issue, comparable to the well-known vertex/facet enumeration or convex hull problems in the classical algebra. It is also a crucial operation in our numerical abstract domain. For this reason, we develop two original algorithms allowing to pass from an external description of tropical polyhedra to an internal description, and vice versa. They are based on a tropical analogue of the double description method introduced by Motzkin et al. We show that they outperform the other existing methods, both in theory and in practice. The cornerstone of these algorithms is a new combinatorial characterization of extreme elements in tropical polyhedra defined by means of inequalities: we have proved that the extremality of an element amounts to the existence of a strongly connected component reachable from any node in a directed hypergraph. We also show that the latter property can be checked in almost linear time in the size of the hypergraph. Moreover, in order to have a better understanding of the intrinsic complexity of tropical polyhedra, we study the problem of determining the maximal number of extreme points in a tropical polyhedron. In the classical case, this problem is addressed by McMullen upper bound theorem. We prove that the maximal number of extreme points in the tropical case is bounded by a similar result. We introduce a class of tropical polyhedra appearing as natural candidates to be maximizing instances. We establish lower and upper bounds on their number of extreme points, and show that the McMullen type bound is asymptotically tight when the dimension tends to infinity and the number of inequalities defining the polyhedra is fixed. Finally, we experiment our tropical polyhedra based static analyzer on programs manipulating strings and arrays. These experimentations show that the analyzer successfully determines precise properties on memory manipulations, and that it scales up to highly disjunctive invariants which could not be computed by the existing methods. The implementation of all the algorithms and abstract domains on tropical polyhedra developed in this work is available in the Tropical Polyhedra Library (TPLib).

Minimization of $\lambda_2(\Omega)$ with a perimeter constraint

Abstract: We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $\gamma$ lower semicontinuous.

Analytic and Probabilistic Problems in Discrete Geometry

Abstract: The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence u1,...,un of norm 1 vectors in a real Hilbert space H , there exists a unit vector \vartheta \epsilon H , such that \sum 1 over [ui, v]2 \leqslant n2. The 2-dimensional case is proved by complex analytic methods. For the higher dimensional extremal cases, we prove a tensorisation result that is similar to F. John's theorem about characterisation of ellipsoids of maximal volume. From this, we deduce that the only full dimensional locally extremal system is the orthonormal system. We also obtain the same result for the weaker, original polarization problem. The second chapter investigates a problem in probabilistic geometry. Take n independent, uniform random points in a triangle T. Convex chains between two fixed vertices of T are defined naturally. Let Ln denote the maximal size of a convex chain. We prove that the expectation of Ln is asymptotically \alpha n1/3, where \alpha is a constant between 1:5 and 3:5 - we conjecture that the correct value is 3. We also prove strong concentration results for Ln, which, in turn, imply a limit shape result for the longest convex chains.

The chord length distribution function for regular polygons

Abstract: In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

Minimization of $\lambda_2(\Omega)$ with a perimeter constraint

Abstract: We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $\gamma$ lower semicontinuous.

Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

Abstract: We consider a variation of a problem stated by Erdos and Szekeres in 1935 about the existence of a number $f^{\mathrm{ES}}(k)$ such that any set $S$ of at least $f^{\mathrm{ES}}(k)$ points in general position in the plane has a subset of $k$ points that are the vertices of a convex $k$-gon. In our setting the points of $S$ are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of $S$ in its interior. We show that any sufficiently large bichromatic set of points in $\mathbb{R}^2$ in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).

Minimum-Cost Load-Balancing Partitions

Abstract: We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let D be the underlying demand region, and let p 1,…,p m be m points representing m facilities. We consider the following problem: Subdivide D into m equal-area regions R 1,…,R m , so that region R i is served by facility p i , and the average distance between a point q in D and the facility that serves q is minimal. We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be convex. As an intermediate result we show how to partition a convex polygon into m equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness of the original polygon. In fact, we prove that our partition is, up to a constant factor, the best one can get if one’s goal is to maximize the fatness of the least fat subregion. We also discuss the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of weights.

Potential based control strategy for arbitrary shape formations of mobile robots

Abstract: In this paper we describe a novel decentralized control strategy to realize formations of mobile robots. We first describe a methodology to obtain a formation with the shape of a regular polygon. Then, applying a bijective coordinates transformation, we show how to obtain a formation with an arbitrary shape. Our control strategy is based on the interaction of some artificial potential fields, but it is not affected by the problem of local minima.

Computation of the integrated flow of particles between polygons

Abstract: To quantify the flow of particles over a heterogeneous area, some models require the integration of a pointwise dispersal function over source and target polygons. This calculation is a non-trivial task and may require a great deal of computing time. In this paper, an efficient and accurate algorithm is presented to integrate general individual dispersal functions between pairs of convex or non-convex polygons. Geometric calculations are performed using standard tools from computational geometry. Numerical integration is then performed either by a grid method or by an adaptive cubature method. The procedure is illustrated with a case study. It is shown that the cubature method is much more efficient than the grid method and that its error estimates are accurate. The algorithm is implemented in a C++ program, Califlopp.

Total curvature and simple pursuit on domains of curvature bounded above

Abstract: We show how circumradius and asymptotic behavior of curves in CAT(0) and CAT(K) spaces(K>0) are controlled by growth rates of total curvature. We apply our results to pursuit and evasion games of capture type with simple pursuit motion, generalizing results that are known for convex Euclidean domains, and obtaining results that are new for convex Euclidean domains and hold on playing fields vastly more general than these.

Empty monochromatic triangles

Abstract: We consider a variation of a problem stated by Erdos and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R^2 in general position determines a super-linear number of empty monochromatic triangles, namely @W(n^5^/^4).

A fast ergodic algorithm for generating ensembles of equilateral random polygons

Abstract: Knotted structures are commonly found in circular DNA and along the backbone of certain proteins. In order to properly estimate properties of these three-dimensional structures it is often necessary to generate large ensembles of simulated closed chains (i.e. polygons) of equal edge lengths (such polygons are called equilateral random polygons). However finding efficient algorithms that properly sample the space of equilateral random polygons is a difficult problem. Currently there are no proven algorithms that generate equilateral random polygons with its theoretical distribution. In this paper we propose a method that generates equilateral random polygons in a 'step-wise uniform' way. We prove that this method is ergodic in the sense that any given equilateral random polygon can be generated by this method and we show that the time needed to generate an equilateral random polygon of length n is linear in terms of n. These two properties make this algorithm a big improvement over the existing generating methods. Detailed numerical comparisons of our algorithm with other widely used algorithms are provided.

Generic spectral simplicity of polygons

Abstract: . We study the Laplace operator with Dirichlet or Neumann boundary conditions on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has a simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form.

Exact moduli space metrics for hyperbolic vortices

Abstract: Exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic N-vortices are derived. These submanifolds, de- notedn,m, are spaces of Cn-invariant vortex configurations with n single vortices at the vertices of a regular polygon and m = N − n coincident vortices at the polygon's centre. The geometric properties ofn,m are investigated, and it is found thatn,n−1 is isometric to the hyperbolic plane of curvature −(3�n) −1 . Geodesic flow onn,m, and a geometrically natural variant of geodesic flow recently proposed by Collie and Tong, are analyzed in detail.

A Local Collision Avoidance Method for Non-strictly Convex Polyhedra

Abstract: This paper proposes a local collision avoidance method for non-strictly convex polyhedra with continuous velocities. The main contribution of the method is that non-strictly convex polyhedra can be used as geometric models of the robot and the environment without any approximation. The problem of the continuous interaction generation between polyhedra is reduced to the continuous constraints generation between polygonal faces and the continuity of those constraints are managed by the combinatorics based on Voronoi regions of a face. A collision-free motion is obtained by solving an optimization problem defined by an objective function which describes a task and linear inequality constraints which do geometrical constraints to avoid collisions. The proposed method is examined using example cases of simple objects and also applied to a humanoid robot HRP-2.

New Approximation Algorithms for Minimum Enclosing Convex Shapes

Abstract: Given $n$ points in a $d$ dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all $n$ points. We give a $O(nd\Qcal/\sqrt{\epsilon})$ approximation algorithm for producing an enclosing ball whose radius is at most $\epsilon$ away from the optimum (where $\Qcal$ is an upper bound on the norm of the points). This improves existing results using \emph{coresets}, which yield a $O(nd/\epsilon)$ greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present a $O(mnd\Qcal/\epsilon)$ approximation algorithm, where $m$ is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of \citet{Nesterov05a} to obtain our results.