Showing papers on "Regular polygon published in 2006"

Discrete conformal mappings via circle patterns

Abstract: We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, that is, 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 supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes, we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples.

Characterization of cheeger sets for convex subsets of the plane

Abstract: Given a planar convex domain Q, its Cheeger set CΩ is defined as the unique minimizer of |∂X|/|(X| among all nonempty open and simply connected subsets X of Ω. We prove an interesting geometric property of C Ω , characterize domains S2 which coincide with C Ω and provide a constructive algorithm for the determination of C Ω .

Monitoring environmental boundaries with a robotic sensor network

Abstract: In this paper we propose and analyze an algorithm to monitor an environmental boundary with mobile sensors. The objective is to optimally approximate the boundary with a polygon. The mobile sensors rely only on sensed local information to position some interpolation points and define an approximating polygon. We design an algorithm that distributes the vertices of the approximating polygon uniformly along the boundary. The notion of uniform placement relies on a metric inspired by known results on approximation of convex bodies. The algorithm is provably convergent for static boundaries and also for slowly-moving boundaries because of certain input-to-state stability properties.

Caging Polygons with Two and Three Fingers.

Abstract: We present algorithms for computing all placements of two and three fingers that cage a given polygonal object with n edges in the plane. A polygon is caged when it is impossible to take the polygon to infinity without penetrating one of the fingers. Using a classification into squeezing and stretching cagings, we provide an algorithm that reports all caging placements of two disc fingers in O(n 2logn) time. Our result extends and improves a recent solution for point fingers. In addition, we construct a data structure requiring O(n 2) storage that can answer in O(logn) whether two fingers in a query placement cage the polygon. We also study caging with three point fingers. Given the placements of two so-called base fingers, we report all placements of the third finger so that the three fingers jointly cage the polygon. Using the fact that the boundary of the set of placements for the third finger consists of equilibrium grasps, we present an algorithm that reports all placements of the third finger in O(n 6log2 n) deterministic time and O(n 6logn(loglogn)3) expected time. Our results extend previous solutions that only apply to convex polygons.

Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes

Abstract: We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the non-degeneracy by generalizing the Alexandrov-Fenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program.

Pseudo-Triangulations - a Survey

Abstract: A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms and applications of pseudo-triangulations.

Strictly convex drawings of planar graphs

Abstract: Every three-connected planar graph with n vertices has a drawing on an O(n(2)) x O(n(2)) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.

Exact and efficient construction of planar Minkowski sums using the convolution method

Abstract: The Minkowski sum of two sets A,B ∈ Rd, denoted A ⊕ B, is defined as {a+b|a ∈ A,b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in R2 using the convolution of the polygon boundaries. This method allows for faster computation of the sum of non-convex polygons in comparison with the widely-used methods for Minkowski-sum computation that decompose the input polygons into convex sub-polygons and compute the union of the pairwise sums of these convex sub-polygon. Partially supported by the IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-006413 (ACS - Algorithms for Complex Shapes), and by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University.

Some ways to reduce the space dimension in polyhedra computations

Abstract: Convex polyhedra are often used to approximate sets of states of programs involving numerical variables. The manipulation of convex polyhedra relies on the so-called double description, consisting of viewing a polyhedron both as the set of solutions of a system of linear inequalities, and as the convex hull of a system of generators, i.e., a set of vertices and rays. The cost of these manipulations is highly dependent on the number of numerical variables, since the size of each representation can be exponential in the dimension of the space. In this paper, we investigate some ways for reducing the dimension: On one hand, when a polyhedron satisfies affine equations, these equations can obviously be used to eliminate some variables. On the other hand, when groups of variables are unrelated with each other, this means that the polyhedron is in fact a Cartesian product of polyhedra of lower dimensions. Detecting such Cartesian factoring is not very difficult, but we adapt also the operations to work on Cartesian products. Finally, we extend the applicability of Cartesian factoring by applying suitable variable change, in order to maximize the factoring.

Convex Lattice Polygons.

Abstract: A lattice is a (rectangular) grid of points, usually pictured as occurring at the intersections of two orthogonal sets of parallel, equally spaced lines The concept of Lattice theory and the area measurement is explained

Locating Sensors in Concave Areas

Abstract: In sensor network localization, multihop based approaches were proposed to approximate the shortest paths to Euclidean distances between pairwise sensors. A good approximation can be achieved when sensors are densely deployed in a convex area, where the shortest paths are close to straight lines connecting pairwise sensors. However, in a concave network, the shortest paths may deviate far away from straight lines, which leads to erroneous distance estimation and inaccurate localization results. In this paper, we propose an improved multihop algorithm which can recognize and filter out the erroneous distance estimation, and therefore achieve accurate localization results even in a concave network.

Isodiametric Problems for Polygons

Abstract: The maximal area of a polygon with n = 2m edges and unit diameter is not known when m ≥ 5, nor is the maximal perimeter of a convex polygon with n = 2m edges and unit diameter known when m ≥ 4. We construct improved polygons in both problems, and show that the values we obtain cannot be improved for large n by more than c1/n3 in the area problem and c2/n5 in the perimeter problem, for certain constants c1 and c2.

The classification of singly periodic minimal surfaces with genus zero and scherk-type ends

Abstract: Given an integer k > 2, let S(k) be the space of complete embedded singly periodic minimal surfaces in R 3 , which in the quotient have genus zero and 2k Scherk-type ends. Surfaces in S(k) can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is'well known. It is also known that S(2) consists of the 1-parameter family of singly periodic Scherk minimal surfaces. We prove that for each k ≥ 3, there exists a natural one-to-one correspondence between S(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M ∈ S(k) the polygon whose edges are the flux vectors at the ends of M (a special polygon is a parallelogram with two sides of length 1 and two sides of length k - 1). As consequence, S(k) reduces to the saddle towers constructed by Karcher.

Spin-orbit induced interference of ballistic electrons in polygon structures

Abstract: We investigate the spin-orbit induced spin-interference pattern of ballistic electrons traveling along any regular polygon It is found that the spin interference depends strongly on the Rashba and Dresselhaus spin-orbit constants as well as on the sidelength and alignment of the polygon We derive the analytical formulas for the limiting cases of either zero Dresselhaus or zero Rashba spin-orbit coupling, including the result obtained for a circle We calculate the nonzero Dresselhaus and Rashba case numerically for the square, triangle, hexagon, and circle and discuss the observability of the spin interference which can potentially be used to measure the Rashba and Dresselhaus coefficients

Finite-difference time-domain simulation of heterostructures with inclusion of arbitrarily complex geometry

Abstract: Currently, there is a great interest in tailoring the polarization properties of composite materials with the goal of controlling the dielectric behavior. This paper reports finite-difference time-domain (FDTD) modeling of the dielectric behavior of two-dimensional (2D) lossless two-phase heterostructures. More specifically, we present extensive results of 2D FDTD computations on the quasistatic effective permittivity of a single inclusion, with arbitrarily complex geometry (regular polygons and fractals), embedded in a plane. The uniaxial perfectly matched layer-absorbing boundary condition is found adequate for truncating the boundary of the 2D space because it leads to only very small backreflections. The effectiveness of the method is demonstrated by the variety of geometries modeled, i.e., regular polygons and fractals, and permittivity contrast ratios which allows us to distinguish between effects of surface fraction and effects of morphology. Our calculations show that geometrical effects can give ...

Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space

Abstract: We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.

Real-Time Regular Polygonal Sign Detection

Abstract: In this paper, we present a new adaptation of the regular polygon detection algorithm for real-time road sign detection for autonomous vehicles. The method is robust to partial occlusion and fading, and insensitive to lighting conditions. We experimentally demonstrate its application to the detection of various signs, particularly evaluating it on a sequence of roundabout signs taken from the ANU/NICTA vehicle. The algorithm runs faster than 20 frames per second on a standard PC, detecting signs of the size that appears in road scenes, as observed from a camera mounted on the rear-vision mirror. The algorithm uses the symmetric nature of regular polygonal shapes, we also use the constrained appearance of such shapes in the road scene to the car in order to facilitate their fast, robust detection.

The Expressivity of Quantifying over Regions

Abstract: We categorize in recursion-theoretic terms the expressivity of a number of first-order languages that allow quantification over regions in Euclidean space. Specifically we show the following: 1. Let U be any class of closed regions in Euclidean space that includes all simple polygons. Let C(x, y) be the relation, “Region x is connected to region y,” and let Convex(x) be the property, “Region x is convex.” Then any relation over U that is analytical and invariant under affine transformations is first-order definable in the structure 〈U , C, Convex〉. 2. Let U be as in (1), and let Closer(x, y, z) be the relation “Region x is closer to y than to z.” Then any relation over U that is analytical and invariant under orthogonal transformations is first-order definable in the structure 〈U , Closer〉. 3. Let U be the class of finite unions of intervals in the real line. Then any relation over U that is analytical and invariant under linear transformations is first-order definable in the structure 〈U , Closer〉. 4. If the class of regions is restricted to be polygons with rational vertices, then results analogous to (1-3) hold, substituting “arithmetical relation” for “analytical relation”.

Structure of shape derivatives around irregular domains and applications

Abstract: In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in R N . This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.

Gaussian marginals of convex bodies with symmetries

Abstract: We prove Gaussian approximation theorems for specific $k$-dimensional marginals of convex bodies which possess certain symmetries. In particular, we treat bodies which possess a 1-unconditional basis, as well as simplices. Our results extend recent results for 1-dimensional marginals due to E. Meckes and the author.

Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

Abstract: Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S' that contains C. More precisely, for any e > 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q| > (1 - e)|S| and we find an axially symmetric convex polygon Q' containing C with area |Q'| < (1 + e)|S'|. We assume that C is given in a data structure that allows to answer the following two types of query in time TC: given a direction u, find an extreme point of C in direction u, and given a line l, find C ∩ l. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then TC = O(logn). Then we can find Q and Q' in time O(e-1/2TC + e-3/2). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O(e-1/2TC).

Trees with Convex Faces and Optimal Angles

Abstract: We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular resolution of the drawing. We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding. In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths.

Approximate convex decomposition and its applications

Abstract: Geometric computations are essential in many real-world problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly used to partition complex models into simpler components. Whereas decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this work, we have developed an approximate technique, called Approximate Convex Decomposition (ACD), which decomposes a given polygon or polyhedron into "approximately convex" pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficiently. Indeed, for many applications, an ACD can represent the important structural features of the model more accurately by providing a mechanism for ignoring less significant features, such as wrinkles and surface texture. Our study of a wide range of applications shows that in addition to providing computational efficiency, ACD also provides natural multi-resolution or hierarchical representations. In this dissertation, we provide some examples of ACD's many potential applications, such as particle simulation, mesh generation, motion planning, and skeleton extraction.

Cutting out polygons with lines and rays

Abstract: We present approximation algorithms for cutting out a polygon P with n vertices from another convex polygon Q with m vertices by line cuts and ray cuts. For line cuts we require both P and Q are convex while for ray cuts we require Q is convex and P is ray cuttable. Our results answer a number of open problems and are either the first solutions or significantly improve over previously known solutions. For the line cutting version, we prove a key property that leads to a simple, constant factor approximation algorithm. For the ray cutting version, we prove it is possible to compute in almost linear time a cutting sequence that is an O(log2 n)-factor approximation of an optimal cutting sequence. No algorithms were previously known for the ray cutting version.

Four-point Fermat location problems revisited. New proofs and extensions of old results

Abstract: What is the point at which the sum of (euclidean) distances to four fixed points in the plane is minimised? This extension of the celebrated location question of Fermat about three points was solved by Fagnano and others around 1750, giving the following simple geometric answer: when the fixed points form a convex quadrangle it is the intersection point of both diagonals, and otherwise it is the fixed point in the triangle formed by the three other fixed points. We show that the first case extends and generalizes to general metric spaces, while the second case extends to any planar norm, any ellipsoidal norm in higher dimensional spaces, and to the sphere.

Structure of shape derivatives around irregular domains and applications

Abstract: In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.