Showing papers on "Regular polygon published in 2007"

Concave hull: a k-nearest neighbours approach for the computation of the region occupied by a set of points

Abstract: This paper describes an algorithm to compute the envelope of a set of points in a plane, which generates convex or non-convex hulls that represent the area occupied by the given points The proposed algorithm is based on a k-nearest neighbours approach, where the value of k, the only algorithm parameter, is used to control the “smoothness” of the final solution The obtained results show that this algorithm is able to deal with arbitrary sets of points, and that the time to compute the polygons increases approximately linearly with the number of points

A Finite Element Method on Convex Polyhedra

Abstract: We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.

Product Gauss cubature over polygons based on Green’s integration formula

Abstract: We have implemented in Matlab a Gauss-like cubature formula over convex, nonconvex or even multiply connected polygons. The formula is exact for polynomials of degree at most 2n-1 using N∼mn 2 nodes, m being the number of sides that are not orthogonal to a given line, and not lying on it. It does not need any preprocessing like triangulation of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented.

Exact Minkowksi sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces

Abstract: We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by CGAL. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small amount of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron.

Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

Abstract: If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing.

Notes on the Rigidity of Graphs

Abstract: The first reference to the rigidity of frameworks in the mathematical literature occurs in a problem posed by Euler in 1776, see [8]. Consider a polyhedron P in 3-space. We view P as a ‘ panel-and-hinge framework’ in which the faces are 2-dimensional panels and the edges are 1-dimensional hinges. The panels are free to move continuously in 3-space, subject to the constraints that the shapes of the panels and the adjacencies between pairs of panels are preserved, and that the relative motion between pairs of adjacent panels is a rotation about their common hinge. The polyhedron P is rigid if every such motion results in a polyhedron which is congruent to P . Euler’s conjecture was that every polyhedron is rigid. The conjecture was verified for the case when P is convex by Cauchy [3] in 1813. Indeed Cauchy proved an even stronger result. Suppose P1 and P2 are two convex polyhedra. If there is a bijection between the faces of P1 and P2 which preserves both the shapes of faces and the adjacencies between pairs of faces, then P1 and P2 are congruent. Cauchy’s strengthening of Euler’s conjecture is not true for all polyhedra, however. Consider the icosahedron, P1. We can reflect one of the vertices of P1 in the plane containing it’s five neighbouring vertices to obtain a non-convex polyhedron P2 with the same faces and adjacencies between faces as P1. Clearly P1 and P2 are not congruent. This example is not a counterexample to Euler’s original conjecture since the reflection is not a continous motion from P1 to P2. Gluck [9] showed in 1975 that Euler’s conjecture is true when P is a ‘generic’ polyhedron i.e. there are no algebraic dependencies between the coordinates of the vertices of P . It follows that ‘almost all’ polyhedra are

Ball-Polyhedra

Abstract: We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.

Erratum: Integral equation‐based overlapped domain decomposition method for the analysis of electromagnetic scattering of 3D conducting objects

Abstract: An integral equation-based overlapped domain decomposition method (IE-ODDM) is presented for the analysis of three-dimensional (3D) electromagnetic scattering problems. Compared with the forward and backward buffer region iterative method proposed by Brennan et al., the convergence of the IE-ODDM is faster due to the edge-effect of the current in each subdomain being effectively depressed. Moreover, the iterative sequence among the subdomains of IE-ODDM is more flexible, which means that asynchronous iteration and even parallel computing are doable. Simulated RCS results of several convex and concave 3D conducting objects verify the speedup and validity of this method. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 265–274, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22110

Convex projective structures on Gromov–Thurston manifolds

Abstract: We consider Gromov–Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n ≥ 4 some of the Gromov–Thurston manifolds admit strictly convex real–projective structures.

Extremal problems for convex polygons

Abstract: Consider a convex polygon V n with n sides, perimeter P n , diameter D n , area A n , sum of distances between vertices S n and width W n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.

Convex Digital Polygons, Maximal Digital Straight Segments and Convergence of Discrete Geometric Estimators

Abstract: Discrete geometric estimators approach geometric quantities on digitized shapes without any knowledge of the continuous shape. A classical yet difficult problem is to show that an estimator asymptotically converges toward the true geometric quantity as the resolution increases. For estimators of local geometric quantities based on Digital Straight Segment (DSS) recognition this problem is closely linked to the asymptotic growth of maximal DSS for which we show bounds both about their number and sizes on Convex Digital Polygons. These results not only give better insights about digitized curves but indicate that curvature estimators based on local DSS recognition are not likely to converge. We indeed invalidate a conjecture which was essential in the only known convergence theorem of a discrete curvature estimator. The proof involves results from arithmetic properties of digital lines, digital convexity, combinatorics and continued fractions.

Embedding $S^n$ into $R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $S^n$

Abstract: In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.

Minimization of convex functionals over frame operators

Abstract: We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one.

Relaxation in nonconvex optimal control problems with subdifferential operators

Abstract: We consider the minimization problem of an integral functional in a separable Hilbert space with integrand not convex in the control defined on solutions of the control system described by nonlinear evolutionary equations with mixed nonconvex constraints. The evolutionary operator of the system is the subdifferential of a proper, convex, lower semicontinuous function depending on time.

Periodic Solutions in the Planar (n 1) Ring Problem with Oblateness

Abstract: In the N-body ring problem, the motion of an infinitesimal particle attracted by the gravitational field of (n + 1) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of n primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity w. Another primary of mass m 0 = βm (β ≥ 0 parameter) is placed at the center of the ring. Moreover, we assume that the central body may be an ellipsoid, or a radiation source, which introduces a new parameter ∈. In this case, the dynamics are found to be much richer than the classical problem due to the different equilibria and bifurcation characteristics. We find families of periodic orbits and make an analysis of the orbits by studying their evolution and stability along the family for several values of the new parameter introduced.

Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra

Abstract: Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in $\mathbb{R}^3$ with a total of $n$ edges consists of $\Theta(n^2)$ connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily degenerate scenes. More generally, we show that a set of $k$ possibly intersecting convex polyhedra with a total of $n$ edges admits, in the worst case, $\Theta(n^2k^2)$ connected components of maximal free line segments tangent to at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present an $O(n^2 k^2 \log n)$ time and $O(nk^2)$ space algorithm that, given a scene of $k$ possibly intersecting convex polyhedra, computes all the minimal free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines.

Computing non-visibility of convex polygonal facets on the surface of a polyhedral CAD model

Abstract: Visibility has found wide applications in manufacturing operations planning and computer vision and graphics. The motivation of this paper is to accurately calculate visibility for objects whose surface is represented by polygonal facets. In this paper, the authors focus on determining non-visibility cones, which are the complementary sets of visibility. This is accomplished by determining sliding planes that comprise the boundaries of a non-visibility cone. The approach presented in this paper directly evaluates the boundaries of the non-visibility cone of an arbitrary convex planar polygon due to the visibility blocked by obstacle polygons. The method is capable of calculating visibility for convex polygons with any number of sides, not limited to triangular facetted models. Implementation is demonstrated in this paper for three to six sided polygonal models.

Aperiodic orbits of piecewise rational rotations of convex polygons with recursive tiling

Abstract: We study piecewise rational rotations of convex polygons with a recursive tiling property. For these dynamical systems, the set Σ of discontinuity-avoiding aperiodic orbits decomposes into invariant subsets endowed with a hierarchical symbolic dynamics (Vershik map on a Bratteli diagram). Under conditions which guarantee a form of asymptotic temporal scaling, we prove minimality and unique ergodicity for each invariant component. We study the multi-fractal properties of the model with respect to recurrence times, deriving a method of successive approximations for the generalized dimensions α( q ). We consider explicit examples in which the trace of the rotation matrix is a quadratic or cubic irrational, and evaluate numerically, with high precision, the function α( q ) and its Legendre transform.

Bregman distances and Chebyshev sets

Abstract: A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.

Numerical index of some polyhedral norms on the plane

Abstract: We give explicit formulae for the numerical index of some (real) polyhedral spaces of dimension two. Concretely, we calculate the numerical index of a family of hexagonal norms, two families of octagonal norms and the family of norms whose unit balls are regular polygons with an even number of vertices.

Selected Topics in Geometry with Classical vs. Computer Proving

Abstract: Automatic Theorem Proving Generalization of the Formula of Heron Simson-Wallace Theorem Transversals in a Polygon Petr-Douglas-Neumann's Theorem Geometric Inequalities Regular Polygons.

Preserving computational topology by subdivision of quadratic and cubic Bézier curves

Abstract: Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bezier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C 2 cubic Bezier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper.

Geometry Reconstruction of Conducting Cylinders Using Genetic Programming

Abstract: A genetic programming-based method for the imaging of two-dimensional conductors is presented. Geometry is encoded in this scheme using a tree-shaped chromosome to represent the Boolean combination of convex polygons into an arbitrary two-dimensional geometry. The polygons themselves are encoded as the convex hull of variable-length lists of points that reside in the terminal nodes of the tree. A set of genetic operators is defined for efficiently solving the inverse scattering problem. Specifically, the encoding scheme allows for a standard genetic programming crossover operator, and several mutation operators are designed in consideration of the encoding scheme. Several results are presented that demonstrate the method on a number of different shapes