Showing papers on "Regular polygon published in 1997"

First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof

Abstract: Let λe be a Dirichlet eigenvalue of the ‘periodically, rapidly oscillating’ elliptic operator –∇·(a(x/e)∇) and let ∇ be a corresponding (simple) eigenvalue of the homogenised operator –∇·(A∇). We characterise the possible limit points of the ratio (λe–λ)/e as e→0. Our characterisation is quite explicit when the underlying domain is a (planar) convex, classical polygon with sides of rational or infinite slopes. In particular, in this case it implies that there is often a continuum of such limit points.

An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications

Abstract: In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an $\epsilon$-optimal solution can be computed efficiently using interior-point algorithms. As applications to this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using {\em Gaussian elimination on leaves of a tree}, we present an algorithm which computes an $\epsilon$-optimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in $O(N \sqrt{N}(\log(\bar c/\epsilon)+\log N))$ arithmetic operations where $\bar c$ is the largest pairwise distance among the given points. The previous best-known result on this problem is a graphical algorithm which requires O(N2) arithmetic operations under certain conditions.

Elastic Fields in a Polygon-Shaped Inclusion With Uniform Eigenstrains

Abstract: In this paper the elastic fields in an arbitrary, convex polygon-shaped inclusion with uniform eigenstrains are investigated under the condition of plane strain. Closed-form solutions are obtained for the elastic fields in a polygon-shaped inclusion. The applications to the evaluation of the effective elastic properties of composite materials with polygon-shaped reinforcements are also investigated for both dilute and dense systems. Numerical examples are presented for the strain field, strain energy, and stiffness of the composites with polygon shaped fibers. The results are also compared with some existing solutions.

Finding the largest area axis-parallel rectangle in a polygon

Abstract: This paper considers the geometric optimization problem of finding the Largest area axis-parallel Rectangle (LR) in an n-vertex general polygon. We characterize the LR for general polygons by considering different cases based on the types of contacts between the rectangle and the polygon. A general framework is presented for solving a key subproblem of the LR problem which dominates the running time for a variety of polygon types. This framework permits us to transform an algorithm for orthogonal polygons into an algorithm for non-orthogonal polygons. Using this framework, we show that the LR in a general polygon (allowing holes) can be found in O(n log2 n) time. This matches the running time of the best known algorithm for orthogonal polygons. References are given for the application of the framework to other types of polygons. For each type, the running time of the resulting algorithm matches the running time of the best known algorithm for orthogonal polygons of that type. A lower bound of time in Ω(n log n) is established for finding the LR in both self-intersecting polygons and general polygons with holes. The latter result gives us both a lower bound of Ω(n log n) and an upper bound of O(n log2 n) for general polygons.

Enumeration of three-dimensional convex polygons

Abstract: A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤ d withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤ d are defined up to a translation, and the relevant statistic is their perimeter. A SAP on ℤ d is said to beconvex if its perimeter is “minimal”, that is, is exactly twice the sum of the side lengths of the smallest hyper-rectangle containing it. In 1984, Delest and Viennot enumerated convex SAPs on the square lattice [6], but no result was available in a higher dimension. We present an elementar approach to enumerate convex SAPs in any dimension. We first obtain a new proof of Delest and Viennot's result, which explains combinatorially the form of the generating function. We then compute the generating function for convex SAPs on the cubic lattice. In a dimension larger than 3, the details of the calculations become very cumbersome. However, our method suggests that the generating function for convex SAPs on ℤ d is always a quotient ofdifferentiably finite power series.

Circle Packings of Maps in Polynomial Time

Abstract: The Andreev?Koebe?Thurston circle packing theorem is generalized and improved in two ways. First, we obtain simultaneous circle packings of the map and its dual map so that, in the corresponding straight-line representations of the map and the dual, any two edges dual to each other are perpendicular. Necessary and sufficient condition for a map to have such a primal?dual circle packing representation in a surface of constant curvature is that its universal cover is 3-connected (the map has no `planar? 2-separations). Second, an algorithm is obtained that given a mapMand a rational number ? < 0 finds an ?-approximation for the radii and the co-ordinates of the centres for the primal?dual circle packing representation ofM. The algorithm is polynomial in |E(M)| and log(1/?). In particular, for a map without planar 2-separations on an arbitrary surface, we have a polynomial time algorithm for simultaneous geodesic convex representations of the map and its dual, so that only edges dual to each other cross, and the angles at the crossings are arbitrarily close to ?/2.

Autonomous mobile robot global motion planning and geometric beacon collection using traversability vectors

Abstract: Approaches in global motion planning (GMP) and geometric beacon collection (for self-localization) using traversability vectors have been developed and implemented in both computer simulation and actual experiments on mobile robots. Both approaches are based on the same simple, modular, and multifunctional traversability vector (t-vector). Through implementation it has been found that t-vectors reduce the computational requirements to detect path obstructions, Euclidean optimal via-points, and geometric beacons, as well as to identify which features are visible to sensors. Environments can be static or dynamic and polygons are permitted to overlap (i.e., intersect or be nested). While the t-vector model does require that polygons be convex, it is a much simpler matter to decompose concave polygons into convex polygon sets than it is to require that polygons not overlap, which is required for many other GMP models. T-vectors also reduce the data size and complexity of standard V-graphs and variations thereof. This paper presents the t-vector model so that the reader can apply it to mobile robot GMP and self-localization.

Orthogonal polyhedra as geometric bounds in constructive solid geometry

Abstract: Set membership classification and, specifically, the evaluation of a CSG tree are problems of a certain complexity. Several techniques to speed up these processes have been proposed such as Active Zones, Geometric Bounds and the Extended Convex Differences Tree. Boxes are the most common geometric bounds studied but other bounds such as spheres, convex hulls and prisms have also been proposed. In this work we propose orthogonal polyhedra as geometric bounds in the CSG model. CSG primitives are approximated by orthogonal polyhedra and the orthogonal bound of the object is obtained by applying the corresponding boolean algebra. A specific model for orthogonal polyhedra is presented that allows a simple and robust boolean operations algorithm between orthogonal polyhedra. This algorithm has linear complexity (is based on a merging process) and avoids floating-point computation.

A Schlafli-type formula for convex cores of hyperbolic 3-manifolds

Abstract: In 3-dimensional hyperbolic geometry, the classical Schlafli formula expresses the variation of the volume of a hyperbolic polyhedron in terms of the length of its edges and of the variation of its dihedral angles. We prove a similar formula for the variation of the volume of the convex core of a geometrically finite hyperbolic 3--manifold M, as we vary the hyperbolic metric of M. In this case, the pleating locus of the boundary of the convex core is not constant any more, but we showed in an earlier paper that the variation of the bending of the boundary of the convex core is described by a geodesic lamination with a certain transverse distribution. We prove that the variation of the volume of the convex core is then equal to 1/2 the length of this transverse distribution.

A Discrete Isoperimetric Problem

Abstract: We prove that the perimeter of any convex n-gons of diameter 1 is at most n2nsin (π/2n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal n-gons. In fact, the convex n-gons of diameter 1 and perimeter n2n sin (π/2n) are in bijective correspondence with the solutions of a diophantine problem.

The VSBUFFER: visibility ordering of unstructured volume primitives by polygon drawing

Abstract: Different techniques have been proposed for rendering volumetric scalar data sets. Usually these approaches are focusing on orthogonal cartesian grids, but in the last years research did also concentrate on arbitrary structured or even unstructured topologies. In particular, direct volume rendering of these data types is numerically complex and mostly requires sorting the whole database. We present a new approach to direct rendering of convex, voluminous polyhedra on arbitrary grid topologies, which efficiently use hardware assisted polygon drawing to support the sorting procedure. The key idea of this technique lies in a two pass rendering approach. First, the volume primitives are drawn in polygon mode to obtain their cross sections in the VSBUFFER orthogonal to the viewing plane. Second, this buffer is traversed in front to back order and the volume integration is performed. Thus, the complexity of the sorting procedure is reduced. Furthermore, any connectivity information can be completely neglected, which allows for the rendering of arbitrary scattered, convex polyhedra.

Method for performing polygon set operations on polygonal regions

Abstract: A method, apparatus, and article of manufacture for performing a polygon set operation on two or more polygons. A trace direction is determined based on the polygonal set operation, and an initial intersection point between first and second polygons is selected. The perimeter of either the first or second polygons is traced beginning from the selected intersection point and continuing in the determined trace direction. The trace switches between the perimeters of the first and second polygons as additional intersection points are reached during the tracing. The tracing is terminated when the selected intersection point is encountered by the tracing step. These steps are then repeated until all intersection points between the first and second polygons have been encountered.

Neural networks for convex hull computation

Abstract: Computing convex hull is one of the central problems in various applications of computational geometry. In this paper, a convex hull computing neural network (CHCNN) is developed to solve the related problems in the N-dimensional spaces. The algorithm is based on a two-layered neural network, topologically similar to ART, with a newly developed adaptive training strategy called excited learning. The CHCNN provides a parallel online and real-time processing of data which, after training, yields two closely related approximations, one from within and one from outside, of the desired convex hull. It is shown that accuracy of the approximate convex hulls obtained is around O[K/sup -1/(N-1/)], where K is the number of neurons in the output layer of the CHCNN. When K is taken to be sufficiently large, the CHCNN can generate any accurate approximate convex hull. We also show that an upper bound exists such that the CHCNN will yield the precise convex hull when K is larger than or equal to this bound. A series of simulations and applications is provided to demonstrate the feasibility, effectiveness, and high efficiency of the proposed algorithm.

Decomposing the Boundary of a Nonconvex Polyhedron

Abstract: We show that the boundary of a three-dimensional polyhedron withr reflex angles and arbitrary genus can be subdivided intoO(r) connected pieces, each of which lies on the boundary of its convex hull. A remarkable feature of this result is that the number of these convex-like pieces is independent of the number of vertices. Furthermore, it is linear inr, which contrasts with a quadratic worst-case lower bound in the number of convex pieces needed to decompose the polyhedron itself. The number of new vertices introduced in the process isO(n). The decomposition can be computed inO(n+rlogr) time.

A crystalline motion: uniqueness and geometric properties

Abstract: The author studies a model of crystalline motion in the plane. Existence and uniqueness of local in time solutions are shown. Geometric properties of the flow are studied, assuming that the Wulff shape is a regular N-sided polygon. The author shows that a small, convex grain of ice immersed in the cold melt shrinks, provided that the flow does not overly deform the initial interface. The author is able to verify the last condition only if the initial interface is a scaled Wulff shape. In this case, the free boundary will remain a scaled Wulff shape at later times. This is shown by using an isoperimetric inequality.

Non-convexity of the space of dihedral angles of hyperbolic polyhedra

Abstract: We prove that the space of dihedral angles of hyperbolic three-dimensional polyhedra, or of compact hyperbolic polyhedra, of a given combinatorial type is not convex.

A fixed pointtheorem for non-self set-valued mappings

Abstract: Let X be a complete, metrically convex metric space, K a closed convex subset of X, CB(X) the set of closed and bounded subsets of X. Let F" K CB(X) satisfying definition (1) below, with the added condition that Fx C_ K for each x OK. Then F has a fixed point in

On the ground state of regular polygonal billiards

Abstract: The Helmholtz equation for regular polygons is investigated by means of the conformal mapping from the circle, which provides an expansion parameter for the approximate evaluation of the lowest eigenvalue and the corresponding eigenvector.

Cartographic line simplication and polygon CSG formulae in O ( n log * n ) time

Abstract: The cartographers' favorite line simplification algorithm recursively selects from a list of data points those to be used to represent a linear feature, such as a coastline, on a map. A constructive solid geometry (CSG) conversion for a polygon takes a list of vertices and produces a formula representing the polygon as an intersection and union of primitive halfspaces. By using a data structure that supports splitting convex hulls and finding extreme points, both were known to have O(n log n) time solutions in the worst-case. This paper shows that both are easier than sorting by presenting an O(n log*n) algorithm for maintaining convex hulls under splitting at extreme points. It opens the question of whether there is a practical, linear-time solution.

Monte carlo approximation of form factors with error bounded a priori

Abstract: The exchange of radiant energy (e.g., visible light, infrared radiation) in simple macroscopic physical models is sometimes approximated by the solution of a system of linear equations (energy transport equations). A variable in such a system represents the total energy emitted by a discrete surface element. The coefficients of these equations depend on the form factors between pairs of surface elements. A form factor is the fraction of energy leaving a surface element which directly reaches another surface element. Form factors depend only on the geometry of the physical model. Determining good approximations of form factors is the most time-consuming step in these methods, when the geometry of the model is complex due to occlusions. In this paper, we introduce a new characterization of form factors based on concepts from integral geometry. Using this characterization, we develop a new and asymptotically efficient Monte Carlo method for the simultaneous approximation of all form factors in an occluded polyhedral environment. The approximation error is bounded without recourse to special hypothesis. This algorithm is, for typical scenes, one order of magnitude faster than methods based on the hemisphere paradigm or on Monte Carlo ray-shooting. Let A be any set of convex nonintersecting polygons in R3 with a total of n edges and vertices. Let ź be the error parameter and let ź be the confidence parameter. We compute an approximation of each nonzero form factor such that with probability at least 1 - ź the absolute approximation error is less than ź. The expected running time of the algorithm is O((ź-2 log ź-1 )(n log2n + K log n)), where K is the expected number of regular intersections for a random projection of A. The number of regular intersections can range from 0 to quadratic in n, but for typical applications it is much smaller than quadratic. The expectation is with respect to the random choices of the algorithm and the result holds for any input.

Newton polygons, curves on torus surfaces, and the converse weil theorem

Abstract: ContentsIntroduction §1. Newton polygons, the Pascal relation, and the Vieta relation §2. Newton polygons and Weil numbers §3. Parametrization of one-dimensional orbits on a torus surface §4. Curves on a torus surface §5. Polygons without interior integer-valued points §6. Polygon ΔD with an interior integer-valued point §7. Polygon ΔD without interior integer-valued points §8. Meromorphic vector functions on compact curves §9. Refinement of Weil's theorem §10. Abel's theorem and the Vieta relation §11. Singularities of the characteristic curve and Newton polygons Bibliography

Constrained Regular Approximation of Logic Programs

Abstract: Regular approximation is a well-known and useful analysis technique for conventional logic programming. Given the existence of constraint solving techniques, one may wish to obtain more precise approximations of programs while retaining the decidable properties of the approximation. Greater precision could increase the effectiveness of applications that make use of regular approximation, such as the detection of useless clauses and type analysis. In this paper, we introduce arithmetic constraints, based on convex polyhedra, into regular approximation. In addition, Herbrand constraints can be introduced to capture dependencies among arguments.

A near-optimal heuristic for minimum weight triangulation of convex polygons

Abstract: A linear-time heuristic for minimum weight triangulation of convex polygons is presented. This heuristic produces a triangulation of length within a factor 1 + {epsilon} from the optimum, where {epsilon} is an arbitrarily small positive constant. This is the first sub-cubic algorithm which guarantees such an approximation factor, and it has interesting applications.

Optimal Shortest Path and Minimum-Link Path Queries between Two Convex Polygons inside a Simple Polygonal Obstacle

Abstract: We present efficient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O(logh + logn) (plus O(k) time for reporting the k edges of the path) using a data structure with O(n) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O(logh + logn) (plus O(k) to report the k links), with O(n3) space and preprocessing time. We also extend our results to the dynamic case, and give a unified data structure that supports both queries for convex polygons in the same region of a connected planar subdivision . The update operations consist of insertions and deletions of edges and vertices. Let n be the current number of vertices in . The data structure uses O(n) space, supports updates in O(log2 n) time, and performs shortest-path and minimum-link-path queries in times O(log h+ log2n) (plus O(k) to report the k edges of the path) and O(log h + k log2 n), respectively. Performing shortest-path queries is a variation of the well-studied separation problem, which has not been efficiently solved before in the presence of obstacles. Also, it was not previously known how to perform minimum-link-path queries in a dynamic environment, even for two-point queries.