Showing papers on "Regular polygon published in 1995"

A Gauss-Newton method for convex composite optimization

Abstract: An extension of the Gauss—Newton method for nonlinear equations to convex composite optimization is described and analyzed. Local quadratic convergence is established for the minimization ofh ο F under two conditions, namelyh has a set of weak sharp minima,C, and there is a regular point of the inclusionF(x) ∈ C. This result extends a similar convergence result due to Womersley (this journal, 1985) which employs the assumption of a strongly unique solution of the composite functionh ο F. A backtracking line-search is proposed as a globalization strategy. For this algorithm, a global convergence result is established, with a quadratic rate under the regularity assumption.

Separation and approximation of polyhedral objects

Abstract: Given a family of disjoint polygons P 1 , P 2 ,…, P k in the plane, and an integer parameter m , it is NP -complete to decide if the P i 's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R 1 , R 2 ,…, R k with pairwise-disjoint boundaries such that P i ⊆ R i and Σ | R i |≤ m . In three dimensions, the problem is NP -complete even for two nested convex polyhedra. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present efficient approximation algorithms for constructing separating families of near-optimal size. Our main results are as follows. In two dimensions, we give an O ( n log n ) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we show how to separate a convex polyhedrom from a nonconvex polyhedron with a polyhedral surface whose facet-complexity is O (log n ) times the optimal, where n =| P |+| Q | is the complexity of the input polyhedra. Our algorithm runs in O ( n 4 ) time, but improves to O ( n 3 ) time if the two polyhedra are nested and convex. Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d ≥4, the facet-complexity of the approximation polyhedron is O ( d log n ) times the optimal, and the algorithm runs in O ( n d +1 ) time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by non-polyhedral surfaces, such as spherical patches.

Computational Geometry and Computer Graphics in C

Abstract: I. BASICS. 1. Introduction. Framework. Our Use of the C++ Language. Robustness. 2. Analysis of Algorithms. Models of Computation. Complexity Measures. Asymptotic Analysis. Analysis of Recursive Algorithms. Problem Complexity. Chapter Notes. Exercises. 3. Data Structures. What are Data Structures? Linked Lists. Lists. Stacks. Binary Search Trees. Braided Binary Search Trees. Randomized Search Trees. Chapter Notes. Exercises. 4. Geometric Data Structures. Vectors. Points. Polygons. Edges. Geometric Objects in Space. Finding the Intersection of a Line and a Triangle. Chapter Notes. Exercises. II. APPLICATIONS. 5. Incremental Insertion. Insertion Sort. Finding Star-Shaped Polygons. Finding Convex Hulls: Insertion Hull. Point Enclosure: The Ray-Shooting Method. Point Enclosure: The Signed Angle Method. Line Clipping: The Cyrus-Beck Algorithm. Polygon Clipping: The Sutherland-Hodgman Algorithm. Triangulating Monotone Polygons. Chapter Notes. Exercises. 6. Incremental Selection. Selection Sort. Finding Convex Hulls: Gift-Wrapping. Finding Complex Hulls: Graham Scan. Removing Hidden Surfaces: The Depth-Sort Algorithm. Intersection of Convex Polygons. Finding Delaunay Triangulations. Chapter Notes. Exercises. 7. Plane-Sweep Algorithms. Finding the Intersections of Line Segments. Finding Convex Hulls: Insertion Hull Revisited. Contour of the Union of Rectangles. Decomposing Polygons into Monotone Pieces. Chapter Notes. Exercises. 8. Divide-and-Conquer Algorithms. Merge Sort. Computing the Intersection of Half-Planes. Finding the Kernel of a Polygon. Finding Voronoi Regions. Merge Hull. Closest Points. Polygon Triangulation. Chapter Notes. Exercises. 9. Spatial Subdivision Methods. The Range Searching Problem. The Grid Method. Quadtrees. Two-Dimensional Search Trees. Removing Hidden Surfaces: Binary Space Partition Trees. Chapter Notes. Exercises. Bibliography. Index.

Computing the visibility graph via pseudo-triangulations

Abstract: We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k + n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudo-triangulations, whose combinatorial properties are crucial for our method.

The Knotting of Equilateral Polygons in R3

Abstract: It was proved in [4] that the knotting probability of a Gaussian random polygon goes to 1 as the length of the polygon goes to infinity. In this paper, we prove the same result for the equilateral random polygons in R3. More precisely, if EPn is an equilateral random polygon of n steps, then we have provided that n is large enough, where ∊ is some positive constant.

A global optimization method for the Stackelberg problem with convex functions via problem transformation and concave programming

Abstract: A global optimization method is proposed for solving the Stackelberg problem through problem transformation and concave programming. The proposed method is applicable to a broad class of the Stackelberg problem, in which each function in upper-level is convex or a difference of two convex functions, and that in lower-level is convex.

Optimal geometric structures of force/torque sensors

Abstract: In controlling manipulators interacting with the external en vironment, an important role is played by the force/torque sensors. In recent years, a number of constructs for such sen sors have been proposed, and some criteria for their evaluation have been introduced. This article presents a systematic analy sis of force sensors at the design stage, which has not been undertaken in the literature. A generalized model of the sensors is developed on the basis of static and kinematic equations, and a block form of the resulting strain compliance matrix of the sensor is obtained. The model developed is then applied to the analysis of two- and three-dimensional sensor schemes cor responding to regular polygons and polyhedrons, respectively. The condition number of the normalized strain compliance matrix of the sensor is used as a performance index. The im possibility of reaching the optimal values of this criterion is stated for the regular polygon form-based sensors. Partial solutions of the optimization probl...

Tentative Prune-And-Search For Computing Fixed-Points With Applications To Geometric Computation

Abstract: Motivated by problems in computational geometry, this paper investigates the complexity of finding a fixed-point of the composition of two or three continuous functions that are defined piecewise. It shows that certain cases require nested binary search taking Θ(log2 n) time. Other cases can be solved in logarithmic time by using a prune-and-search technique that may make tentative discards and later revoke or certify them. This work finds application in optimal subroutines that compute approximations to convex polygons, dense packings, and Voronoi vertices for Euclidean and polygonal distance functions.

Construction of do loops from systems of affine constraints

Abstract: Most parallelization techniques for $\mathtt{DO}$ loop nests are based on reindexation. Reindexation yields a new iteration space, which is a convex integer polyhedron defined by a set of affine co...

The limit shape of convex lattice polygons

Abstract: It is proved here that, asn??, almost all convex (1/n)?2-lattice polygons lying in the square [?1, 1]2 are very close to a fixed convex set.

Flag-Homogeneous Compact Connected Polygons

Abstract: All flag-homogeneous compact connected polygons are classified explicitly. It turns out that these polygons are precisely the compact connected Moufang polygons.

Linear Stability Analysis of Some Symmetrical Classes of Relative Equilibria

Abstract: The linear stability of several classes of symmetrical relative equilibria of the Newtonian n-body problem are studied. Most turn out to be unstable; however, a ring of at least seven small equal masses around a sufficiently large central mass is stable.

Compaction algorithms for non-convex polygons and their applications

Abstract: Given a two-dimensional, non-overlapping layout of convex and non-convex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial two-dimensional packing applications, compaction can improve the material utilization of already tightly packed layouts. Efficient algorithms for compacting a layout of non-convex polygons are not previously known. This dissertation offers the first systematic study of compaction of non-convex polygons. We start by formalizing the compaction problem as that of planning a motion that minimizes some linear objective function of the positions. Based on this formalization, we study the complexity of compaction and show it to be PSPACE-hard. The major contribution of this dissertation is a position-based optimization model that allows us to calculate directly new polygon positions that constitute a locally optimum solution of the objective via linear programming. This model yields the first practically efficient algorithm for translational compaction-compaction in which the polygons can only translate. This compaction algorithm runs in almost real time and improves the material utilization of production quality human-generated layouts from the apparel industry. Several algorithms are derived directly from the position-based optimization model to solve related problems arising from manual or automatic layout generation. In particular, the model yields an algorithm for separating overlapping polygons using a minimal amount of motion. This separation algorithm together with a database of human-generated markers can automatically generate markers that approach human performance. Additionally, we provide several extensions to the position-based optimization model. These extensions enables the model to handle small rotations, to offer flexible control of the distances between polygons and to find optimal solution to two-dimensional packing of non-convex polygons. This dissertation also includes a compaction algorithm based on existing physical simulation approaches. Although our experimental results showed that it is not practical for compacting tightly packed layouts, this algorithm is of interest because it shows that the simulation can speed up significantly if we use geometrical constraints to replace physical constraints. It also reveals the inherent limitations of physical simulation algorithms in compacting tightly packed layouts. Most of the algorithms presented in this dissertation have been implemented on a SUN SparcStation$\sp{\rm TM}$ and have been included in a software package licensed to a CAD company.

Stacking models of vesicles and compact clusters

Abstract: We investigate three simple lattice models of two dimensional vesicles. These models differ in their behavior from the universality class of partially convex polygons, which has been recently established. They do not have the tricritical scaling of those models, and furthermore display a surprising feature: their (perimeter) free energy is discontinuous with an isolated value at zero pressure. We give the full asymptotic descriptions of the generating functions in area and perimeter variables from theq-series solutions and obtain the scaling functions where applicable.

A heuristic proof of a long-standing conjecture of d. g. kendall concerning the shapes of certain large random polygons

Abstract: In the early 1940s David Kendall conjectured that the shapes of the 'large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

Quadrangulations of Planar Sets

Abstract: Given a set S such as a polygon or a set of points, a quadrangulation of S is a partition of the interior of S, if S is a polygon, or the interior of the convex hull of S, if S is a set of points, into quadrangles (quadrilaterals) obtained by inserting edges between pairs of points (diagonals between vertices of the polygon) such that the edges intersect each other only at their end points. Not all polygons or sets of points admit quadrangulations, even when the quadrangles are not required to be convex (convex quadrangulations). In this paper we briefly survey some recent results concerning the characterization of those planar sets that always admit quadrangulations (convex and non-convex) as well as some related computational problems.

Aluminum beverage can

Abstract: An aluminum beverage can the top wall of which, and preferably the bottom wall as well, are substantially in the form of (1) a regular polygon of at least four sides, (2) a Reuleaux triangle, (3) an extended Reuleaux triangle, (4) a symmetrical curve of constant width derived from a regular polygon having an odd number of sides at least five in number, or (5) an extended symmetrical curve of constant width derived in the same way

An optimal algorithm for roundness determination on convex polygons

Abstract: In tolerancing, the Out-Of-Roundness factor determines the relative circularity of planar shapes. The measurement of concern in this work is the Minimum Radial Separation, as recommended by the American National Standards Institute (ANSI). Here presented is a further clarification of the complexity of a previously presented algorithm of Van-Ban Le and D. T. Lee to determine the Minimum Radial Separation of simple polygons, which is found to be Θ(n2). Secondly, an optimal O(n) time algorithm to compute the Minimum Radial Separation of convex polygons is presented, which represents not only a factor n improvement over the previously best known algorithm, but also a factor of log n improvement over Le and Lee's conjectured complexity for the problem.

Boundary value problems for second order impulsive differential equations using set-valued maps

Abstract: In this paper, we establish some existence results for boundary value problems for second order impulsive differential equations. Our results rely on a version of the Nonlinear Alternative of A. Granas and on a version of the Schauder Fixed Point Theorem for compact, u.s.c., convex values maps.

Geometric lower bounds for parametric matroid optimization

Abstract: We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: k-sets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: ˜.nr 1=3 / for a general n-element matroid with rank r, and ˜.mfi.n// for the special case of parametric graph minimum spanning trees. The only previous lower bound was˜.n log r/ for uniform matroids; upper bounds of O.mn 1=2 / for arbitrary matroids and O.mn 1=2 = log ⁄ n/ for uniform matroids were also known.

Fast Polygon-Cube Intersection Testing

Abstract: Publisher Summary This chapter presents a method for fast polygon-cube intersection testing. Efficient polygon-cube intersection testing is an important problem in computer graphics. Renderers can profit from fast polygon tests against display volumes, often avoiding the more expensive clipping operation. Likewise, bounding volume techniques can utilize such fast tests on the faces of polyhedral volumes. This chapter generalizes previous triangle–cube intersection methods to support arbitrary n-gons. It highlights that convex, concave, self-intersecting, and degenerate polygons are fully treated, and the new algorithm is more efficient and robust. The approach presented is a hybrid of the two techniques. It contains only a single intersection calculation, which is rarely performed because of the trivial tests that precede it. The rest of the calculations are of the same sort of fast inequality tests. This new implementation, however, is not restricted to convex figures. The implementation uses the C vector macro library vec.h.

How to Hold a Convex Body

Abstract: Convex bodies are often used for mathematical tests. They occasionally try to escape. Can the testing mathematician hold them still by using a circle? Rarely not.

Algebraic decomposition of non-convex polyhedra

Abstract: Any arbitrary polyhedron P/spl sube/R/sup d/ can be written as algebraic sum of simple terms, each an integer multiple of the intersection of d or fewer half-spaces defined by facets of P. P can be non-convex and can have holes of any kind. Among the consequences of this result are a short boolean formula for P, a fast parallel algorithm for point classification, and a new proof of the Gram-Sommerville angle relation.

Vertical decomposition of arrangements of hyperplanes in four dimensions

Abstract: We show that, for any collection ? ofn hyperplanes in ?4, the combinatorial complexity of thevertical decomposition of the arrangementA(?) of ? isO(n4 logn). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them.