Showing papers on "Orthogonal convex hull published in 2005"

Strong CHIP, normality, and linear regularity of convex sets

Abstract: We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a. characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at x is bounded away from 0 uniformly over all points in the intersection of these convex sets.

On the Fermat--Weber center of a convex object

Abstract: We show that for any convex object Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at least @D(Q)/7, where @D(Q) is the diameter of Q, and that there exists a convex object P for which this distance is @D(P)/6. We use this result to obtain a linear-time approximation scheme for finding an approximate Fermat-Weber center of a convex polygon Q.

Strong CHIP for Infinite System of Closed Convex Sets in Normed Linear Spaces

Abstract: For a general (possibly infinite) system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property. One set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interior-point conditions together with appropriate continuity of the associated set-valued function on the (topologized) index set I. In the special case when I is finite and X is finite dimensional, one of these results reduces to a classical result of Rockafellar.

On the minimum size of a point set containing two non-intersecting empty convex polygons

Abstract: Let n(k,l) be the smallest integer such that any set of n(k, I) l) points in the plane, no three collinear, contains both an empty convex k-gon and an empty convex l-gon, which do not intersect. We show that n(3, 5) = 10, 12 < n(4, 5) < 14, 16 < n(5, 5) < 20.

System and method for presenting independent variable geometric relationships

Abstract: A pair of convex clusters are selected. Each convex cluster has a center of mass located at an original fixed distance from a common origin, and is oriented along a vector formed at a fixed angle from a common polar axis. A span is measured between the centers of mass of each convex cluster. A segment is measured from the center of mass of each convex shape to a point closest to the other convex shape along the span. A new fixed distance from the common origin for the center of mass for one of the convex clusters located along the vector for that convex cluster is evaluated if the span is less than the sum of the segments of the convex clusters. The pair of convex clusters are displayed rendered using at least the new fixed distance for the center of mass of the one convex cluster.

Reconfirmation of two results on disjoint empty convex polygons

Abstract: For k ≥ 3, let m(k, k + 1) be the smallest integer such that any set of m(k, k + 1) points in the plane, no three collinear, contains two different subsets Q1 and Q2, such that CH(Q1) is an empty convex k-gon, CH(Q2) is an empty convex (k + 1)-gon, and CH(Q1) ∩ CH(Q2) = 0, where CH stands for the convex hull. In this paper, we revisit the case of k = 3 and k = 4, and provide new proofs.

Optimal convex partitions of point sets with few inner points.

Abstract: We present a fixed-parameter algorithm for the Minimum Convex Partition and the Minimum Weight Convex Partition problem. On a set P of n points the algorithm runs in O(2kn +n logn) time. The parameter k is the number of points in P lying in the interior of the convex hull of P .

Minimization of convex functions on the convex hull of a point set

Abstract: A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in Rn is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal to m. The method is based on a new procedure for finding the projection of the gradient of the objective function onto a simplicial cone in Rm, which is the tangent cone at the current point to the simplex defined by the usual constraints on barycentric coordinates. It is shown that this projection can be computed in O(m log m) operations. For strictly convex quadratic functions, the basic method can be refined to a noniterative method terminating with the optimal solution.

The Randomized Integer Convex Hull

Abstract: Let K ⊂ Rd be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull IL(K) = conv (K ⋂ L), where L is a randomly translated and rotated copy of the integer lattice Zd. We estimate the expected number of vertices of IL(K), whose behaviour is similar to the expected number of vertices of the convex hull of Vol K random points in K. In the planar case we also describe the expectation of the missed area Vol (K \ IL(K)). Surprisingly, for K a polygon, the behaviour in this case is different from the convex hull of random points.

On the Distribution of the Number of Vertices of a Random Polygon

Abstract: Assume that n points P1; . . . ;Pn are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P1; . . . ;Pn, and (1, 0). Denote by Nn the number of those points P1; . . . ;Pn which are vertices. Let p ðnÞ k ðk 1⁄4 1; . . . ; nÞ be the probability that Nn 1⁄4 k. B AR ANY, ROTE, STEIGER, and ZHANG [1] proved that pðnÞ n 1⁄4 2=1⁄2n!ðnþ 1Þ! . We derive for k 1⁄4 1; . . . ; n 1 the values of p k and thus obtain the exact distribution of Nn. Knowing this distribution provides the key to the answer of some long-standing questions in geometrical probability, e.g., to the distribution of the number of vertices of the convex hull of n points distributed independently and uniformly in a convex polygon. Mathematics Subject Classification (2000): 52A22, 60D05.

Convex drawings of plane graphs of minimum outer apices

Abstract: In a convex drawing of a plane graph G, every facial cycle of G is drawn as a convex polygon. A polygon for the outer facial cycle is called an outer convex polygon. A necessary and sufficient condition for a plane graph G to have a convex drawing is known. However, it has not been known how many apices of an outer convex polygon are necessary for G to have a convex drawing. In this paper, we show that the minimum number of apices of an outer convex polygon necessary for G to have a convex drawing is, in effect, equal to the number of leaves in a triconnected component decomposition tree of a new graph constructed from G, and that a convex drawing of G having the minimum number of apices can be found in linear time.

Stacking and bundling two convex polygons

Abstract: Given two compact convex sets C1 and C2 in the plane, we consider the problem of finding a placement ϕC1 of C1 that minimizes the area of the convex hull of ϕC1∪C2. We first consider the case where ϕC1 and C2 are allowed to intersect (as in “stacking” two flat objects in a convex box), and then add the restriction that their interior has to remain disjoint (as when “bundling” two convex objects together into a tight bundle). In both cases, we consider both the case where we are allowed to reorient C1, and where the orientation is fixed. In the case without reorientations, we achieve exact near-linear time algorithms, in the case with reorientations we compute a (1+e)-approximation in time O(e−1/2 log n+e−3/2 log e−1/2), if two sets are convex polygons with n vertices in total.

On convex quadrangulations of point sets on the plane

Abstract: Let Pn be a set of n points on the plane in general position, n ≥ 4. A convex quadrangulation of Pn is a partitioning of the convex hull Conv(Pn) of Pn into a set of quadrilaterals such that their vertices are elements of Pn, and no element of Pn lies in the interior of any quadrilateral. It is straightforward to see that if P admits a quadrilaterization, its convex hull must have an even number of vertices. In [6] it was proved that if the convex hull of Pn has an even number of points, then by adding at most •n/• Steiner points in the interior of its convex hull, we can always obtain a point set that admits a convex quadrangulation. The authors also show that n/• Steiner points are sometimes necessary. In this paper we show how to improve the upper and lower bounds of [6] to •n/• +2 and to n/• respectively. In fact, in this paper we prove an upper bound of n, and with a long and unenlightening case analysis (over fifty cases!) we can improve the upper bound to •n/• + 2, for details see [9].

On the local structure of rank-one convex hulls

Abstract: In this note we prove that if K is a compact set of m x n matrices containing an isolated point X with no rank-one connection into the convex hull of K \ {X}, then the rank-one convex hull separates as K rc = (K \ {X}) rc ∪{X}. This is an extension of a result of P. Pedregal, which holds for 2 × 2 matrices.

On the relative distances of six points in a plane convex body

Abstract: Let C be a convex body in the Euclidean plane By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body

A fixed parameter algorithm for the minimum number convex partition problem

Abstract: Given an input consisting of an n-vertex convex polygon with k hole vertices or an n-vertex planar straight line graph (PSLG) with k holes and/or reflex vertices inside the convex hull, the parameterized minimum number convex partition (MNCP) problem asks for a partition into a minimum number of convex pieces. We give a fixed-parameter tractable algorithm for this problem that runs in the following time complexities: - linear time if k is constant, - time polynomial in n if k = O(log n / log log n), or, to be exact, in O(n . k 6k-5 . 2 16k ) time.

An Improved Algorithm to Determine the Convex Hull of 2-D Points Set

Abstract: Convex hull is the basic topic in computational geometry.Convex hull of point set and that of a polygon are two classes of it.Based on quickqull algorithm,an improved algorithm to determine the convex hull of 2-D point set was proposed in this paper.After finding points at eight defined extremal positions,some vertices of convex hull can be exactly determined,thus the coarse approximation of convex hull was formed.Refinement was implemented by scanning the coarse result to find the missed vertices.This can be achieved with such data structure as linklist or stack.In general case,the whole process has a computational complexity near linear time.The proposed algorithm has been applied in control points-based image registration effectively.

Generating Random Points (or Vectors) Controlling the Percentage of them that Are Extreme in their Convex (or Positive) Hull

Abstract: This paper presents a technique to generate random data in dimensional space m such that their convex (or positive) hull contains a specific percentage of extreme points (or vectors), determined by the analyst or generator of the data. The methodology strives to remove symmetry, regularity, or predictability, which may be desirable in data used to test or compare algorithms or heuristics. There are numerous applications for this methodology.

Convex hull and closure of a fuzzy set

Abstract: Convex fuzzy set and closed fuzzy set are important concepts in the fuzzy set theory. In this paper, concepts of convex hull, closure, convex closure and closed convex hull of a fuzzy set are presented. Some properties of these concepts are studied, and the relations among these concepts are also investigated. Furthermore, the expression theorems for points in level sets of convex hull and closure of a fuzzy set are given.

Sensor-based Planning for Planar Multi-Convex Rigid Bodies

Abstract: This paper presents a method for a planar rigid body consisting of multiple convex bodies to explore an unknown planar workspace, i.e., an unknown configuration space diffeomorphic to SE(2). This method is based on a roadmap termed concave hierarchical generalized Voronoi graph (concave-HGVG). Just as in our previous work, we decompose the free configuration space into contractible cells in which we define the concave generalized Voronoi graphs (concave-GVG), and then connect these graphs using an additional structure termed one-tangent edges. Since the robot consists of multiple convex bodies, the one-tangent edges are defined using the diameter function of the convex hull of the convex bodies as well as the individual convex bodies. These two structures together form the concave-HGVG, which is a one-dimensional roadmap of the multi-convex bodies in plane. Both components are defined in terms of workspace distance measurement, and thus the concave-HGVG can readily be constructed in a sensor-based way.

New proofs about the number of empty convex 4-gons and 5-gons in a planar point set

Abstract: The present article discusses the number of empty convex 4-gons and empty convex 5-gons in a finite planar point set. New proofs are provided for the two related important results.

Jump Systems and Laminated Manhattan Sets

Abstract: A jump system is a set of lattice points satisfying a certain " two-step " axiom. A Manhattan set is the convex hull of a two-dimensional jump system. Taking multiple Manhattan sets, in layers, forms a three-dimensional object. We determine under what conditions this object is, in turn, a jump system.

Combinatorial problems at the interface of discrete and convex geometry

Abstract: This thesis consists of three chapters. The first two chapters concern lattice points and convex sets. In the first chapter we consider convex lattice polygons with minimal perimeter. Let n be a positive integer and any norm in R2. Denote by B the unit ball of and Vb,u the class of convex lattice polygons with n vertices and least -perimeter. We prove that after suitable normalisation, all members of Vb,u tend to a fixed convex body, as n > oo. In the second chapter we consider maximal convex lattice polygons inscribed in plane convex sets. Given a convex compact set K CM2 what is the largest n such that K contains a convex lattice n-gon We answer this question asymptotically. It turns out that the maximal n is related to the largest affine perimeter that a convex set contained in K can have. This, in turn, gives a new characterisation of Ko, the convex set in K having maximal affine perimeter. In the third chapter we study a combinatorial property of arbitrary finite subsets of Rd. Let X C Rd be a finite set, coloured with J colours. Then X contains a rainbow subset 7 CX, such that any ball that contains Y contains a positive fraction of the points of X.