Showing papers on "Orthogonal convex hull published in 2014"

On the volume of the convex hull of two convex bodies

Abstract: In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean \(n\)-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.

Packing and Covering with Centrally Symmetric Convex Disks

Abstract: Given a convex disk K (a convex compact planar set with nonempty interior), let źL(K) and źL(K) denote the lattice packing density and the lattice covering density of K, respectively. We prove that for every centrally-symmetric convex disk K we have that $$ 1\le\delta_L(K)\theta_L(K)\le1.17225\ldots $$ The left inequality is tight and it improves a 10-year old result.

An Efficient Convex Hull Algorithm Using Affine Transformation in Planar Point Set

Abstract: When trying to find the convex hull (CH) of a point set, humans can neglect most non-vertex points by an initial estimation of the boundary of the point set easily. The proposed CH algorithm imitates this characteristic of visual attention, starts by constructing an initial convex polygon (ICP), and measures the width and length of ICP through a shape estimation step. It then maps the point set into the new one by an affine transformation and makes most of the new points exist in a new initial convex polygon (NICP) which approximated to a regular convex polygon. Next, it discards the interior points in NICP by an inscribed circle and processes the remaining points outside NICP by Quickhull. Finally, the algorithm outputs the vertex set of CH. Two theorems are also proposed to solve an unconstrained optimization problem instead of the iteration method. Compared with four popular CH algorithms, the proposed algorithm can generate CH much faster than them and achieve a better performance.

Rapid preconditioning of data for accelerating convex hull computations

Abstract: Given a dataset of two-dimensional points in the plane with integer coordinates, the method proposed reduces a set of n points down to a set of s points s ≤ n, such that the convex hull on the set of s points is the same as the convex hull of the original set of n points. The method is O(n). It helps any convex hull algorithm run faster. The empirical analysis of a practical case shows a percentage reduction in points of over 98%, that is reflected as a faster computation with a speedup factor of at least 4.

Concentration of random polytopes around the expected convex hull

Abstract: We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a corresponding strong law of large numbers.

Finding convex hull vertices in metric space

Abstract: The convex hull has been extensively studied in computational geometry and its applications have spread over an impressive number of fields. How to find the convex hull is an important and challenging problem. Although many algorithms had been proposed for that, most of them can only tackle the problem in two or three dimensions and the biggest issue is that those algorithms rely on the samples' coordinates to find the convex hull. In this paper, we propose an approximation algorithm named FVDM, which only utilizes the information of the samples' distance matrix to find the convex hull. Experiments demonstrate that FVDM can effectively identify the vertices of the convex hull.

Laminated structure and laminated structure production method

Abstract: A laminated structure includes an anti-reflection structure having periodic concavo-convex parts on a surface thereof, and a transparent conductive layer formed on the concavo-convex parts. An arbitrary convex part, excluding a convex part located at an outermost side, and six convex parts having distances from the arbitrary convex part that amount to a smallest sum, are arranged to satisfy a condition requiring a connecting part to exist between the arbitrary convex part and each of four convex parts amongst the six convex parts, and a condition requiring a concave part to exist between the arbitrary convex part and each of two remaining convex parts amongst the six convex parts.

Convex hull of face vectors of colored complexes

Abstract: In this paper we verify a conjecture by Kozlov [D.N. Kozlov, Convex Hulls of f- and @b-vectors, Discrete Comput. Geom. 18 (1997) 421-431], which describes the convex hull of the set of face vectors of r-colorable complexes on n vertices. As part of the proof we derive a generalization of Turan's graph theorem.

Technical note: Precise convex hull computation for freeform models using a hierarchical Gauss map and a Coons bounding volume hierarchy

Abstract: We present an interactive-speed algorithm for computing the precise convex hull of freeform geometric models. The algorithm is based on two pre-built data structures: (i) a Gauss map organized in a hierarchy of normal pyramids and (ii) a Coons bounding volume hierarchy (CBVH) which effectively approximates freeform surfaces with a hierarchy of bilinear surfaces. For the axis direction of each normal pyramid, we sample a point on the convex hull boundary using the CBVH. The sampled points together with the hierarchy of normal pyramids serve as a hierarchical approximation of the convex hull, with which we can eliminate the majority of redundant surface patches. We compute the precise trimmed surface patches on the convex hull boundary using a numerical tracing technique and then stitch them together in a correct topology while filling the gaps with tritangent planes and bitangent developable scrolls. We demonstrate the effectiveness of our algorithm using experimental results.

A note on the upper bound for disjoint convex partitions

Abstract: Let n(k, l,m), k ≤ l ≤ m, be the smallest integer such that any finite planar point set which has at least n(k, l,m) points in general position, contains an empty convex k-hole, an empty convex l-hole and an empty convex m-hole, in which the three holes are pairwise disjoint. In this article, we prove that n(4, 4, 5) ≤ 16.

The Convex Hull of Points on a Sphere is a Spanner.

Abstract: Let S be a finite set of points on the unit-sphere S. In 1987, Raghavan suggested that the convex hull of S is a Euclidean t-spanner, for some constant t. We prove that this is the case for t = 3π(π/2 + 1)/2. Our proof consists of generalizing the proof of Dobkin et al. [2] from the Euclidean Delaunay triangulation to the spherical Delaunay triangulation.

On the intersection of random rotations of a symmetric convex body

Abstract: Let C be a symmetric convex body of volume 1 in R. We provide general estimates for the volume and the radius of C ∩ U(C) where U is a random orthogonal transformation of R. In particular, we consider the case where C is in the isotropic position or C is the volume normalized Lq-centroid body Zq(μ) of an isotropic log-concave measure μ on R.

Turning Orthogonally Convex Polyhedra into Orthoballs

Abstract: As a step towards characterizing the graphs of orthogonally convex polyhedra, we show that for any simple orthogonally convex polyhedron there is an orthoball that is equivalent in the sense that it has the same graph and the same face normals. An orthoball is a simple orthogonally convex polyhedron with a point inside that sees the whole interior (informally, it is \round"). The consequence for reconstructing polyhedra from graphs is that if we start from a 3-regular planar graph labelled with face normals, and wish to nd a corresponding or

A generator of random convex polygons in a disc

Abstract: We propose an algorithm that generates a random polygon as a convex hull of n points uniformly and independently distributed in a disc without explicitly generate all the points.

DRCH: A Method for 3D Convex Hull

Abstract: A novel three-dimensional (3D) convex hull method is proposed, which is called dimensionality reduction convex hull method (DRCH).Through having 3d point set map to 2d plane, most initial 3D points in the convex hull are removed. Then, the remaining points are to generate 3D convex hull using any convex hull algorithm. The experiment demonstrates 3D DRCH is faster than general 3D convex hull algorithms. Its time complexity is O(r log r), where r is the number of points not in the hull. And DRCH can be generalized to higher-dimensional problems.

Convex Hulls and Metric Gabriel Graphs

Abstract: The convex hulls construction is mostly known from the point of view of 2D Euclidean geometry where it associates to a given set of points called seeds, the smallest convex polygon containing these seeds. For the cellular automata case, different adaptations of the definition and associated constructions have been proposed to fit with the discreteness of the cellular spaces. We review some of these propositions and show the link with the famous majority and voting rules. We then unify all these definitions in a unique framework using metric spaces and provide a general solution to the problem. This will lead us to an understanding of the convex hull construction as a chase for shortest paths. This emphases the importance of Voronoi diagrams and its related proximity graphs: Delaunay and Gabriel graphs. Indeed, the central problem to be solved is that of connecting arbitrary sets of seeds, in a local and finite-state way, while remaining inside the desired convex hull, i.e by shortest paths. This is exactly what will be made possible by a suitable generalization of Gabriel graphs from Euclidean to arbitrary metric spaces and the study of its construction by cellular automata. The general solution therefore consists of two levels: a connecting level using the metric Gabriel graphs and a level completing the convex hull locally as the majority rule does. Both levels can be generalized to compute the convex hull, when the seeds are moving.

A new method of minimum convex hull in computer applications

Abstract: The minimum convex hull is one of the widely studied problems in computational geometry, as well as widely applied in many fields such as in GIS. This paper, through analysis of the disadvantage of the traditional minimum convex hull serial algorithm, puts forward a new method of the minimum convex hull. It introduces three new concepts such as 'absolute vertex', 'absolutely wrong vertex' and 'possible vertex' in minimum convex hull, which can improve the algorithm computing efficiency, and increase the applicability of minimum convex hull in large number of spatial data. In the end, through comparative analysis, compared to the traditional serial algorithm, this paper's algorithm has obvious advantages on seeking minimum convex hull vertex in large number of spatial data, this algorithm execution efficiency is higher than the traditional serial algorithm for minimum convex hull, and it also can improved the applicability of minimum convex hull.

A chaotic random convex hull

Abstract: The asymptotic behavior of the expected size of the convex hull of uniformly random points in a convex body in Rd is polynomial for a smooth body and polylogarithmic for a polytope. We construct a body whose expected size of the convex hull oscillates between these two behaviors when the number of points increases.

Determining Curves in Convex Hull from a Set of Planar Closed Convex Curves

Abstract: Convex hull, a minimum enclosing convex envelope is a popular construction in the field of computational geometry. Typical convex hull include not just the points in the hull but also the connectivity between them. However, at times, it may be sufficient to get only points contributing to the hull, which can then be employed in constructions such as positive α-hull. Algorithms for convex hull primarily focus on point-set as input. Very few algorithms handle input curves as such, without discretization. In this paper, algorithm for determining curves that belong to convex hull using minimum spanning tree (MST) is proposed. The edges of the MST can be considered as a rubber-band connecting all the curves with zero enclosed area. Valency of nodes in the MST is used to define curve triplets. Maximum inscribed circle (MIC) for all triplets is identified (and stored in a triplet matrix) and its minimum is then picked to identify the starting triplet. The triplet is then deleted, updating the triplet matrix. The...

Probabilities of hitting a convex hull Probabilit es d'atteinte d'une envelopple convexe

Abstract: In this note, we consider the non-negative least square method with a random matrix. This problem has connections with the probability that the origin is not in the convex hull of many random points. As related problems, suitable estimates are obtained as well on the probability that a small ball does not hit the convex hull.

Implementation of bottleneck non cross matching for a set of convex points using convex hull and development of an image search algorithim

Abstract: The convex hull of a set X of points in the Euclidean plane is the smallest convex set that contains X. Let P be a set of 2n points in the plane which are in convex position. In this paper we propose to compute in O (n3) time and O(n2) space a bottleneck non-crossing matching of P. We compute the longest edge of bottleneck matching via dynamic programming. We extend our idea towards several applications of convex hull which includes Pattern matching and shape based image retrieval. The results shown in this paper illustrate the various applications of convex hull especially on shape based image retrieval and implements the bottleneck non cross matching on a set of convex points in O (n3) time and O(n2) space.