Set Of Half-spaces Research Articles

Boundary-sampled halfspaces

We present a novel representation of solid models for shape design. Like Constructive Solid Geometry (CSG), the solid shape is constructed from a set of halfspaces without the need for an explicit boundary structure. Instead of using Boolean expressions as in CSG, the shape is defined by sparsely placed samples on the boundary of each halfspace. This representation, called Boundary-Sampled Halfspaces (BSH), affords greater agility and expressiveness than CSG while simplifying the reverse engineering process. We discuss theoretical properties of the representation and present practical algorithms for boundary extraction and conversion from other representations. Our algorithms are demonstrated on both 2D and 3D examples.

On the local existence and blow-up for generalized SQG patches

We study patch solutions of a family of transport equations given by a parameter \(\alpha \), \(0< \alpha <2\), with the cases \(\alpha =0\) and \(\alpha =1\) corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for \(H^{2}\) patches in the half-space setting for \(0<\alpha < 1/3\), allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of \(\alpha \) for which finite time singularities have been shown in Kiselev et al. (Commun Pure Appl Math 70(7):1253–1315, 2017) and Kiselev et al. (Ann Math 3:909–948, 2016). Second, we establish that patches remain regular for \(0<\alpha <2\) as long as the arc-chord condition and the regularity of order \(C^{1+\delta }\) for \(\delta >\alpha /2\) are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in Córdoba et al. (Proc Natl Acad Sci USA 102:5949–5952, 2005) and Scott and Dritschel (Phys Rev Lett 112:144505, 2014) for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in Gancedo (Adv Math 217(6):2569–2598, 2008) and in Chae et al. (Commun Pure Appl Math 65(8):1037–1066, 2012), giving local existence for patches in \(H^{2}\) for \(0<\alpha < 1\) and in \(H^3\) for \(1<\alpha <2\).

Zarankiewicz’s problem for semilinear hypergraphs

AbstractA bipartite graph$H = \left (V_1, V_2; E \right )$with$\lvert V_1\rvert + \lvert V_2\rvert = n$issemilinearif$V_i \subseteq \mathbb {R}^{d_i}$for some$d_i$and the edge relationEconsists of the pairs of points$(x_1, x_2) \in V_1 \times V_2$satisfying a fixed Boolean combination ofslinear equalities and inequalities in$d_1 + d_2$variables for somes. We show that for a fixedk, the number of edges in a$K_{k,k}$-free semilinearHis almost linear inn, namely$\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$for any$\varepsilon&gt; 0$; and more generally,$\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$for a$K_{k, \dotsc ,k}$-free semilinearr-partiter-uniform hypergraph.As an application, we obtain the following incidence bound: given$n_1$points and$n_2$open boxes with axis-parallel sides in$\mathbb {R}^d$such that their incidence graph is$K_{k,k}$-free, there can be at most$O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces.We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy ino-minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

Tolerance Analysis With Polytopes in HV-Description

This article proposes the use of polytopes in HV-description to solve tolerance analysis problems. Polytopes are defined by a finite set of half-spaces representing geometric, contact, or functional specifications. However, the list of the vertices of the polytopes is useful for computing other operations as Minkowski sums. Then, this paper proposes a truncation algorithm to obtain the V-description of polytopes in ℝn from its H-description. It is detailed how intersections of polytopes can be calculated by means of the truncation algorithm. Minkowski sums as well can be computed using this algorithm making use of the duality property of polytopes. Therefore, a Minkowski sum can be calculated intersecting some half-spaces in the dual space. Finally, the approach based on HV-polytopes is illustrated by the tolerance analysis of a real industrial case using the open source software politocat and politopix.

Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space

We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated $C^{1,\alpha}$-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are based on the existence of homogenization correctors adapted to the half-space setting, which we construct---by an entirely deterministic argument---as a modification of the homogenization corrector on the whole space. This adaption procedure is carried out inductively on larger scales, crucially relying on the regularity theory already established on smaller scales.

On the no-signaling approach to quantum nonlocality

The no-signaling approach to nonlocality deals with separable and inseparable multiparty correlations in the same set of probability states without conflicting causality. The set of half-spaces describing the polytope of no-signaling probability states that are admitted by the most general class of Bell scenarios is formulated in full detail. An algorithm for determining the skeleton that solves the no-signaling description is developed upon a new strategy that is partially pivoting and partially incremental. The algorithm is formulated rigorously and its implementation is shown to be effective to deal with the highly degenerate no-signaling descriptions. Several applications of the algorithm as a tool for the study of quantum nonlocality are mentioned. Applied to a large set of bipartite Bell scenarios, we found that the corresponding no-signaling polytopes have a striking high degeneracy that grows up exponentially with the size of the Bell scenario.

Tolerance analysis by polytopes: Taking into account degrees of freedom with cap half-spaces

To determine the relative position of any two surfaces in a system, one approach is to use operations (Minkowski sum and intersection) on sets of constraints. These constraints are made compliant with half-spaces of Rn where each set of half-spaces defines an operand polyhedron. These operands are generally unbounded due to the inclusion of degrees of invariance for surfaces and degrees of freedom for joints defining theoretically unlimited displacements. To solve operations on operands, Minkowski sums in particular, “cap” half-spaces are added to each polyhedron to make it compliant with a polytope which is by definition a bounded polyhedron. The difficulty of this method lies in controlling the influence of these additional half-spaces on the topology of polytopes calculated by sum or intersection. This is necessary to validate the geometric tolerances that ensure the compliance of a mechanical system in terms of functional requirements.

Polyhedral particles for the discrete element method

The geometry of convex polyhedra is described by a set of half spaces. This geometry representation is used in the discrete element method to model polyhedral particles. An algorithm for contact detection and the calculation of the interaction forces for these particles is presented. Finally the presented model is exemplified by simulating the particle flow through a hopper.

Mean value property and a Berezin-type transform on the half-space

Let B be the open unit ball in R n . Liu (2007) [6] has shown that if f ∈ C ( B ¯ ) then the iterates of a Berezin-type transform B B k f of f converge to the Poisson extension of the boundary values of f, as k → ∞ . In this paper, we extend this to the half-space setting. First, we obtain the mean value property for harmonic functions on the half-space H. Based on this property, we define a Berezin-type transform B H and investigate the limit of the iterates of B H .

Double integral characterizations of harmonic Bergman spaces

Recently Li et al. have characterized, except for a critical case, the weighted Bergman spaces over the complex ball by means of integrability conditions of double integrals associated with difference quotients of holomorphic functions. In this paper we extend those characterizations to the case of weighted harmonic Bergman spaces over the real ball and complement their results by providing a characterization for the missing critical case. We also investigate the possibility of extensions to the half-space setting. Our observations reveal an interesting half-space phenomenon caused by the unboundedness of the half-space.

Existence of prescribed mean curvature graphs in hyperbolic space

In this paper we are concerned with questions of existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space ?n+1. In the half-space setting, we will study radial graphs over the totally geodesic hypersurface \(\). We prove the following existence result: Let \(\) be a bounded domain of class \(\) and let \(\). If \(\) everywhere on \(\), where \(\) denotes the hyperbolic mean curvature of the cylinder over \(\), then for every \(\) there is a unique graph over \(\) with mean curvature \(\) attaining the boundary values \(\) on \(\). Further we show the existence of smooth boundary data such that no solution exists in case of \(\) for some \(\) under the assumption that \(\) has a sign.

Efficient algorithms for the minimum diameter bridge problem

We give efficient algorithms for constructing a bridge between two convex regions in a fixed dimensional space so that the diameter of the bridged region is minimized. If both the set of vertices and the set of halfspaces defining the facets of the convex regions are given, we have an optimal linear time algorithm. If only vertices are given, we give a subquadratic time algorithm, and if only halfspaces are given, we give a quadratic time algorithm.

Inverse problems for homogeneous transport equations:I. The one-dimensional case

This paper and a companion paper by Bal (Bal G 2000 Inverse Problems 16 1013) are parts I and II of a series dealing with thereconstruction from boundary measurements of the scatteringoperator of homogeneous linear transport equations. This firstpart addresses the one-dimensional half-space setting, for both time-dependent and steady-state problems. We propose areconstruction procedure based on a Riccati equation for theresponse operator. In the steady-state case, we show that thereconstruction is a well-posed inverse problem provided thescattering operator satisfies enough symmetry constraints. Theproblem becomes ill posed when some of these constraints aredropped. We show that ill-posed problems in the steady-statecase become well posed in the time-dependent setting with time-independent scattering operators.

Incremental Convex Hull Algorithms Are Not Output Sensitive

A polytope is the bounded intersection of a finite set of half-spaces of $ \Bbb R$ $^d$ . Every polytope can also be represented as the convex hull conv $ \cal V $ of its vertices (or extreme points) $ \cal V $ . The convex hull problem is to convert from the vertex representation to the half-space representation or (equivalently by geometric duality) vice versa. Given an ordering v 1 . . . v n of the input vertices, after some initialization an incremental convex hull algorithm constructs half-space descriptions $ \cal H $ n-k . . . $\cal H$ n where $\cal H$ i is the half-space description of conv {v 1 . . . v i } . Let m i denote | $ \cal H$ i |, and let m denote m n . Let φ (d) denote $ d/\lceil \sqrt{d} \rceil-1 $ ; in this paper we give families of polytopes for which m n-1 ∈ Ω m φ(d) ) for any ordering of the input. We also give a family of 0/1 -polytopes with a similar blowup in intermediate size. Since m n-1 is not bounded by any polynomial in m , n , and d , incremental convex hull algorithms cannot in any reasonable sense be considered output sensitive. It turns out that the same families of polytopes are also hard for the other main types of convex hull algorithms known.

Primal—Dual Methods for Vertex and Facet Enumeration

Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (resp. vertex) to the vertex (resp. halfspace) representation is called vertex enumeration (resp. facet enumeration ). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (resp. facet) enumeration problem is the facet (resp. vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal—dual algorithms use a solution to the easy direction as an oracle to help solve the seemingly hard direction.

On stabbling lines for convex polyhedra in 3D

A line ℓ is called a stabbling line for a set B of convex polyhedra in R 3 if it intersects every polyhedron of B . This paper presents an upper bound of O( n 3log n) on the complexity of the space of stabbling lines for B , where n is the number of edges in the polyhedra of B . We solve a more general problem that counts the number of faces in a set of convex polyhedra, which are implicitly defined by a set of half-spaces and a set of hyperplanes. We show that the former problem is a special case of the latter problem. We also apply this technique to obtain an upper bound on the number of distinct faces that ever appear on the intersection of a set of half-spaces as we insert or delete half-spaces dynamically.

Efficient CSG Representations of Two-Dimensional Solids

Good methods are known for converting a Constructive Solid Geometry (CSG) representation of a solid into a boundary representation (b-rep) of the solid, but not for performing the inverse conversion, b-rep→CSG, which is the subject of this paper. Important applications of b-rep→CSG conversion arise in solid modeling, image processing, and elsewhere. The problem can be divided into two tasks: (1) finding a set of halfspaces that is necessary and sufficient (but not unique) to represent a given solid, and (2) constructing an efficient CSG representation using those halfspaces. This paper solves the problem for curved planar solids, i.e., r-sets in E2, with or without holes, whose boundary is given by a collection of edges. The edges may be subsets of straight lines or convex curves (i.e., curves which intersect any line in at most two points). We prove a number of results and describe algorithms that have been fully implemented for solids bounded by line segments and circular arcs. Empirical results show that the computed CSG representations are superior to those produced by earlier algorithms, and produce superior three-dimensional CSG representations for mechanical parts defined by contour sweeping. A companion paper generalizes the results to higher dimensional solids.

Construction and optimization of CSG representations

Boundary representations (B-reps) and constructive solid geometry (CSG) are widely used representation schemes for solids. While the problem of computing a B-rep from a CSG representation is relatively well understood, the inverse problem of B-rep to CSG conversion has not been addressed in general. The ability to perform B-rep to CSG conversion has important implications for the architecture of solid modelling systems and, in addition, is of considerable theoretical interest.The paper presents a general approach to B-rep to CSG conversion based on a partition of Euclidean space by surfaces induced from a B-rep, and on the well known fact that closed regular sets and regularized set operations form a Boolean algebra. It is shown that the conversion problem is well defined, and that the solution results in a CSG representation that is unique for a fixed set of halfspaces that serve as a ‘basis’ for the representation. The ‘basis’ set contains halfspaces induced from a B-rep plus additional non-unique separating halfspaces.An important characteristic of B-rep to CSG conversion is the size of a resulting CSG representation. We consider minimization of CSG representations in some detail and suggest new minimization techniques.While many important geometric and combinatorial issues remain open, a companion paper shows that the proposed approach to B-rep to CSG conversion and minimization is effective in E2, In E3, an experimental system currently converts natural-quadric B-reps in PARASOLID to efficient CSG representations in PADL-2.

Construction and optimization of CSG representations

Boundary representations (B-reps) and constructive solid geometry (CSG) are widely used representation schemes for solids. While the problem of computing a B-rep from a CSG representation is relatively well understood, the inverse problem of B-rep to CSG conversion has not been addressed in general. The ability to perform B-rep to CSG conversion has important implications for the architecture of solid modelling systems and, in addition, is of considerable theoretical interest. The paper presents a general approach to B-rep to CSG conversion based on a partition of Euclidean space by surfaces induced from a B-rep, and on the well known fact that closed regular sets and regularized set operations form a Boolean algebra. It is shown that the conversion problem is well defined, and that the solution results in a CSG representation that is unique for a fixed set of halfspaces that serve as a ‘basis’ for the representation. The ‘basis’ set contains halfspaces induced from a B-rep plus additional non-unique separating halfspaces. An important characteristic of B-rep to CSG conversion is the size of a resulting CSG representation. We consider minimization of CSG representations in some detail and suggest new minimization techniques. While many important geometric and combinatorial issues remain open, a companion paper shows that the proposed approach to B-rep to CSG conversion and minimization is effective in E 2, In E 3, an experimental system currently converts natural-quadric B-reps in PARASOLID to efficient CSG representations in PADL-2.

Maintenance of configurations in the plane

For a number of common configurations of points (lines) in the plane, we develop data structures in which insertions and deletions of points (or lines, respectively) can be processed rapidly, without sacrificing much of the efficiency of query answering which known static structures for these configurations attain. As a main result we establish a fully dynamic maintenance algorithm for convex hulls that can process insertions and deletions of single points in only O(log 2 n) steps per transaction, where n is the number of points currently in the set. The algorithm has several intriguing applications, including the fact that the “trimmed” mean of a set of n points in the plane can be determined in only O( n log 2 n) steps. Likewise, efficient algorithms are obtained for dynamically maintaining the common intersection of a set of half-spaces and for dynamically maintaining the maximal elements of a set of points. The results are all derived by means of one master technique, which is applied repeatedly and which captures an appropriate notion of “decomposability” for configurations closely related to the existence of divide-and-conquer solutions.