Points In R2 Research Articles

Rectilinear convex hull of points in 3D and applications

Let P be a set of n points in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} time and O(n) space algorithm to compute RCH(P). We also obtain an efficient O(nlog2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log ^2 n)$$\\end{document} time and O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}. Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Redundantly globally rigid braced triangulations

By mapping the vertices of a graph G to points in ℝ3, and its edges to the corresponding line segments, we obtain a three-dimensional realization of G. A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.

An optimal algorithm for the minimum-width cubic shell problem

Given a set of n points in R3, the minimum-width cubic shell problem asks to find a thinnest cubic shell that encloses the input points, where a cubic shell refers to as a closed volume between two concentric axis-aligned cubes. In this paper, we improve the previous O(nlog2⁡n)-time algorithm presented in Bae (2019) [6] to O(nlog⁡n) worst-case time. This is the first optimal-time algorithm to the problem.

A new proof of Atiyah’s conjecture on configurations of four points

In the article entitled Atiyah, “The geometry of classical particles,” Surv. Differ. Geom. 7, 1–15 (2000) and in the article entitled Atiyah, “Configurations of points,” Philos. Trans. R. Soc., A 359, 1375–1387 (2001), Sir Michael Atiyah introduced what is known as the Atiyah problem on configurations of points, which can be briefly described as the conjecture that the n polynomials (each defined up to a phase factor) associated geometrically to a configuration of n distinct points in R3 are always linearly independent. The first “hard” case is for n = 4 points, for which the linear independence conjecture was proved by Eastwood and Norbury- in the article entitled “A proof of Atiyah’s conjecture on configurations of four points in Euclidean three-space,” Geom. Topol. 5(2), 885–893 (2001). We present a new proof of Atiyah’s linear independence conjecture on configurations of four points, i.e., of Eastwood and Norbury’s theorem. Our proof consists in showing that the Gram matrix of the four polynomials associated with a configuration of four points in Euclidean 3-space is always positive definite. It makes use of 2-spinor calculus and the theory of Hermitian positive semidefinite matrices.

On the construction of polynomial minimal surfaces with Pythagorean normals

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

Geodesic spanners for points in [formula omitted] amid axis-parallel boxes

Let P be a set of n points in R3 amid a bounded number of obstacles. Consider the metric space M=(P,dM) where dM(p,q) is the geodesic distance of p and q, i.e., the length of a shortest path from p to q avoiding obstacles. When obstacles are axis-parallel boxes, we prove that M admits an 83-spanner with O(nlog3⁡n) edges. In other words, let S be a complete graph on n vertices where each node corresponds to a point p∈P. For nodes u and v of S corresponding to p,q∈P, the edge (u,v) is associated with the geodesic distance of p and q as its weight. We indeed prove that S admits a near linear-size t-spanner for some constant t.

Computing a minimum-width cubic and hypercubic shell

This paper addresses the problem of computing a minimum-width axis-aligned cubic shell enclosing a given set of n points in R3. A cubic shell is a closed volume between two concentric and face-parallel cubes. Prior to this work, there was no known algorithm for this problem in the literature. We present the first nontrivial algorithm whose running time is O(nlog2n) in the worst case. Our approach extends to higher dimension d>3, resulting in an algorithm in O(n⌊d∕2⌋logd−1n) expected time.

The partial order competition dimensions of bipartite graphs

Choi et al. (2016) introduced the notion of the partial order competition dimension of a graph and characterized the graphs with partial order competition dimension d in terms of homothetic regular (d−1)-simplices in Rd which are contained in the hyperplane {x∈Rd∣x⋅1=0}. In this paper, we introduce a useful notion “order types for two points in R3” and give an upper bound of the partial order competition dimension of a graph in terms of its chromatic number, which is achieved via constructing a matrix in a somewhat clever way, to study partial order competition dimensions of bipartite graphs and planar graphs.

Counterpoint Theory and the Novel Function Able to Classify All Continuous Solutions to the Nash Bargaining Problem

This paper solves the long-standing bargaining problem of how to define the unique equilibrium solution that represents how two firms optimally share a surplus they jointly create in repeated games. A unique equilibrium point is determined, representative of an infinite number of points in R3 that cannot be reached by any two vector combinations. The unique solution is modeled using the continuous bi-continuous function f: J-->R3 of J, an open Mobius band (Vaughan 1977). Applying this unique continuous bi-continuous function, I show how to classify all continuous solutions to Nash’s Bargaining Problem. I introduce the concept of integrated dealmaking without compromise in repeated games called Counterpoint Theory. Applying the theory and function, I solve the long-standing problem of how to model corruption without bribes in repeated games and explain new market creation based on alternative states of the world.

Characterization of Rectifying and Sphere Curves in ℝ3

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves. This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point in ℝ3. Secondly, some formulas of curvature and torsion for sphere curves are identified. KEYWORDS: Space Curve; Rectifying Curve; Curvature; Torsion; Rectifying Plane; Tangent Vector; Normal Vector; Binormal Vector

Enumeration of complex and real surfaces via tropical geometry

Abstract We prove a correspondence theorem for singular tropical surfaces in ℝ3, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin’s lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in ℝ3. As application we show that there are pencils of real surfaces of degree d in ℙ3 containing at least (3/2)d 3 + O(d 2) singular surfaces, which is asymptotically comparable to the number 4(d − 1)3 of all complex singular surfaces in the pencil. Our result relies on the classification of singular tropical surfaces [12].

Equivariant cohomology and the Varchenko–Gelfand filtration

The cohomology of the configuration space of n points in R3 is isomorphic to the regular representation of the symmetric group, which acts by permuting the points. We give a new proof of this fact by showing that the cohomology ring is canonically isomorphic to the associated graded of the Varchenko–Gelfand filtration on the cohomology of the configuration space of n points in R1. Along the way, we give a presentation of the equivariant cohomology ring of the R3 configuration space with respect to a circle acting on R3 via rotation around a fixed line. We extend our results to the settings of arbitrary real hyperplane arrangements (the aforementioned theorems correspond to the braid arrangement) as well as oriented matroids.

A Randomized Approach to Volume Constrained Polyhedronization Problem

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

Minimum number of affine simplices of given dimension

In this paper we formulate and solve extremal problems in the Euclidean space Rd and further in hypergraphs, originating from problems in stoichiometry and elementary linear algebra. The notion of affine simplex is the bridge between the original problems and the presented extremal theorem on set systems. As a sample corollary, it follows that if no triple is collinear in a set S of n points in R3, then S contains at least n4−cn3 affine simplices for some constant c. A function related to Sperner’s Theorem and its well-known extension to reciprocal sums is also considered and its relation to Turán’s hypergraph problems is discussed.

Surface embedding narrow volume reconstruction from unorganized points

In this paper, we present a novel fast and accurate numerical method for the surface embedding narrow volume reconstruction from unorganized points in R3. Though the level set method prevails in the image processing, it requires a redistancing procedure to maintain a desired shape of the level set function. On the other hand, our method is based on the Allen–Cahn equation, which has been applied in image segmentation due to its motion by mean curvature property. We modify the original Allen–Cahn equation by multiplying a control function to restrict the evolution within a narrow band around the given surface data set. To improve the numerical stability of our proposed model, we split the governing equation into linear and nonlinear terms and use an operator splitting technique. The linear equation is solved by the multigrid method which is a fast solver and the nonlinear equation is solved analytically. The unconditional stability of the proposed scheme is also proved. Various numerical results are presented to demonstrate the robustness and accuracy of the proposed method.

On the Geometric Structures with n Points and k Distances

In this paper the geometric structures M(n, k) containing n points in R3 having k distinct distances are investigated. We will show that up to similarity there exist uncountable models M(n, k) and up to pseudo-equivalence the number of possible models for M(n, k) is finite.

The 2-center problem in three dimensions

Let P be a set of n points in R3. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present a randomized algorithm for computing a 2-center of P that runs in O(β(r⁎)n2log4nloglogn) expected time; here β(r)=1/(1−r/r0)3, r⁎ is the radius of the 2-center balls of P, and r0 is the radius of the smallest enclosing ball of P. The algorithm is near quadratic as long as r⁎ is not too close to r0, which is equivalent to the condition that the centers of the two covering balls be not too close to each other. This improves an earlier slightly super-cubic algorithm of Agarwal, Efrat, and Sharir (2000) [2] (at the cost of making the algorithm performance depend on the center separation of the covering balls).

Matroid Enumeration for Incidence Geometry

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ3 and that of Motzkin on ≤12 lines in ℝ2, extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.

Incidences in Three Dimensions and Distinct Distances in the Plane

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new set-up, but are still unable to fully resolve them.Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes, Kaplan and Sharir [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in ℝ3. Applying these bounds, we obtain, among several other results, the upper bound O(s3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s3/k12/7).One of our unresolved conjectures is that this number is O(s3/k2), for k ≥ 2. If true, it would imply the lower bound Ω(s/logs) on the number of distinct distances in the plane.