Hyperbolic Voronoi Diagram

Voronoi diagrams belong to frequently used structures in computational geometry with application in many fields of science. The properties of Voronoi diagram are already studied in various metric spaces – Euclidean, Manhattan, Minkowski, Hausdorff, or Karlsruhe and also in the hyperbolic metric. In this paper, we focus on the Voronoi diagram and its dual in the Poincaré ball model of the three-dimensional hyperbolic space. We first present some basic tools from the Poincaré ball model needed to construct a Voronoi diagram and for a closer observation of its properties. We have determined certain conditions for the position of the generators controlling the behavior of the hyperbolic Voronoi diagram. In the last section, we demonstrate this effect on the dual graph of a hyperbolic Voronoi diagram, i. e., on hyperbolic Delaunay tessellations.KeywordsPoincaré ball modelhyperbolic spaceVoronoi diagramDelaunay tessellation

SummaryVoronoi diagram is a versatile geometric structure used in solving many computational geometry problems in the field of image analysis, computer vision, engineering, medicine, molecular biology and various other disciplines. As a result it is a popular research domain. The literature boasts of many elegant and efficient methods for the construction of the Voronoi diagram in the real space. But these algorithms often fail to perform well when applied to the discrete domain. We present in this paper a novel algorithm for the construction of discrete Voronoi diagram using digital geometric concepts that takes care of the geometric perspective of the Voronoi diagram in the discrete domain. Although our technique is more suited for the CPU but by careful use of some cut‐off heuristics we are able to design a parallel programming approach for the construction of the Voronoi diagram making use of the GPU computational power. The speedup of our algorithm is compatible with the popular GPU technique for Voronoi diagram construction, namely, Jump flood algorithm. But since our algorithm uses the geometric perspective for the construction of Voronoi diagram it gives more accurate output and hence requires no correction step as is the case for the Jump flood algorithm. Furthermore, our approach focuses on extensibility, allowing for easy amenability for construction of higher dimensional Voronoi diagram. This ensures that the Voronoi diagram generation process can be adapted and expanded upon to meet specific requirements or incorporate future enhancements. We present an implementation of the proposed algorithm on a GPU using TAICHI language and describe its performance benefits on multiple input sites.

Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry. The first part of the book studies classical problems and techniques that refer to polyhedral structures. The authors include a study on algorithms for computing convex hulls as well as the construction of Voronoi diagrams and Delone triangulations. The second part of the book develops the primary concepts of (non-linear) computational algebraic geometry. Here, the book looks at Grbner bases and solving systems of polynomial equations. The theory is illustrated by applications in computer graphics, curve reconstruction and robotics. Throughout the book, interconnections between computational geometry and other disciplines (such as algebraic geometry, optimization and numerical mathematics) are established. Polyhedral and Algebraic Methods in Computational Geometry is directed towards advanced undergraduates in mathematics and computer science, as well as towards engineering students who are interested in the applications of computational geometry.

This work has three main parts: the bisectors of curves in the plane, the bisectors of surfaces in \(\mathbb{R}^{3}\), and the Voronoï diagram of a finite family of parallel half-lines in \(\mathbb{R}^{3}\). These subjects are closely related, and have applications in CAD/CAGD and Computational Geometry.We present a new approach to determine, using Cramer’s rule and suitable elimination steps, an exact algebraic parameterization (rational or non rational) of the bisector curve of two given planar rational curves. The approach is, then, generalized by using the generalized Cramer’s rule to determine an exact algebraic parameterization of the bisector surface of two low degree rational surfaces. We apply the method to obtain parametrizations of the bisector of two rational plane curves, when one of them is a circle or a line. On the other hand, we show how to easily obtain parametrizations of the bisector of the following pairs of surfaces: plane-quadric, plane-torus, circular cylinder-non developable quadric, circular cylinder-torus, cylinder-cylinder, cylinder-cone and cone-cone. These parametrizations are rational in most cases. In the remaining cases the parametrization involves one square root which is well-suited for a good approximation of the bisector.In addition, a different approach for the bisector curve problem will be presented. This new method uses dynamic color in GeoGebra for the computation of a geometric and numerical characterization of the bisector of two planar geometric objects (two curves, or a curve and a point). Even if it does not provide an algebraic representation, the method could lead to the computation of an approximate representation of the bisector curve.The Voronoï diagram (VD) is a fundamental data structure in computational geometry with various applications in theoretical and practical areas. We consider the VD of a set of parallel half-lines constrained to a compact domain \(\mathcal{D}_{0} \subset \mathbb{R}^{3}\), with respect to the Euclidean distance. This new kind of VD can be applied to the solution of some problems in the drilling industry, such as hydraulic or mining. We present an efficient approximate algorithm for computing such VD, using a box subdivision process, which produces a mesh representing the topology of the VD.KeywordsComputational GeometryInterval ArithmeticRational CurfRational SurfaceCompact DomainThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

Geostationary (GEO) satellites used for civilian, military and various other strategic requirements are critical space assets for a country. Given a satellite of interest, an assessment of the area of influence of neighborhood satellite operations is critical for the safety of the satellite asset. Nearest neighbor search is a fundamental problem in computational geometry. Voronoi diagram is a geometric structures that is based on proximity and can be used for nearest neighbour identification. A solution methodology for assessment of threat perception is presented using Voronoi diagrams, Delaunay triangulation, and power diagram. The intuitive properties based on the topology and geometry of Voronoi diagrams, and the adjacency relations of Delaunay triangulation help in describing some kind of region of influence around each satellite. The influence exerted by non-cooperative GEO satellites over the satellite of interest, aid in estimating the threat perception. Once the threat is narrowed down, the costly and limited number of ground surveillance sensors like telescopes and radars will be optimally reoriented to closely monitor the non-cooperative satellites of interest.

Accurately defining the logistics park hinterland is the basis of the rational planning of the logistics park as well as further improving the logistics efficiency. However, currently, the parameters of the methods and models for defining the logistics park hinterland are still not perfect, which lead those reasonableness and accuracy are also not ideal. Subsequently, based on the Voronoi diagram basic theory in computational geometry which reflects the advantage of the continuity of space division as well as by introducing both the ‘scale’ parameter of the space object in the breaking-point model and the heterogeneity of the spatial medium caused by both the transportation network and the regional comprehensive strength factors into Voronoi diagram, the model based on the analysis of spatial medium effect applying to weighted Voronoi diagram to define the logistics park hinterland was built. Then the plug-in used in GIS software was programmed. Finally, this model and plug-in were practically applied to the planning example of Xinjin logistics park in Sichuan Province. Practice has proved that this model overcome the defects existing in the conventional space division theories and the ordinary Voronoi diagram theory, which makes the definition of the logistics park hinterland more reasonable and scientific.

We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram ofn sites in the plane can be computed in ?(n) time when these sites form the vertices of a convex polygon in, say, counterclockwise order. This settles an open problem in computational geometry. Our techniques can also be used to obtain linear-time algorithms for computing the furthest-site Voronoi diagram and the medial axis of a convex polygon and for deleting a site from a general planar Voronoi diagram.

The Voronoi tessellation is a natural way of space segmentation, which has many applications in various fields of science and technology, as well as in social sciences and visual art. The varieties of the Voronoi tessellation methods are commonly used in computational fluid dynamics, computational geometry, geolocation and logistics, game dev programming, cartography, engineering, liquid crystal electronic technology, machine learning, etc. The very innovative results were obtained in astronomy, namely for a large-scale galaxy distribution and cosmic web pattern, for revealing the quasi-periodicity in a pencil-beam survey, for a description of constraints on the isotropic cosmic microwave background and the explosion scenario likely supernova events, for image processing, adaptive smoothing, segmentation, for signal-to-noise ratio balancing, for spectrography data analysis as well as in the moving-mesh cosmology simulation. We briefly describe these results, paying more attention to the practical application of the Voronoi tessellation related to the spatial large-scale galaxy distribution.

In the last years, the field of non-linear Computational Geometry has produced very relevant algorithmic advances, for instance, by adapting to low-degree curved objects techniques dealing with linear primitives. Moreover the introduction of new algorithms and approaches for dealing efficiently with curves and surfaces has shown a considerable impact in fields such as Computer-Aided Design where final users are continuously demanding more efficiency, accuracy, robustness, etc. New approaches for solving geometric problems, coming from Computer-Aided Geometric Design and involving curves and surfaces with a Computational Geometry flavour is the main focus of this special issue. Contributions were asked in areas such as: arrangements of low degree curves and surfaces; Voronoi diagrams and medial axes involving curved ob jects; offset computations and their topology; distance predicates for curves and surfaces; closed formulae for geometric queries involving curves and surfaces; intersection detection for moving curved ob jects; adapting algebraic or geometrical techniques for solving algorithmic problems involving curves and surfaces. This special issue is a direct consequence of the Second International Conference on Mathematical Aspects of Computer and Information Sciences - MACIS 2007 (see http:// www- spiral.lip6.fr/ MACIS2007) held in Paris in December 2007 and where Computational Geometry and Computer Aided Geometric Design were one of the main topics (among other two, Polynomial System Solving and Algorithms and Complexity). Finally six papers (out of eight papers submitted by the March 2009 deadline) were finally accepted. The two papers by C. D'Andrea et al. and I. Z. Emiris et al. deal with the problem of analyzing the Newton polygon of the implicit equation of a rational curve starting from its parameterization: both papers are devoted to study from different viewpoints the properties of the set of monomials appearing in the implicit equation of the considered curve without computing it. The two papers by E. Berberich et al. introduce a new approach to deal with the problem of determining the shape of an arrangement of curves on several classes of parametric surfaces which is then applied to compute, for example, the Minkowski sum of polytopes or the envelope of surfaces. The two papers by D. Lazard and J. Cheng et al. deal with the problem of computing the topology of a semialgebraic set (i.e. a set in $\mathbb{R}^n$ defined by polynomial equations and inequations): the first one shows how to deal with the general case by using the so called Cylindrical Algebraic Decomposition and the second one introduces an extremely fast algorithm computing the topology of a real algebraic plane curve defined implicitely. We would like finally to thank all the reviewers who definitively helped with their work to improve the papers that appear into this special issue.

This paper establishes a relation between the field of study of Geometric Design of the Structures known as and Computational Geometry. According to some properties of Voronoi Diagrams and Convex Hulls, and by means of two transformations: one inversive, the other projective, a way for generating polyhedra approximating spheres and other second order surfaces (ellipsoids, paraboloids and hyperboloids), starting from a simple Voronoi Diagram, is shown. The paper concludes with the geometric design of C-TANGENT domes: a new way for generating Spatial Meshes that can become a possible field of application for Computational Geometry.

Chapter 7 A survey of computational geometry

INTRODUCTION: In integrating Spiral Coverage into Cellular Decomposition, which combines structured grid-based techniques with flexible, quick spiral traversal, time efficiency is increased.OBJECTIVES: In the field of robotics and computational geometry, the study proposes a comparative exploration of two prominent path planning methodologies—Boustrophedon Cellular Decomposition and the innovative Spiral Coverage. Boustrophedon coverage has limitations in time efficiency due to its back-and-forth motion pattern, which can lead to lengthier coverage periods, especially in congested areas. Nevertheless, it is useful in some situations. It is critical to address these time-related issues to make Boustrophedon algorithms more useful in practical settings. METHODS: The research centres on achieving comprehensive cell coverage, addressing the complexities arising from confined spaces and intricate geometries. While conventional methods emphasise route optimization between points, the coverage path planning approach seeks optimal paths that maximize coverage and minimize associated costs. This study delves into the theory, practical implementation, and application of Spiral Coverage integrated with established cellular decomposition techniques.RESULTS: Through comparative analysis, it illustrates the advantages of spiral coverage over boustrophedon coverage in diverse robotics and computational applications. The research highlights Spiral Coverage's superiority in terms of path optimization, computational efficiency, and adaptability, proposing a novel perspective into cell decomposition. The methodology integrates the Spiral Coverage concept, transcending traditional techniques reliant on grids or Voronoi diagrams. Rigorous evaluation validates its potential to enhance path planning, exemplifying a substantial advancement in robotics and computational geometry.CONCLUSION: Our findings show that spiral coverage is on an average 45% more efficient than conventional Boustrophedon coverage. This paper set the basis for the future work on how different algorithms can traverse different shapes more efficiently.

Given a complex polynomial p(z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B m (z), m≥2. Let θ be a simple root of p(z), V(θ) its Voronoi cell, and A m (θ) its basin of attraction with respect to B m (z). We prove that given any closed subset C of V(θ), including any homothetic copy of V(θ), there exists m 0 such that for all m≥m 0, C is also a subset of A m (θ). This implies that when all roots of p(z) are simple, the basins of attraction of B m (z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B m (z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry.

The Voronoi diagram is one of the most fundamental data structures in computational geometry, which is concerned with the design and analysis of algorithms for geometrical problems. In this paper, a parallel algorithm for constructing the Voronoi diagram on CREW (Concurrent Read and Exclusive Write) model is proposed. This is an improved algorithm based on Preilowski and Mumbeck’s work. In their algorithm, they apply the Neighbor-Point-Theorem and present a parallel approach to check neighbor points. In this article, we propose an improved approach, Wave-Front algorithm, which is a quite different way to check neighbor points. The algorithm is then implemented in both sequential and multithreaded models.Since the Wave-Front algorithm has inherently concurrent tasks that can be executed simultaneously, multithreaded version was executed to observe the performance. Computational results indicate the effectiveness of the threaded model.KeywordsParallel AlgorithmVoronoi DiagramNeighbor PointConcurrent TaskVoronoi PolygonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.