Computational Geometry and Spatial Meshes

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.

Chapter 7 A survey of computational geometry

The farthest line-segment Voronoi diagram shows properties surprisingly different from the farthest point Voronoi diagram: Voronoi regions may be disconnected and they are not characterized by convex-hull properties. In this paper we introduce the farthest line-segment hull and its Gaussian map, a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram similarly to the way an ordinary convex hull characterizes the regions of the farthest-point Voronoi diagram. We also derive tighter bounds on the (linear) size of the farthest line-segment Voronoi diagram. With the purpose of unifying construction algorithms for farthest-point and farthest line-segment Voronoi diagrams, we adapt standard techniques for the construction of a convex hull to compute the farthest line-segment hull in O(n logn) or output-sensitive O(n logh) time, where n is the number of segments and h is the size of the hull (number of Voronoi faces). As a result, the farthest line-segment Voronoi diagram can be constructed in output sensitive O(n logh) time.

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.

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.

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.

This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also likely to be symmetrically shaped, which can be useful for object recognition in computer vision or pattern recognition. The focus is primarily on two-dimensional algorithms, including well-known methods like Gift Wrapping, Graham Scan, Divide and Conquer, QuickHull, TORCH, Kirkpatrick–Sediel, and Chan’s algorithms. These algorithms vary in terms of time complexity and scalability to higher dimensions. This study is extended to three-dimensional convex hull algorithms, such as NAW, randomized insertion, and parallelized versions, such as CudaHull and CudaChain. This study aimed to elucidate the operational principles, step-by-step procedures, and comparative time complexities of each algorithm. The implementation in Python facilitates a detailed comparison of the algorithmic performance through stepwise analysis and graphical outputs. The ultimate goal is to provide insights into the strengths and weaknesses of each algorithm under various scenarios, thereby offering a comprehensive guide for practical implementation.

Voronoi diagrams are among the most extensively studied objects in computational geometry with useful applications in different areas of science. To understand impacts of non-Euclidean geometry on computational geometry, this paper investigates the Voronoi diagram in hyperbolic space specially the one in the Poincare hyperbolic disk, which is a 2-dimensional manifold with negative curvature. We first prove some lemma in Poincare hyperbolic disk and then give an incremental algorithm to construct 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

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.

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.

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.

Machine learning algorithm based on convex hull analysis

In the paper, we present and prove Castle Theorem of Convex Hull, design and realize the Fast Rampart Searching Algorithm. The algorithm can be treated as the preprocess of Convex Hull calculation of very large planar point set. When calculating the Convex Hull of a very large planar point set, we can use the Fast Rampart Searching Algorithm to get a very small part points as candidate point set, and then we can get the Convex Hull of the whole planar point set from the candidate point set through other algorithms. Keywords-fast rampart searching algorithm; castle theorem; convex hull; computational geometry