FLOATING-POINT ARITHMETIC FOR COMPUTATIONAL GEOMETRY PROBLEMS WITH UNCERTAIN DATA

The algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating-point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, it is not common knowledge. There is no paper that systematically discusses what can go wrong and provides simple examples for the di.erent ways in which floating-point implementations can fail. Due to this lack of examples, instructors of computational geometry have little material for demonstrating the inadequacy of floating-point arithmetic for geometric computations, students of computational geometry and implementers of geometric algorithms still underestimate the seriousness of the problem, and researchers in our and neighboring disciplines still believe that simple approaches are able to overcome the problem. In this paper, we study simple algorithms for two simple geometric problems, namely computing convex hulls and triangulations of point sets, and show how they can fail and explain why they fail when executed with floating-point arithmetic.

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 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.

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.

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.

In this paper we use interval arithmetic tools for the computation of componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces. We define a mapping for which under certain assumptions we can construct an interval vector which is mapped into itself. Using Brouwer's fixed point theorem we conclude the existence of a solution of the original equation in this interval vector. Under different assumptions we can construct an interval vector such that the range of the mapping has no point in common with this interval vector. This implies that there is no solution in this interval vector. Furthermore we consider an iteration method which improves componentwise errorbounds for a solution of a quadratic. The theoretical results of this paper are demonstrated by some numerical examples using the algebraic eigenvalue problem which is probably the best known example of a quadratic equation.

We examine several methods for numerically solving linear equations that arise in the study of Markov chains. These methods are Gaussian elimination, state-reduction, closed-form matrix solutions, and some hybrid methods. The emphasis is on moments of first-passage times and times to absorption. We compare the methods on the basis of accuracy and computation. We conclude that state-reduction is the most accurate and that the matrix solutions have the least computation time. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Numerical solution of linear equations with Toeplitz and Vector Toeplitz matrices

17 - Engineering mathematics

The first step in the analysis of a complex structure is typically a spatial discretization of the continuum equations into a finite element, finite difference or a similar model. The analysis problem then requires the solution of algebraic equations (static response), algebraic eigenvalue problems (buckling or vibration) or ordinary differential equations (transient response). The sensitivity calculation is then equivalent to the mathematical problem of obtaining the derivatives of the solutions of those equations with respect to their coefficients. This is the main subject of the present chapter.

This paper investigates the stress distribution on the teeth and jaw during bruxism that may provide additional insights to the consequences of bruxism, splint design considerations, a basis for possible therapy protocols, and a baseline for future bruxism studies. A three-dimensional (3D) solid model of a complete jaw comprised of teeth, periodontal ligaments, and maxilla bone is reconstructed based on a Cone Beam Computed Tomography (CBCT) scan of a patient. A finite element model of the jaw is then constructed using the geometry imported from the solid model that has been reworked to remove imperfections during reconstruction and to allow proper contact between teeth. A linear elastic finite element analyses were then performed to simulate the bruxism phenomenon which includes biting and grinding. Numerical analysis is conducted only to the maxilla, with forces are placed on the teeth’s surface where contact occurs. The analysis is conducted for all stages of bruxism and possible movements patterns. Results suggest that high stresses occur during grinding, indicating a strong correlation between bruxism and dental issues such as tooth wear and alveolar resorption.

The number of guillotine partitions in d dimensions

COMBINATORIAL AND DISCRETE GEOMETRY Finite Point Configurations, J. Pach Packing and Covering, G. Fejes Toth Tilings, D. Schattschneider and M. Senechal Helly-Type Theorems and Geometric Transversals, R. Wenger Pseudoline Arrangements, J.E. Goodman Oriented Matroids, J. Richter-Gebert and G.M. Ziegler Lattice Points and Lattice Polytopes, A. Barvinok New! Low-Distortion Embeddings of Finite Metric Spaces, P. Indyk and J. Matousek New! Geometry and Topology of Polygonal Linkages, R. Connelly and E.D. Demaine New! Geometric Graph Theory, J. Pach Euclidean Ramsey Theory, R.L. Graham Discrete Aspects of Stochastic Geometry, R. Schneider Geometric Discrepancy Theory and Uniform Distribution, J.R. Alexander, J. Beck, and W.W.L. Chen Topological Methods, R.T. Zivaljevic Polyominoes, S.W. Golomb and D.A. Klarner POLYTOPES AND POLYHEDRA Basic Properties of Convex Polytopes, M. Henk, J. Richter-Gebert, and G.M. Ziegler Subdivisions and Triangulations of Polytopes, C.W. Lee Face Numbers of Polytopes and Complexes, L.J. Billera and A. Bjoerner Symmetry of Polytopes and Polyhedra, E. Schulte Polytope Skeletons and Paths, G. Kalai Polyhedral Maps, U. Brehm and E. Schulte ALGORITHMS AND COMPLEXITY OF FUNDAMENTAL GEOMETRIC OBJECTS Convex Hull Computations, R. Seidel Voronoi Diagrams and Delaunay Triangulations, S. Fortune Arrangements, D. Halperin Triangulations and Mesh Generation, M. Bern Polygons, J. O'Rourke and S. Suri Shortest Paths and Networks, J.S.B. Mitchell Visibility, J. O'Rourke Geometric Reconstruction Problems, S.S. Skiena New! Curve and Surface Reconstruction, T.K. Dey Computational Convexity, P. Gritzmann and V. Klee Computational Topology, G. Vegter Computational Real Algebraic Geometry, B. Mishra GEOMETRIC DATA STRUCTURES AND SEARCHING Point Location, J. Snoeyink New! Collision and Proximity Queries, M.C. Lin and D. Manocha Range Searching, P.K. Agarwal Ray Shooting and Lines in Space, M. Pellegrini Geometric Intersection, D.M. Mount New! Nearest Neighbors in High-Dimensional Spaces, P. Indyk COMPUTATIONAL TECHNIQUES Randomization and Derandomization, O. Cheong, K. Mulmuley, and E. Ramos Robust Geometric Computation, C.K. Yap Parallel Algorithms in Geometry, M.T. Goodrich Parametric Search, J.S. Salowe New! The Discrepancy Method in Computational Geometry, B. Chazelle APPLICATIONS OF DISCRETE AND COMPUTATIONAL GEOMETRY Linear Programming, M. Dyer, N. Megiddo, and E. Welzl Mathematical Programming, M.H. Todd Algorithmic Motion Planning, M. Sharir Robotics, D. Halperin, L.E. Kavraki, and J.-C. Latombe Computer Graphics, D. Dobkin and S. Teller New! Modeling Motion, L.J. Guibas Pattern Recognition, J. O'Rourke and G.T. Toussaint Graph Drawing, R. Tamassia and G. Liotta Splines and Geometric Modeling, C.L. Bajaj New! Surface Simplification and 3D Geometry Compression, J. Rossignac Manufacturing Processes, R. Janardan and T.C. Woo Solid Modeling, C.M. Hoffmann New! Computation of Robust Statistics: Depth, Median, and Related Measures, P.J. Rousseeuw and A. Struyf New! Geographic Information Systems, M. van Kreveld Geometric Application of the Grassmann-Cayley Algebra, N.L. White Rigidity and Scene Analysis, W. Whiteley Sphere Packing and Coding Theory, G.A. Kabatiansky and J.A. Rush Crystals and Quasicrystals, M. Senechal New! Biological Applications of Computational Topology, H. Edelsbrunner New! GEOMETRIC SOFTWARE Software, J. Joswig Two Computation Geometry Libraries: LEDA and CGAL, L. Kettner and S. Naher Index of Defined Terms New! Index of Cited Authors

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