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• Part 1 Preliminaries: fields
• vector spaces
• linear transformations
• cosets of a vector space
• invariant subspaces. Part 2 Symmetric bilinear forms: symmetric bilinear forms
• congruence
• orthogonal complements
• orthogonal bases
• Witt's cancellation theorem
• isotropic and anisotropic spaces
• functions on inner product spaces. Part 3 Plane geometries: the affine plane
• the affine group
• postulates for the Euclidean plane
• inner product planes
• projective planes
• conic sections. Part 4 Homogeneous spaces in Rn: topological groups
• homogeneous spaces
• geometry on homogeneous spaces
• the Riemann sphere
• the Poincare upper half-plane
• differentiable manifolds. Part 5 Topics in computer graphics: a first graphics programme
• a computer graphics system overview
• geometric mappings in a CG system
• the line-drawing algorithm
• the wing-edge object representation
• the conic sections
• Bezier curves and B-splines
• hidden surface removal
• texture mapping
• quadric intermediate surfaces
• Koch systems. Appendices: equivalence relations - basics
• the Jordan canonical form - proof of Jordan's theorem
• GraphLib documentation - types, procedures and functions.
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Stressing the interplay between theory and its practice, this text presents the construction of linear models that satisfy geometric postulate systems and develops geometric topics in computer graphics. It includes a computer graphics utility library of specialized subroutines on a 3.5" disk, designed for use with Turbo PASCAL 4.0 (or later version) - an effective means of computer-aided instruction for writing graphics problems. Providing instructors with maximum flexibility that allows for the mathematics or computer graphics sections to be taught independently, this book: reviews linear algebra and notation, focusing on ideas of geometric significance that are often omitted in general purpose linear algebra courses; develops symmetric bilinear forms through classical results, including the inertia theorem, Witt's cancellation theorem and the unitary diagonalization of symmetric matrices; examines the Klein Erlanger programm, constructing models of geometries, and studying associated transformation groups; clarifies how to construct geometries from groups, encompassing topological notions; and introduces topics in computer graphics, including geometric modeling, surface rendering and transformation groups.
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Verdict is a collection of subroutines for evaluating the geometric qualities of triangles, quadrilaterals, tetrahedra, and hexahedra using a variety of metrics. A metric is a real number assigned to one of these shapes depending on its particular vertex coordinates. These metrics are used to evaluate the input to finite element, finite volume, boundary element, and other types of solvers that approximate the solution to partial differential equations defined over regions of space. The geometric qualities of these regions is usually strongly tied to the accuracy these solvers are able to obtain in their approximations. The subroutines are written in C++ and have a simple C interface. Each metric may be evaluated individually or in combination. When multiple metrics are evaluated at once, they share common calculations to lower the cost of the evaluation.

The Common Geometry Module (CGM) is a code library which provides geometry functionality used for mesh generation and other applications. This functionality includes that commonly found in solid modeling engines, like geometry creation, query and modification; CGM also includes capabilities not commonly found in solid modeling engines, like geometry decomposition tools and support for shared material interfaces. CGM is built upon the ACIS solid modeling engine, but also includes geometry capability developed beside and on top of ACIS. CGM can be used as-is to provide geometry functionality for codes needing this capability. However, CGM can also be extended using derived classes in C++, allowing the geometric model to serve as the basis for other applications, for example mesh generation. CGM is supported on Sun Solaris, SGI, HP, IBM, DEC, Linux and Windows NT platforms. CGM also includes support for loading ACIS models on parallel computers, using MPI-based communication. Future plans for CGM are to port it to different solid modeling engines, including Pro/Engineer or SolidWorks. CGM is being released into the public domain under an LGPL license; the ACIS-based engine is available to ACIS licensees on request.

• Ch. 1. Geometry, Algebra, and Algorithms. 1. Polynomials and Affine Space. 2. Affine Varieties. 3. Parametrizations of Affine Varieties. 4. Ideals. 5. Polynomials of One Variable
• Ch. 2. Groebner Bases. 2. Orderings on the Monomials in [actual symbol not reproducible]. 3. A Division Algorithm in [actual symbol not reproducible]. 4. Monomial Ideals and Dickson's Lemma. 5. The Hilbert Basis Theorem and Groebner Bases. 6. Properties of Groebner Bases. 7. Buchberger's Algorithm. 8. First Applications of Groebner Bases. 9. (Optional) Improvements on Buchberger's Algorithm.

This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960s. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years and new algorithms, coupled with the power of fast computers, have led to some interesting applications, for example in robotics and in geometric theorem proving. This book is an introduction to algebraic geometry and commutative algebra aimed primarily at undergraduates. Emphasizing applications and the computational and algorithmic aspects of the subject, the text has less abstract flavour than standard treatments. With few prerequisites, it should also a useful introduction to the subject for computer scientists.
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Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. Contains a new section on Axiom and an update about MAPLE, Mathematica and REDUCE.
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• Introduction and Overview
• Geometric Fundamentals
• Polytopes and Polyhedra
• Linear Programming
• Computation of Convex Hulls
• Voronoi Diagrams
• Delone Triangulations
• Algebraic and Geometric Foundations
• Gröbner Bases and Buchberger's Algorithm
• Solving Systems of Polynomial Equations Using Gröbner Bases
• Reconstruction of Curves
• Plücker coordinates and lines in space
• Applications of non-linear computational geometry.

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 Grobner 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.
(source: Nielsen Book Data)

Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two. Discrete and Computational Geometry offers a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincare conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language. This richly illustrated textbook also features numerous exercises and unsolved problems. The essential introduction to discrete and computational geometry Covers traditional topics as well as new and advanced material Features numerous full-color illustrations, exercises, and unsolved problems Suitable for sophomores in mathematics, computer science, engineering, or physics Rigorous but accessible An online solutions manual is available (for teachers only). To obtain access, please e-mail: Vickie_Kearn@press.princeton.edu.
(source: Nielsen Book Data)

• Computational Geometry: Introduction.- Line Segment Intersection: Thematic Map Overlay.- Polygon Triangulation: Guarding an Art Gallery.- Linear Programming: Manufacturing with Molds.- Orthogonal Range Searching: Querying a Database.- Point Location: Knowing Where You Are.- Voronoi Diagrams: The Post Office Problem.- Arrangements and Duality: Supersampling in Ray Tracing.- Delaunay Triangulations: Height Interpolation.- More Geometric Data Structures: Windowing.- Convex Hulls: Mixing Things.- Binary Space Partitions: The Painter's Algorithm.- Robot Motion Planning: Getting Where You Want to Be.- Quadtrees: Non-Uniform Mesh Generation.- Visibility Graphs: Finding the Shortest Route.- Simplex Range Searching: Windowing Revisited.- Bibliography.- Index.
• (source: Nielsen Book Data)

This well-accepted introduction to computational geometry is a textbook for high-level undergraduate and low-level graduate courses. The focus is on algorithms and hence the book is well suited for students in computer science and engineering. Motivation is provided from the application areas: all solutions and techniques from computational geometry are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. For students, this motivation will be especially welcome. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement. All the basic techniques and topics from computational geometry, as well as several more advanced topics, are covered. The book is largely self-contained and can be used for self-study by anyone with a basic background in algorithms. In the second edition, besides revisions to the first edition, a number of new exercises have been added.
(source: Nielsen Book Data)

In this introduction to computational geometry the the text focuses on algorithms. All solutions and techniques described in the text are from computational geometry and are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems.
(source: Nielsen Book Data)

• Contents: Introduction.- Geometric Searching.- Convex Hulls: Basic Algorithms.- Convex Hulls: Extensions and Applications.- Proximity: Fundamental Algorithms.- Proximity: Variants and Generalizations.- Intersections.- The Geometry of Rectangles.- References.- Author Index.- Subject Index.
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From the reviews: 'This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two.''...is remarkable book is a comprehensive and systematic study on research results obtained especially in the last ten years. The very clear presentation concentrates on basic ideas, fundamental combinatorial structures, and crucial algorithmic techniques. The plenty of results is clever organized following these guidelines and within the framework of some detailed case studies. A large number of figures and examples also aid the understanding of the material.Therefore, it can be highly recommended as an early graduate text but it should prove also to be essential to researchers and professionals in applied fields of computer-aided design, computer graphics, and robotics' - "Biometrical Journal".
(source: Nielsen Book Data)

• How solid is solid modeling?
• Robustness issues in geometric algorithms
• Implementing geometric algorithms robustly
• Robustness in geometric algorithms
• Applications of computational geometry in mechanical engineering design and manufacture
• On some applications of computational geometry in manufacturing and virtual environments
• Visualizing geometric algorithms
• State of the art
• Geometric algorithm visualization, current status and future
• Position paper for panel discussion
• Designing the computational geometry algorithms library CGAL
• The computational geometry impact task force report: An executive summary
• Geometric manipulation of flexible ligands
• Ray-representation formalism for geometric computations on protein solid models
• Column-based strip packing using ordered and compliant containment
• Computing a flattest, undercut-free parting line for a convex polyhedron, with application to mold design
• Geometric problems in machine learning
• Matching convex polygons and polyhedra, allowing for occlusion
• Stably placing piecewise smooth objects
• A beam-tracing algorithm for prediction of indoor radio propagation
• Extracting geometric information from architectural drawings
• Using the visibility complex for radiosity computation
• The CGAL kernel: A basis for geometric computation
• Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator.

Content Description #Anthology selected from contributions to the First ACM Workshop on Applied Computational Geometry.#Includes bibliographical references and index.

• On 5-Gons and 5-Holes / Oswin Aichholzer, Thomas Hackl and Birgit Vogtenhuber
• On Reversibility among Parallelohedra / Jin Akiyama, Ikuro Sato and Hyunwoo Seong
• A History of Flips in Combinatorial Triangulations / Prosenjit Bose and Sander Verdonschot
• Tangled Thrackles / János Pach, Radoš Radoičić and Géza Tóth
• Open Guard Edges and Edge Guards in Simple Polygons / Csaba D. Tóth, Godfried T. Toussaint and Andrew Winslow
• String-Wrapped Rotating Disks / Joseph O'Rourke
• The Chromatic Number of the Convex Segment Disjointness Graph / Ruy Fabila-Monroy and David R. Wood
• Continuous Flattening of Convex Polyhedra / Jin-ichi Itoh, Chie Nara and Costin Vîlcu
• Convexifying Monotone Polygons while Maintaining Internal Visibility / Oswin Aichholzer, Mario Cetina, Ruy Fabila-Monroy, Jesús Leaños and Gelasio Salazar, et al.
• On the Number of Radial Orderings of Colored Planar Point Sets / José M. Díaz-Báñez, Ruy Fabila-Monroy and Pablo Pérez-Lantero
• Notes on the Twisted Graph / Elsa Omaña-Pulido and Eduardo Rivera-Campo
• Locating a Service Facility and a Rapid Transit Line / José Miguel Díaz-Báñez, Matias Korman, Pablo Pérez-Lantero and Inmaculada Ventura
• Simultaneously Flippable Edges in Triangulations / Diane L. Souvaine, Csaba D. Tóth and Andrew Winslow.

• Spiral Serpentine Polygonization of a Planar Point Set / Justin Iwerks and Joseph S.B. Mitchell
• The 1-Center and 1-Highway Problem / José Miguel Díaz-Báñez, Matias Korman, Pablo Pérez-Lantero and Inmaculada Ventura
• Compact Grid Representation of Graphs / José Cáceres, Carmen Cortés, Clara Isabel Grima, Masahiro Hachimori and Alberto Márquez, et al.
• On the Heaviest Increasing or Decreasing Subsequence of a Permutation, and Paths and Matchings on Weighted Point Sets / Toshinori Sakai and Jorge Urrutia
• A Generalization of the Source Unfolding of Convex Polyhedra / Erik D. Demaine and Anna Lubiw
• Large Angle Crossing Drawings of Planar Graphs in Subquadratic Area / Patrizio Angelini, Giuseppe Di Battista, Walter Didimo, Fabrizio Frati and Seok-Hee Hong, et al.
• Connecting Red Cells in a Bicolour Voronoi Diagram / Manuel Abellanas, Antonio L. Bajuelos, Santiago Canales, Mercè Claverol and Gregorio Hernández, et al.
• Covering Islands in Plane Point Sets / Ruy Fabila-Monroy and Clemens Huemer
• Rectilinear Convex Hull with Minimum Area / Carlos Alegría-Galicia, Tzolkin Garduño, Areli Rosas-Navarrete, Carlos Seara and Jorge Urrutia
• Separated Matchings and Small Discrepancy Colorings / Viola Mészáros
• A Note on the Number of Empty Triangles / Alfredo García
• Meshes Preserving Minimum Feature Size / Greg Aloupis, Erik D. Demaine, Martin L. Demaine, Vida Dujmović and John Iacono
• Geometric Graphs in the Plane Lattice / Mikio Kano and Kazuhiro Suzuki.

This Festschrift volume is published in honor of Ferran Hurtado on the occasion of his 60th birthday; it contains extended versions of selected communications presented at the XIV Spanish Meeting on Computational Geometry, held at the University of Alcala, Spain, in June 2011. Ferran Hurtado has played a central role in the Spanish community of Computational Geometry since its very beginning, and the quantity and quality of the international participants in the conference is an indisputable proof of his relevance in the international level. The 26 revised full papers were carefully reviewed and selected from numerous submissions. The papers present original research in computational geometry, in its broadest sense. Topics included are discrete and combinatorial geometry, linear programming applied to geometric problems, geometric algorithms and data structures, theoretical foundations of computational geometry, questions of interest in the implementation of geometric algorithms, and applications of computational geometry.
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• Functions and Transformations.- Functions and Graphs.- Rigid Motions (Isometries).- Affine and Projective Transformations.- Mobius Transformations.- Curves and Surfaces.- Examples of Curves.- Geometry of Curves.- Geometry of Surfaces.- Examples of Surfaces.- Piecewise Curves and Surfaces.
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This text on geometry is devoted to various central geometrical topics including: graphs of functions, transformations, (non-)Euclidean geometries, curves and surfaces as well as their applications in a variety of disciplines. This book presents elementary methods for analytical modeling and demonstrates the potential for symbolic computational tools to support the development of analytical solutions. The author systematically examines several powerful tools of MATLAB(R) including 2D and 3D animation of geometric images with shadows and colors and transformations using matrices. With over 150 stimulating exercises and problems, this text integrates traditional differential and non-Euclidean geometries with more current computer systems in a practical and user-friendly format. This text is an excellent classroom resource or self-study reference for undergraduate students in a variety of disciplines.
(source: Nielsen Book Data)

• Preface.-
• 1. Fundamentals.-
• 2. Models of Line Space.-
• 3. Linear Complexes.-
• 4. Approximation in Line Space.-
• 5. Ruled Surfaces.-
• 6. Developable Surfaces.-
• 7. Line Congruences and Line Complexes.-
• 8. Linear Line Mappings - Computational Kinematics.- References.- List of Symbols.- Index.- Color Plates.
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• Fundamentals.- Models of Line Space.- Linear Complexes.- Approximation in Line Space.- Ruled Surfaces.- Developable Surfaces.- Line Congruences and Line Complexes.- Linear Line Mappings - Computational Kinematics.
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The geometry of lines occurs naturally in such different areas as sculptured surface machining, computation of offsets and medial axes, surface reconstruction for reverse engineering, geometrical optics, kinematics and motion design, and modeling of developable surfaces. This book covers line geometry from various viewpoints and aims towards computation and visualization. Besides applications, it contains a tutorial on projective geometry and an introduction into the theory of smooth and algebraic manifolds of lines. It will be useful to researchers, graduate students, and anyone interested either in the theory or in computational aspects in general, or in applications in particular.
(source: Nielsen Book Data)
The geometry of lines occurs naturally in such different areas as sculptured surface machining, computation of offsets and medial axes, surface reconstruction for reverse engineering, geometrical optics, kinematics and motion design, and modeling of developable surfaces. This book covers line geometry from various viewpoints and aims towards computation and visualization. Besides applications, it contains a tutorial on projective geometry and an introduction into the theory of smooth and algebraic manifolds of lines. It will be useful to researchers, graduate students, and anyone interested either in the theory or in computational aspects in general, or in applications in particular. From the reviews : "The authors have combined results from the classical parts of geometry with computational methods. This results in a unique and fascinating blend, which is shown to be useful for a variety of applications, including robotics, geometrical optics, computer animation, and geometric design. The contents of the book are visualized by a wealth of carefully chosen illustrations, making the book a sheer pleasure to read, or even just browse in. The book will help to bring the concepts and techniques of line geometry, which have been shown to be useful for various applications in geometric design and engineering, to the attention of a wider audience." B.Juttler, MATHEMATICAL REVIEWS Clippings 2002f "...There is a vast amount of fascinating geometry of all sorts in this book. The topics are perhaps somewhat eclectic - they mirror the primary interests of the authors - but, because the motivation is to develop the geometry that applies to real world problems, the subject is far from monolithic and is open to interpretation. The ideas here build up layer upon layer. In the end, the authors have been mostly successful in sustaining their central theme, despite the need to weave together projective, differential, algebraic and metric geometry. They have also presented the mathematics in a predominantly modern way. That is important because there exist in the engineering literature archaeological remnants of outdated notation and concepts. [...] The large number (264) of line diagrams are of very good quality and considerably enhance one's understanding. [...] a book which is without doubt an important contribution to this growing branch of geometrical research." P. Donelan - New Zealand Mathematical Society Newsletter 87, 2003 "...Overall I recommend this text to anyone who wants to learn about line geometry, projective geometry and the geometric side of some algebra. The book fills a niche that has been neglected for long and should benefit researchers interested in geometric methods...It covers a body of knowledge that is underrepresented in the literature and deserves to be known more widely. The authors wrote a clearly developed and beautifully illustrated book that fills a gaping hole in the contemporary literature." ACM SIGACT News 36:3, 2005.
(source: Nielsen Book Data)

• Computational Geometry: Introduction.- Line Segment Intersection: Thematic Map Overlay.- Polygon Triangulation: Guarding an Art Gallery.- Linear Programming: Manufacturing with Molds.- Orthogonal Range Searching: Querying a Database.- Point Location: Knowing Where You Are.- Voronoi Diagrams: The Post Office Problem.- Arrangements and Duality: Supersampling in Ray Tracing.- Delaunay Triangulations: Height Interpolation.- More Geometric Data Structures: Windowing.- Convex Hulls: Mixing Things.- Binary Space Partitions: The Painter's Algorithm.- Robot Motion Planning: Getting Where You Want to Be.- Quadtrees: Non-Uniform Mesh Generation.- Visibility Graphs: Finding the Shortest Route.- Simplex Range Searching: Windowing Revisited.- Bibliography.- Index.
• (source: Nielsen Book Data)

This well-accepted introduction to computational geometry is a textbook for high-level undergraduate and low-level graduate courses. The focus is on algorithms and hence the book is well suited for students in computer science and engineering. Motivation is provided from the application areas: all solutions and techniques from computational geometry are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. For students this motivation will be especially welcome. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement. All the basic techniques and topics from computational geometry, as well as several more advanced topics, are covered. The book is largely self-contained and can be used for self-study by anyone with a basic background in algorithms. In this third edition, besides revisions to the second edition, new sections discussing Voronoi diagrams of line segments, farthest-point Voronoi diagrams, and realistic input models have been added.
(source: Nielsen Book Data)