Modern Geometries

Non-Euclidean, Projective, and Discrete

Author: Michael Henle

Publisher: Pearson College Division

ISBN: 9780130323132

Category: Mathematics

Page: 389

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Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.

Computer Graphics Through OpenGL®

From Theory to Experiments

Author: Sumanta Guha

Publisher: CRC Press

ISBN: 0429874839

Category: Computers

Page: 732

View: 3842

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COMPREHENSIVE COVERAGE OF SHADERS AND THE PROGRAMMABLE PIPELINE From geometric primitives to animation to 3D modeling to lighting, shading and texturing, Computer Graphics Through OpenGL®: From Theory to Experiments is a comprehensive introduction to computer graphics which uses an active learning style to teach key concepts. Equally emphasizing theory and practice, the book provides an understanding not only of the principles of 3D computer graphics, but also the use of the OpenGL® Application Programming Interface (API) to code 3D scenes and animation, including games and movies. The undergraduate core of the book takes the student from zero knowledge of computer graphics to a mastery of the fundamental concepts with the ability to code applications using fourth-generation OpenGL®. The remaining chapters explore more advanced topics, including the structure of curves and surfaces, applications of projective spaces and transformations and the implementation of graphics pipelines. This book can be used for introductory undergraduate computer graphics courses over one to two semesters. The careful exposition style attempting to explain each concept in the simplest terms possible should appeal to the self-study student as well. Features • Covers the foundations of 3D computer graphics, including animation, visual techniques and 3D modeling • Comprehensive coverage of OpenGL® 4.x, including the GLSL and vertex, fragment, tessellation and geometry shaders • Includes 180 programs with 270 experiments based on them • Contains 750 exercises, 110 worked examples, and 700 four-color illustrations • Requires no previous knowledge of computer graphics • Balances theory with programming practice using a hands-on interactive approach to explain the underlying concepts

Automated Deduction in Geometry

8th International Workshop, ADG 2010, Munich, Germany, July 22-24, 2010, Revised Papers

Author: Pascal Schreck,Julien Narboux,Jürgen Richter-Gebert

Publisher: Springer Science & Business Media

ISBN: 3642250696

Category: Computers

Page: 259

View: 2656

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This book constitutes the thoroughly refereed post-workshop proceedings of the 8th International Workshop on Automated Deduction in Geometry, ADG 2010, held in Munich, Germany in July 2010. The 13 revised full papers presented were carefully selected during two rounds of reviewing and improvement from the lectures given at the workshop. Topics addressed by the papers are incidence geometry using some kind of combinatoric argument; computer algebra; software implementation; as well as logic and proof assistants.

Topology and Its Applications

Author: William F. Basener

Publisher: John Wiley & Sons

ISBN: 1118626222

Category: Mathematics

Page: 384

View: 1266

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Discover a unique and modern treatment of topology employing across-disciplinary approach Implemented recently to understand diverse topics, such as cellbiology, superconductors, and robot motion, topology has beentransformed from a theoretical field that highlights mathematicaltheory to a subject that plays a growing role in nearly all fieldsof scientific investigation. Moving from the concrete to theabstract, Topology and Its Applications displays both the beautyand utility of topology, first presenting the essentials oftopology followed by its emerging role within the new frontiers inresearch. Filling a gap between the teaching of topology and its modernuses in real-world phenomena, Topology and Its Applications isorganized around the mathematical theory of topology, a frameworkof rigorous theorems, and clear, elegant proofs. This book is the first of its kind to present applications incomputer graphics, economics, dynamical systems, condensed matterphysics, biology, robotics, chemistry, cosmology, material science,computational topology, and population modeling, as well as otherareas of science and engineering. Many of these applications arepresented in optional sections, allowing an instructor to customizethe presentation. The author presents a diversity of topological areas, includingpoint-set topology, geometric topology, differential topology, andalgebraic/combinatorial topology. Topics within these areasinclude: Open sets Compactness Homotopy Surface classification Index theory on surfaces Manifolds and complexes Topological groups The fundamental group and homology Special "core intuition" segments throughout the book brieflyexplain the basic intuition essential to understanding severaltopics. A generous number of figures and examples, many of whichcome from applications such as liquid crystals, space probe data,and computer graphics, are all available from the publisher's Website.

Mathematics

Author: Barry Max Brandenberger

Publisher: MacMillan Reference Library

ISBN: N.A

Category: Juvenile Nonfiction

Page: 4

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This new series offers the most comprehensive views of key areas in the world of science. Each set explores all facets of the topic, offering not only descriptive and analytical information, but also cultural and ethical issues, and career opportunities in many fields of science.

Über die Hypothesen, welche der Geometrie zu Grunde liegen

Author: Bernhard Riemann,Hermann Weyl

Publisher: Springer-Verlag

ISBN: 3662248611

Category: Mathematics

Page: 50

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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Geometries

Author: Alekseĭ Bronislavovich Sosinskiĭ

Publisher: American Mathematical Soc.

ISBN: 082187571X

Category: Mathematics

Page: 301

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The book is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein's Erlangen Program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal--although some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms ``toy geometries'', the geometries of Platonic bodies, discrete geometries, and classical continuous geometries. The text is based on first-year semester course lectures delivered at the Independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean geometry, albeit those lacking this knowledge may refer to a compendium in Chapter 0. Per the author's predilection, the book contains very little regarding the axiomatic approach to geometry (save for a single chapter on the history of non-Euclidean geometry), but two Appendices provide a detailed treatment of Euclid's and Hilbert's axiomatics. Perhaps the most important aspect of this course is the problems, which appear at the end of each chapter and are supplemented with answers at the conclusion of the text. By analyzing and solving these problems, the reader will become capable of thinking and working geometrically, much more so than by simply learning the theory. Ultimately, the author makes the distinction between concrete mathematical objects called ``geometries'' and the singular ``geometry'', which he understands as a way of thinking about mathematics. Although the book does not address branches of mathematics and mathematical physics such as Riemannian and Kahler manifolds or, say, differentiable manifolds and conformal field theories, the ideology of category language and transformation groups on which the book is based prepares the reader for the study of, and eventually, research in these important and rapidly developing areas of contemporary mathematics.

Vorlesungen über Grundlagen der Geometrie

Author: Kurt Reidemeister

Publisher: Springer-Verlag

ISBN: 3662400782

Category: Mathematics

Page: 150

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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Anschauliche Geometrie

Author: David Hilbert,Stefan Cohn-Vossen

Publisher: Springer-Verlag

ISBN: 3662366851

Category: Mathematics

Page: 312

View: 6611

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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Grundlagen der Geometrie

Author: David Hilbert

Publisher: SEVERUS Verlag

ISBN: 3863479475

Category: Mathematics

Page: 248

View: 9162

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Seine Erkenntnisse beeinflussen bis heute die Forschung: David Hilbert baut in seinen „Grundlagen der Geometrie“ auf Euklids Lehre ein Grundsatzsystem auf, von dem ausgehend er wichtige geometrische Sätze ableitet. Die erstmals 1899 erschienene Abhandlung machte Hilbert zu einem der wichtigsten Mathematiker der Neuzeit, der auch den Formalismus entscheidend prägte.

Geometrie und Billard

Author: Serge Tabachnikov

Publisher: Springer-Verlag

ISBN: 3642319254

Category: Mathematics

Page: 165

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Wie bewegt sich ein Massenpunkt in einem Gebiet, an dessen Rand er elastisch zurückprallt? Welchen Weg nimmt ein Lichtstrahl in einem Gebiet mit ideal reflektierenden Rändern? Anhand dieser und ähnlicher Fragen stellt das vorliegende Buch Zusammenhänge zwischen Billard und Differentialgeometrie, klassischer Mechanik sowie geometrischer Optik her. Dabei beschäftigt sich das Buch unter anderem mit dem Variationsprinzip beim mathematischen Billard, der symplektischen Geometrie von Lichtstrahlen, der Existenz oder Nichtexistenz von Kaustiken, periodischen Billardtrajektorien und dem Mechanismus für Chaos bei der Billarddynamik. Ergänzend wartet dieses Buch mit einer beachtlichen Anzahl von Exkursen auf, die sich verwandten Themen widmen, darunter der Vierfarbensatz, die mathematisch-physikalische Beschreibung von Regenbögen, der poincaresche Wiederkehrsatz, Hilberts viertes Problem oder der Schließungssatz von Poncelet.​

Elementare Differentialgeometrie

Author: Christian Bär

Publisher: Walter de Gruyter

ISBN: 3110224593

Category: Mathematics

Page: 356

View: 1519

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This textbook presents an introduction to the differential geometry of curves and surfaces. This second, revised edition has been expanded to include solutions and applications in cartography. Topics include Euclidean geometry, curve theory, surface theory, curvature concepts, minimal surfaces, Riemann geometry and the Gauss-Bonnet theorem.

Analysis II

Author: Wolfgang Walter

Publisher: Springer-Verlag

ISBN: 3642967922

Category: Mathematics

Page: 398

View: 4401

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Dem erfolgreichen Konzept von Analysis I folgend, wird auch im zweiten Teil dieses zweibändigen Analysis-Werkes viel Wert auf historische Zusammenhänge, Ausblicke und die Entwicklung der Analysis gelegt. Zu den Besonderheiten, die über den kanonischen Stoff des zweiten und dritten Semesters einer Analysisvorlesung hinausgehen, gehört das Lemma von Marston Morse. Die Grundtatsachen über die verschiedenen Integralbegriffe werden allesamt aus Sätzen über verallgemeinerte Limites (Moore-Smith-Konvergenz) abgeleitet. Die C?-Approximation von Funktionen (Friedrich Mollifiers) wird ebenso behandelt, wie die Theorie der absolut stetigen Funktionen. Bei den Fourierreihen wird die klassische Theorie in Weiterführung einer von Chernoff und Redheffer entwickelten Methode behandelt. Zahlreiche Beispiele, Übungsaufgaben und Anwendungen, z.B. aus der Physik und Astronomie runden dieses Lehrbuch ab.