Classical And Discrete Differential Geometry

Author : David Xianfeng Gu,Emil Saucan
Publisher : CRC Press
Page : 690 pages
File Size : 51,8 Mb
Release : 2023-01-31
Category : Computers
ISBN : 9781000804461

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Classical and Discrete Differential Geometry by David Xianfeng Gu,Emil Saucan Pdf

This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.

Author : American Mathematical Society. Short Course, Discrete Differential Geometry
Publisher : American Mathematical Soc.
Page : 140 pages
File Size : 53,6 Mb
Release : 2020-09-02
Category : Education
ISBN : 9781470446628

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An Excursion Through Discrete Differential Geometry by American Mathematical Society. Short Course, Discrete Differential Geometry Pdf

Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. It aims to translate concepts from classical differential geometry into a language that is purely finite and discrete, and can hence be used by algorithms to reason about geometric data. In contrast to standard numerical approximation, the central philosophy of DDG is to faithfully and exactly preserve key invariants of geometric objects at the discrete level. This process of translation from smooth to discrete helps to both illuminate the fundamental meaning behind geometric ideas and provide useful algorithmic guarantees. This volume is based on lectures delivered at the 2018 AMS Short Course ``Discrete Differential Geometry,'' held January 8-9, 2018, in San Diego, California. The papers in this volume illustrate the principles of DDG via several recent topics: discrete nets, discrete differential operators, discrete mappings, discrete conformal geometry, and discrete optimal transport.

Author : Alexander I. Bobenko,Yuri B. Suris
Publisher : American Mathematical Society
Page : 432 pages
File Size : 40,5 Mb
Release : 2023-09-14
Category : Mathematics
ISBN : 9781470474560

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Discrete Differential Geometry by Alexander I. Bobenko,Yuri B. Suris Pdf

An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question “How do we discretize differential geometry?” arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful.

Author : Alexander I. Bobenko,Yuri B. Suris
Publisher : American Mathematical Soc.
Page : 433 pages
File Size : 45,8 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821847008

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Discrete Differential Geometry by Alexander I. Bobenko,Yuri B. Suris Pdf

"An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of Integrable systems. One of the main goals of this book Is to reveal this integrable structure of discrete differential geometry." "The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question "How do we discretize differential geometry?" arising in their specific field."--BOOK JACKET.

Author : Dirk J. Struik
Publisher : Courier Corporation
Page : 254 pages
File Size : 44,6 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9780486138183

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Lectures on Classical Differential Geometry by Dirk J. Struik Pdf

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student. Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there. For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.

Author : David Xianfeng Gu,Emil Saucan
Publisher : CRC Press
Page : 589 pages
File Size : 55,8 Mb
Release : 2023-01-31
Category : Computers
ISBN : 9781000804454

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Classical and Discrete Differential Geometry by David Xianfeng Gu,Emil Saucan Pdf

This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.

Author : Alexander I. Bobenko
Publisher : Springer
Page : 441 pages
File Size : 44,6 Mb
Release : 2016-08-12
Category : Mathematics
ISBN : 9783662504475

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Advances in Discrete Differential Geometry by Alexander I. Bobenko Pdf

This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture. This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.

Author : Dirk Jan Struik
Publisher : Courier Corporation
Page : 254 pages
File Size : 52,6 Mb
Release : 1961-01-01
Category : Mathematics
ISBN : 0486656098

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Lectures on Classical Differential Geometry by Dirk Jan Struik Pdf

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student. Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there. For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.

Author : Károly Bezdek
Publisher : Springer Science & Business Media
Page : 166 pages
File Size : 41,9 Mb
Release : 2010-06-23
Category : Mathematics
ISBN : 9781441906007

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Classical Topics in Discrete Geometry by Károly Bezdek Pdf

Geometry is a classical core part of mathematics which, with its birth, marked the beginning of the mathematical sciences. Thus, not surprisingly, geometry has played a key role in many important developments of mathematics in the past, as well as in present times. While focusing on modern mathematics, one has to emphasize the increasing role of discrete mathematics, or equivalently, the broad movement to establish discrete analogues of major components of mathematics. In this way, the works of a number of outstanding mathema- cians including H. S. M. Coxeter (Canada), C. A. Rogers (United Kingdom), and L. Fejes-T oth (Hungary) led to the new and fast developing eld called discrete geometry. One can brie y describe this branch of geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. This, as a classical core part, also includes the theory of polytopes and tilings in addition to the theory of packing and covering. D- crete geometry is driven by problems often featuring a very clear visual and applied character. The solutions use a variety of methods of modern mat- matics, including convex and combinatorial geometry, coding theory, calculus of variations, di erential geometry, group theory, and topology, as well as geometric analysis and number theory.

Author : Dirk Jan STRUIK
Publisher : Unknown
Page : 221 pages
File Size : 48,7 Mb
Release : 1957
Category : Electronic
ISBN : OCLC:503836650

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Lectures on Classical Differential Geometry. (Second Printing.). by Dirk Jan STRUIK Pdf

Author : E.B. Vinberg
Publisher : Springer Science & Business Media
Page : 263 pages
File Size : 55,9 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662029015

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Geometry II by E.B. Vinberg Pdf

A very clear account of the subject from the viewpoints of elementary geometry, Riemannian geometry and group theory – a book with no rival in the literature. Mostly accessible to first-year students in mathematics, the book also includes very recent results which will be of interest to researchers in this field.

Author : K. Roly Bezdek
Publisher : Unknown
Page : 180 pages
File Size : 41,6 Mb
Release : 2010-06-25
Category : Electronic
ISBN : 1441906010

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Classical Topics in Discrete Geometry by K. Roly Bezdek Pdf

Author : Gary R. Jensen,Emilio Musso,Lorenzo Nicolodi
Publisher : Springer
Page : 571 pages
File Size : 54,9 Mb
Release : 2016-04-20
Category : Mathematics
ISBN : 9783319270760

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Surfaces in Classical Geometries by Gary R. Jensen,Emilio Musso,Lorenzo Nicolodi Pdf

Designed for intermediate graduate studies, this text will broaden students' core knowledge of differential geometry providing foundational material to relevant topics in classical differential geometry. The method of moving frames, a natural means for discovering and proving important results, provides the basis of treatment for topics discussed. Its application in many areas helps to connect the various geometries and to uncover many deep relationships, such as the Lawson correspondence. The nearly 300 problems and exercises range from simple applications to open problems. Exercises are embedded in the text as essential parts of the exposition. Problems are collected at the end of each chapter; solutions to select problems are given at the end of the book. Mathematica®, MatlabTM, and Xfig are used to illustrate selected concepts and results. The careful selection of results serves to show the reader how to prove the most important theorems in the subject, which may become the foundation of future progress. The book pursues significant results beyond the standard topics of an introductory differential geometry course. A sample of these results includes the Willmore functional, the classification of cyclides of Dupin, the Bonnet problem, constant mean curvature immersions, isothermic immersions, and the duality between minimal surfaces in Euclidean space and constant mean curvature surfaces in hyperbolic space. The book concludes with Lie sphere geometry and its spectacular result that all cyclides of Dupin are Lie sphere equivalent. The exposition is restricted to curves and surfaces in order to emphasize the geometric interpretation of invariants and other constructions. Working in low dimensions helps students develop a strong geometric intuition. Aspiring geometers will acquire a working knowledge of curves and surfaces in classical geometries. Students will learn the invariants of conformal geometry and how these relate to the invariants of Euclidean, spherical, and hyperbolic geometry. They will learn the fundamentals of Lie sphere geometry, which require the notion of Legendre immersions of a contact structure. Prerequisites include a completed one semester standard course on manifold theory.

Author : Alexander I. Bobenko
Publisher : Springer
Page : 341 pages
File Size : 46,7 Mb
Release : 2008-01-17
Category : Mathematics
ISBN : 3764386207

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Discrete Differential Geometry by Alexander I. Bobenko Pdf

Presents a collection of essays that cover the main topics of the 2004 Oberwolfach Seminar which focused on discrete differential geometry.

Author : Christian Bär
Publisher : Cambridge University Press
Page : 335 pages
File Size : 45,9 Mb
Release : 2010-05-06
Category : Mathematics
ISBN : 9780521896719

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Elementary Differential Geometry by Christian Bär Pdf

This easy-to-read introduction takes the reader from elementary problems through to current research. Ideal for courses and self-study.