Classical And Discrete Differential Geometry

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Classical And Discrete Differential Geometry

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Author : David Xianfeng Gu
language : en
Publisher: CRC Press
Release Date : 2023-01-31

Classical And Discrete Differential Geometry written by David Xianfeng Gu and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-01-31 with Computers categories.

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.

Discrete Differential Geometry

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Author : Alexander I. Bobenko
language : en
Publisher: American Mathematical Society
Release Date : 2023-09-14

Discrete Differential Geometry written by Alexander I. Bobenko and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-14 with Mathematics categories.

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.

An Excursion Through Discrete Differential Geometry

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Author : American Mathematical Society. Short Course, Discrete Differential Geometry
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-09-02

An Excursion Through Discrete Differential Geometry written by American Mathematical Society. Short Course, Discrete Differential Geometry and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-09-02 with Education categories.

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.

Symmetries And Integrability Of Difference Equations

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Author : Decio Levi
language : en
Publisher: Springer
Release Date : 2017-06-30

Symmetries And Integrability Of Difference Equations written by Decio Levi and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-30 with Science categories.

This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.

Advances In Discrete Differential Geometry

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Author : Alexander I. Bobenko
language : en
Publisher: Springer
Release Date : 2016-08-12

Advances In Discrete Differential Geometry written by Alexander I. Bobenko and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-12 with Mathematics categories.

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.

Foundations Of Arithmetic Differential Geometry

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Author : Alexandru Buium
language : en
Publisher: American Mathematical Society
Release Date : 2023-11-20

Foundations Of Arithmetic Differential Geometry written by Alexandru Buium and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-11-20 with Mathematics categories.

The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is “intrinsically curved”; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.

Discrete Differential Geometry

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Author : Alexander I. Bobenko TU Berlin
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-03-27

Discrete Differential Geometry written by Alexander I. Bobenko TU Berlin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-03-27 with Mathematics categories.

This is the first book on a newly emerging field of discrete differential geometry providing an excellent way to access this exciting area. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. The carefully edited collection of essays gives a lively, multi-facetted introduction to this emerging field.

Differential Geometry And Lie Groups

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Author : Jean Gallier
language : en
Publisher: Springer Nature
Release Date : 2020-08-18

Differential Geometry And Lie Groups written by Jean Gallier and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-08-18 with Mathematics categories.

This textbook explores advanced topics in differential geometry, chosen for their particular relevance to modern geometry processing. Analytic and algebraic perspectives augment core topics, with the authors taking care to motivate each new concept. Whether working toward theoretical or applied questions, readers will appreciate this accessible exploration of the mathematical concepts behind many modern applications. Beginning with an in-depth study of tensors and differential forms, the authors go on to explore a selection of topics that showcase these tools. An analytic theme unites the early chapters, which cover distributions, integration on manifolds and Lie groups, spherical harmonics, and operators on Riemannian manifolds. An exploration of bundles follows, from definitions to connections and curvature in vector bundles, culminating in a glimpse of Pontrjagin and Chern classes. The final chapter on Clifford algebras and Clifford groups draws the book to an algebraic conclusion, which can be seen as a generalized viewpoint of the quaternions. Differential Geometry and Lie Groups: A Second Course captures the mathematical theory needed for advanced study in differential geometry with a view to furthering geometry processing capabilities. Suited to classroom use or independent study, the text will appeal to students and professionals alike. A first course in differential geometry is assumed; the authors’ companion volume Differential Geometry and Lie Groups: A Computational Perspective provides the ideal preparation.

Kinematic Differential Geometry And Saddle Synthesis Of Linkages

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Author : Delun Wang
language : en
Publisher: John Wiley & Sons
Release Date : 2015-05-11

Kinematic Differential Geometry And Saddle Synthesis Of Linkages written by Delun Wang and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-11 with Technology & Engineering categories.

With a pioneering methodology, the book covers the fundamental aspects of kinematic analysis and synthesis of linkage, and provides a theoretical foundation for engineers and researchers in mechanisms design. • The first book to propose a complete curvature theory for planar, spherical and spatial motion • Treatment of the synthesis of linkages with a novel approach • Well-structured format with chapters introducing clearly distinguishable concepts following in a logical sequence dealing with planar, spherical and spatial motion • Presents a pioneering methodology by a recognized expert in the field and brought up to date with the latest research and findings • Fundamental theory and application examples are supplied fully illustrated throughout

Icgg 2018 Proceedings Of The 18th International Conference On Geometry And Graphics

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Author : Luigi Cocchiarella
language : en
Publisher: Springer
Release Date : 2018-07-06

Icgg 2018 Proceedings Of The 18th International Conference On Geometry And Graphics written by Luigi Cocchiarella and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-07-06 with Technology & Engineering categories.

This book gathers peer-reviewed papers presented at the 18th International Conference on Geometry and Graphics (ICGG), held in Milan, Italy, on August 3-7, 2018. The spectrum of papers ranges from theoretical research to applications, including education, in several fields of science, technology and the arts. The ICGG 2018 mainly focused on the following topics and subtopics: Theoretical Graphics and Geometry (Geometry of Curves and Surfaces, Kinematic and Descriptive Geometry, Computer Aided Geometric Design), Applied Geometry and Graphics (Modeling of Objects, Phenomena and Processes, Applications of Geometry in Engineering, Art and Architecture, Computer Animation and Games, Graphic Simulation in Urban and Territorial Studies), Engineering Computer Graphics (Computer Aided Design and Drafting, Computational Geometry, Geometric and Solid Modeling, Image Synthesis, Pattern Recognition, Digital Image Processing) and Graphics Education (Education Technology Research, Multimedia Educational Software Development, E-learning, Virtual Reality, Educational Systems, Educational Software Development Tools, MOOCs). Given its breadth of coverage, the book introduces engineers, architects and designers interested in computer applications, graphics and geometry to the latest advances in the field, with a particular focus on science, the arts and mathematics education.