Classical And Discrete Differential Geometry

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Discrete Differential Geometry

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Author : Alexander I. Bobenko
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Discrete Differential Geometry written by Alexander I. Bobenko 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 2008 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." "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.

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.

Classical Topics In Discrete Geometry

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Author : Károly Bezdek
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-06-23

Classical Topics In Discrete Geometry written by Károly Bezdek 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 2010-06-23 with Mathematics categories.

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.

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.

Introduction To Differential Geometry

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Author : Joel W. Robbin
language : en
Publisher: Springer Nature
Release Date : 2022-01-12

Introduction To Differential Geometry written by Joel W. Robbin and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-12 with Mathematics categories.

This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point. The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.

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.

Elementary Differential Geometry

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Author : Christian Bär
language : en
Publisher: Cambridge University Press
Release Date : 2010-05-06

Elementary Differential Geometry written by Christian Bär and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-05-06 with Mathematics categories.

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

An Introduction To Differential Geometry

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Author : T. J. Willmore
language : en
Publisher: Courier Corporation
Release Date : 2013-05-13

An Introduction To Differential Geometry written by T. J. Willmore and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-05-13 with Mathematics categories.

This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.

Visual Differential Geometry And Forms

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Author : Tristan Needham
language : en
Publisher: Princeton University Press
Release Date : 2021-07-13

Visual Differential Geometry And Forms written by Tristan Needham and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-13 with Mathematics categories.

An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.