Alexander I. Bobenko TU Berlin's Books

Painleve Equations in the Differential Geometry of Surfaces

By: Alexander I. Bobenko TU Berlin,Ulrich Eitner

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Painleve Equations in the Differential Geometry of Surfaces

By: Alexander I. Bobenko TU Berlin,Ulrich Eitner

This book brings together two different branches of mathematics: the theory of Painlev and the theory of surfaces. Self-contained introductions to both these fields are presented. It is shown how some classical problems in surface theory can be solved using the modern theory of Painlev equations. In particular, an essential part of the book is devoted to Bonnet surfaces, i.e. to surfaces possessing families of isometries preserving the mean curvature function. A global classification of Bonnet surfaces is given using both ingredients of the theory of Painlev equations: the theory of isomonodromic deformation and the Painlev property. The book is illustrated by plots of surfaces. It is intended to be used by mathematicians and graduate students interested in differential geometry and Painlev equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.

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Page Count

125

Publisher

Springer

Published Date

ISBN 10

3540444521

ISBN 13

9783540444527

Computational Approach to Riemann Surfaces

By: Alexander I. Bobenko

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Computational Approach to Riemann Surfaces

By: Alexander I. Bobenko

This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces. Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics.

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268

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Springer Science & Business Media

Published Date

ISBN 10

3642174124

ISBN 13

9783642174124

Non-Euclidean Laguerre Geometry and Incircular Nets

By: Alexander I. Bobenko,Carl O.R. Lutz,Helmut Pottmann,Jan Techter

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Non-Euclidean Laguerre Geometry and Incircular Nets

By: Alexander I. Bobenko,Carl O.R. Lutz,Helmut Pottmann,Jan Techter

This textbook is a comprehensive and yet accessible introduction to non-Euclidean Laguerre geometry, for which there exists no previous systematic presentation in the literature. Moreover, we present new results by demonstrating all essential features of Laguerre geometry on the example of checkerboard incircular nets. Classical (Euclidean) Laguerre geometry studies oriented hyperplanes, oriented hyperspheres, and their oriented contact in Euclidean space. We describe how this can be generalized to arbitrary Cayley-Klein spaces, in particular hyperbolic and elliptic space, and study the corresponding groups of Laguerre transformations. We give an introduction to Lie geometry and describe how these Laguerre geometries can be obtained as subgeometries. As an application of two-dimensional Lie and Laguerre geometry we study the properties of checkerboard incircular nets.

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Page Count

142

Publisher

Springer Nature

Published Date

ISBN 10

3030818470

ISBN 13

9783030818470

Discrete Differential Geometry

Integrable Structure

By: Alexander I. Bobenko,Yuri B. Suris

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Discrete Differential Geometry

Integrable Structure

By: Alexander I. Bobenko,Yuri B. Suris

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.

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Page Count

432

Publisher

American Mathematical Society

Published Date

ISBN 10

1470474565

ISBN 13

9781470474560

Computational Approach to Riemann Surfaces

By: Alexander I. Bobenko TU Berlin,Christian Klein

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Computational Approach to Riemann Surfaces

By: Alexander I. Bobenko TU Berlin,Christian Klein

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268

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Springer

Published Date

ISBN 10

3642174132

ISBN 13

9783642174131

Book's by Alexander I. Bobenko TU Berlin

Painleve Equations in the Differential Geometry of Surfaces

Painleve Equations in the Differential Geometry of Surfaces

Computational Approach to Riemann Surfaces

Computational Approach to Riemann Surfaces

Non-Euclidean Laguerre Geometry and Incircular Nets

Non-Euclidean Laguerre Geometry and Incircular Nets

Discrete Differential Geometry

Discrete Differential Geometry

Computational Approach to Riemann Surfaces

Computational Approach to Riemann Surfaces

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