**Graduate Studies in Mathematics**

Volume: 98;
2008;
404 pp;
Hardcover

MSC: Primary 53;
Secondary 51; 37; 39; 52

Print ISBN: 978-0-8218-4700-8

Product Code: GSM/98

List Price: $75.00

Individual Member Price: $60.00

**Electronic ISBN: 978-1-4704-1793-2
Product Code: GSM/98.E**

List Price: $75.00

Individual Member Price: $60.00

#### Supplemental Materials

# Discrete Differential Geometry: Integrable Structure

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*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.

#### Readership

Graduate students and research mathematicians interested in discrete differential geometry and its applications.

#### Reviews & Endorsements

This book gives new life to old concepts of classical differential geometry, and a beautiful introduction to new notions of discrete integrable systems. It should be of interest to researchers in several areas of mathematics (integrable systems, differential geometry, numerical approximation of special surfaces), but also to advanced students interested in a good introduction to several classical areas of mathematics. Parts of it could well be used for graduate or possibly advanced undergraduate courses in mathematics.

-- Mathematical Reviews

It can serve as a very good introduction into contemporary research and it seems to be the first book devoted to the topic. ... The book is well and clearly written.

-- EMS Newsletter

#### Table of Contents

# Table of Contents

## Discrete Differential Geometry: Integrable Structure

- Cover Cover11 free
- Title page iii5 free
- Contents v7 free
- Preface xi13 free
- Introduction xiii15 free
- Classical differential geometry 127 free
- Discretization principles. Multidimensional nets 3157
- Discretization principles. Nets in quadrics 87113
- Special classes of discrete surfaces 127153
- Approximation 187213
- Consistency as integrability 209235
- Discrete complex analysis. Linear theory 291317
- Discrete complex analysis. Integrable circle patterns 311337
- Foundations 331357
- Solutions of selected exercises 369395
- Bibliography 385411
- Notations 399425
- Index 401427 free
- Back Cover Back Cover1433