This textbook provides a comprehensive and rigorous treatment of the mathematical theory underlying rigidity and the flexibility of frameworks. Integrating classical geometry, modern rigidity theory, and topological methods, the authors develop a unified perspective on how geometric constraints determine the possible motions and configurations of planar and spatial structures.

The book begins by discussing the foundations of rigid motions, infinitesimal rotations, and vector fields, establishing the analytical and algebraic tools required for later chapters. The book then advances to a systematic study of unit-bar frameworks, infinitesimal rigidity, the rigidity matrix, rigidity of graphs and the rigidity and flexibility of a polyhedral surface. Each chapter is accompanied by exercises, with complete solutions provided in the final chapter.

Throughout the book, the authors incorporate historical context, classical theorems, and modern applications in robotics and computational geometry. This book is suitable as a graduatelevel textbook for a course on the geometry of frameworks or as a reference for researchers in geometry, combinatorics, and related applied fields.

Discusses the basics of the geometry and rigidity of frameworks Offers a detailed introduction to key concepts on the rigidity of frameworks from a geometric point of view Includes solutions to selected exercises

Autorentext

Hiroshi Maehara is an emeritus professor of mathematics at Ryukyu University in Japan. He obtained his Ph.D. in mathematics from Tokyo University in 1986. He was a guest professor at the Institute of Statistical Mathematics in Tokyo from 1994 to 2005. He also held a professor position at the Research Institute of Education Development of Tokai University in Tokyo from 2009 to 2012. His main research interests lie in discrete geometry and combinatorics, and his hobby, other than mathematics, is the game of go.

Horst Martini is a distinguished retired professor of mathematics from the Chemnitz University of Technology in Germany. He obtained his Ph.D. in mathematics from the University of Dresden in 1984, followed by his habilitation from the University of Jena in 1988. His research interests include convex geometry, discrete geometry, functional analysis, and classical subfields of geometry, as well as applications in optimization and related fields. His scholarly pursuits also extend to certain fields in history of mathematics. He previously served as editor in chief of the Springer journal Contributions to Algebra and Geometry. His passions include travelling to interesting places of geographical and historical significance and studying different genres of good music.

Inhalt

Chapter 1. Motions and velocity vector felds.- Chapter 2. Motions of frameworks.- Chapter 3. Unit-bar frameworks.- Chapter 4. Infinitesimal motions.- Chapter 5. Rigidity of graphs in the plane.- Chapter 6. Rigidity of graphs in R^d.- Chapter 7. Flexible polyhedral surfaces in R^3.- Chapter 8. Addendum.- Chapter 9. Solutions to the exercises.