Geometry of Frameworks

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 graduate-level textbook for a course on the geometry of frameworks or as a reference for researchers in geometry, combinatorics, and related applied fields.