Book contents
- Frontmatter
- Contents
- Preface
- 0 Introduction
- Part I Linkages
- 1 Problem Classification and Examples
- 2 Upper and Lower Bounds
- 3 Planar Linkage Mechanisms
- 4 Rigid Frameworks
- 5 Reconfiguration of Chains
- 6 Locked Chains
- 7 Interlocked Chains
- 8 Joint-Constrained Motion
- 9 Protein Folding
- Part II Paper
- Part III Polyhedra
- Bibliography
- Index
4 - Rigid Frameworks
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Contents
- Preface
- 0 Introduction
- Part I Linkages
- 1 Problem Classification and Examples
- 2 Upper and Lower Bounds
- 3 Planar Linkage Mechanisms
- 4 Rigid Frameworks
- 5 Reconfiguration of Chains
- 6 Locked Chains
- 7 Interlocked Chains
- 8 Joint-Constrained Motion
- 9 Protein Folding
- Part II Paper
- Part III Polyhedra
- Bibliography
- Index
Summary
The area of rigidity theory studies a special class of problems about linkages: can the linkage move at all? This seemingly simple question has a deep theory behind it, with many variations and characterizations.
This chapter gives an overview of some of the key results in 2D rigidity theory, with a focus on the results that we need elsewhere in the book, in particular, in our study of locked linkages in Section 6.6 (p. 96). A full survey is beyond our scope; see Connelly (1993), Graver (2001), Graver et al. (1993), and Whiteley (2004) for more information.
BRIEF HISTORY
While rigidity theory fell out of prominence for the first half of the twentieth century, since then there has been increasing interest and connections made to applications. Perhaps the first result that could be identified as in this area was Augustin-Louis Cauchy's theorem from 1813 on the rigidity of convex polyhedra, settling part of a problem posed by Leonhard Euler in 1766. This and related results on polyhedral rigidity are covered in Section 8.2 (p. 143) and in Part III, Section 23.1 (p. 341). Another early result in the area is James Clerk Maxwell's work from 1864 on planar frameworks, and it is here that the development in this chapter culminates. In some sense, we follow backward in time, building on the new infrastructure and deep understanding of the area, to make it easier to reach some older and more difficult results.
- Type
- Chapter
- Information
- Geometric Folding AlgorithmsLinkages, Origami, Polyhedra, pp. 43 - 58Publisher: Cambridge University PressPrint publication year: 2007