We now turn to chains—and more generally linkages—that restrict joint motions in someway. Many physical linkages have some type of joint-angle constraint, for example, robot arms (Figure 1.2, p. 12). Among the variety of possible constraints, we concentrate in this section on just two: keeping all angles between incident links fixed, and keeping all angles convex. Both have applications, and both have led to a rich collection of results.

FIXED-ANGLE LINKAGES

Introduction and motivation. This section is concerned with fixed-angle linkages: linkages with a fixed angle assigned between each pair of incident edges. (We'll use the term linkage to include both general and fixed-angle linkages.) Fixed-angle chains are of particular interest, because, as mentioned in Section 1.2.4 (p. 14), they may serve as models for protein backbones: each vertex models an atom, and each link an atom-to atom bond. The backbone of the protein (ignoring the “side chains”) is an open chain. The bond angles are nearly fixed, leaving one degree of spinning motion. As this is the primary motivation for fixed-angle chains, we will disallow self-crossings to match the physical constraints. The folding behavior of fixed-angle chains is of intense current interest, constituting the unsolved “protein folding problem.” We will more directly address protein folding via fixed-angle chains in Section 9.1.

Fixed-angle trees arise in the same protein models, either when the side chains cannot be ignored, or for so-called “star” or “dentritic” polymers (Frank-Kamenetskii 1997; Soteros and Whittington 1988).