Configuration Spaces of Linkages in R^n

TL;DR

This paper explores the mathematical structure of linkage configuration spaces in R^n, demonstrating their relation to algebraic sets and characterizing computable functions, with implications for understanding mechanical linkages.

Contribution

It establishes that any compact algebraic set can be realized as a linkage configuration space and characterizes functions computable by linkages.

Findings

Configuration spaces can be isomorphic to algebraic sets.

Flexible edges allow configuration spaces to represent polynomial-defined sets.

Characterization of functions computable by linkages.

Abstract

This paper studies the configuration space of all possible positions of a linkage in R^n. For example, it shows that for every compact algebraic set, there is a linkage whose configuration space is analytically isomorphic to a finite number of copies of the algebraic set. If flexible edges are allowed, any compact set given by polynomial equalities and inequalities is the configuration space of a linkage. This paper also characterizes functions which can be computed by linkages.