Planar Linkages and Algebraic Sets

TL;DR

This paper studies the configuration spaces of planar linkages and cabled linkages, classifying their possible realizations and exploring their geometric properties within a mathematical framework.

Contribution

It extends the understanding of planar linkage configuration spaces by including cabled linkages and classifying these spaces up to analytic isomorphism.

Findings

Classification of configuration spaces of cabled linkages

Extension of Kapovich-Millson's work to flexible cables

Analysis of geometric and algebraic properties of these spaces

Abstract

A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the configuration space of all planar realizations of a linkage (following work of Kapovich-Millson). We also look at configuration spaces of cabled linkages, where some edges are flexible cables. These configuration spaces are classified up to analytic isomorphism.