Symmetric Configuration spaces of linkages

Abstract

A $configuration$ of a linkage $\Gamma$ is a possible positioning of $\Gamma$ in ${\mathbb{R}}^d$ and the collection of all such forms the configuration space $\mathcal{C}(\Gamma )$ of $\Gamma$. We here introduce the notion of the $symmetricconfigurationspace$ of a linkage, in which we identify configurations which are geometrically indistinguishable. We show that the symmetric configuration space of a planar polygon has a regular cell structure, provide some principles for calculating this structure, and give a complete description of the symmetric configuration space of all quadrilaterals and of the equilateral pentagon.

45 pages, 29 figures