Symmetric Configuration spaces of linkages

TL;DR

This paper introduces the symmetric configuration space for linkages, showing it has a regular cell structure for planar polygons, and provides complete descriptions for quadrilaterals and equilateral pentagons.

Contribution

It defines symmetric configuration spaces for linkages, establishes their cell structure, and fully characterizes these spaces for specific polygon cases.

Findings

Symmetric configuration space of planar polygons has a regular cell structure.

Complete description of symmetric configuration space for quadrilaterals.

Complete description of symmetric configuration space for equilateral pentagons.

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 $symmetric configuration space$ 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.