Squared Distance Function on the Configuration Space of a planar Spider with Applications to Hooke Energy and Voronoi Distance

TL;DR

This paper analyzes the configuration space of planar spider mechanisms using Morse theory on the squared distance function, providing new insights into their structure and applications to Hooke energy and Voronoi distance.

Contribution

It introduces a Morse-Bott framework for understanding spider mechanism configuration spaces and derives formulas for critical manifolds and indices, advancing the mathematical understanding of these systems.

Findings

Critical manifolds are described as products of polygon spaces.

A formula for Morse-Bott indices of critical manifolds is derived.

Applications to Hooke energy and Voronoi distance are demonstrated.

Abstract

Spider mechanisms are the simplest examples of arachnoid mechanisms, they are one step more complicated than polygonal linkages. Their configuration spaces have been studied intensively, but are yet not completely understood. In the paper we study them using the Morse theory of the squared distance function from the "body" of the spider to some fixed point in the plane. Generically, it is a Morse-Bott function. We list its critical manifolds, describe them as products of polygon spaces, and derive a formula for their Morse-Bott indices. We apply the obtained results to Hooke energy and Voronoi distance.