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

Maciej Denkowski, Gaiane Panina, Dirk Siersma

TL;DR

This paper develops a Morse-Bott framework for the squared distance function on planar spider configuration spaces, showing that generically the function is Morse-Bott with critical manifolds that decompose as products of polygon spaces. It derives explicit Morse indices for stationary configurations, and demonstrates how these results yield topological information (via Morse-Bott polynomials and Euler characteristics) and concrete insights for Hooke energy and Voronoi distance. The work provides detailed examples (e.g., a tripod and two-leg spiders) and extends the analysis to weighted Hooke energies and the Voronoi distance, including perturbation (Morsification) techniques to obtain Morse-type counts. The methodologies offer a robust toolkit for understanding the topology of spider spaces and have potential applications to higher-dimensional ambient spaces and graph-linkage systems.

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.

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

TL;DR

This paper develops a Morse-Bott framework for the squared distance function on planar spider configuration spaces, showing that generically the function is Morse-Bott with critical manifolds that decompose as products of polygon spaces. It derives explicit Morse indices for stationary configurations, and demonstrates how these results yield topological information (via Morse-Bott polynomials and Euler characteristics) and concrete insights for Hooke energy and Voronoi distance. The work provides detailed examples (e.g., a tripod and two-leg spiders) and extends the analysis to weighted Hooke energies and the Voronoi distance, including perturbation (Morsification) techniques to obtain Morse-type counts. The methodologies offer a robust toolkit for understanding the topology of spider spaces and have potential applications to higher-dimensional ambient spaces and graph-linkage systems.

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.
Paper Structure (27 sections, 21 theorems, 24 equations, 12 figures)

This paper contains 27 sections, 21 theorems, 24 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Basic notions: spider mechanism, its configuration space and work space. Robot arms and polygon spaces
  3. Spider mechanisms and spiders
  4. Robot arms: spiders with one leg
  5. Polygonal linkages and polygon spaces: spiders with two legs
  6. The work map
  7. Discs-Annuli structure on the work space
  8. The Euler characteristic of the spider space
  9. The squared distance function as a Morse function
  10. Squared distance function defined on the spider space.
  11. Critical points
  12. Morse indices
  13. Morse-Bott Polynomials
  14. Examples
  15. A tripod
  16. ...and 12 more sections

Key Result

Proposition 1

(KM) The squared distance between the endpoints of a robot arm $||a-x||^2$ (defined on the configuration space i.e. composed with the work map, cf. Section s:sqmorse) is a Morse-Bott function, whose critical points and critical manifolds are: The Morse index of an aligned configuration equals $M(\mathcal{L})$.

Figures (12)

  • Figure 1: Spider in the plane.
  • Figure 2: Several values of the Morse index
  • Figure 3: Critical circles and work space.
  • Figure 4: Constructions with legs
  • Figure 5: Partition of the plane.
  • ...and 7 more figures

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2: SSB, see also H, Mou
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Definition 4
  • ...and 25 more