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Minimally rigid tensegrity frameworks

Adam D. W. Clay, Tibor Jordán, Sára Hanna Tóth

Abstract

A $d$-dimensional tensegrity framework $(T,p)$ is an edge-labeled geometric graph in ${\mathbb R}^d$, which consists of a graph $T=(V,B\cup C\cup S)$ and a map $p:V\to {\mathbb R}^d$. The labels determine whether an edge $uv$ of $T$ corresponds to a fixed length bar in $(T,p)$, or a cable which cannot increase in length, or a strut which cannot decrease in length. We consider minimally infinitesimally rigid $d$-dimensional tensegrity frameworks and provide tight upper bounds on the number of its edges, in terms of the number of vertices and the dimension $d$. We obtain stronger upper bounds in the case when there are no bars and the framework is in generic position. The proofs use methods from convex geometry and matroid theory. A special case of our results confirms a conjecture of Whiteley from 1987. We also give an affirmative answer to a conjecture concerning the number of edges of a graph whose three-dimensional rigidity matroid is minimally connected.

Minimally rigid tensegrity frameworks

Abstract

A -dimensional tensegrity framework is an edge-labeled geometric graph in , which consists of a graph and a map . The labels determine whether an edge of corresponds to a fixed length bar in , or a cable which cannot increase in length, or a strut which cannot decrease in length. We consider minimally infinitesimally rigid -dimensional tensegrity frameworks and provide tight upper bounds on the number of its edges, in terms of the number of vertices and the dimension . We obtain stronger upper bounds in the case when there are no bars and the framework is in generic position. The proofs use methods from convex geometry and matroid theory. A special case of our results confirms a conjecture of Whiteley from 1987. We also give an affirmative answer to a conjecture concerning the number of edges of a graph whose three-dimensional rigidity matroid is minimally connected.

Paper Structure

This paper contains 15 sections, 29 theorems, 51 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Rigid frameworks and the rigidity matroid
  3. The rigidity matrix and the rigidity matroid
  4. A theorem of Steinitz and its refinements
  5. Preliminary lemmas
  6. The upper bounds
  7. Frameworks with no bars
  8. Circuit decomposition of matroids
  9. Circuit decompositions of frameworks
  10. Upper bounds for generic cable-strut frameworks
  11. Higher dimensions
  12. Minimally ${\cal R}_d$-connected graphs
  13. Further corollaries
  14. Concluding remarks
  15. Acknowledgements

Key Result

Theorem 2.1

RW Let $(T, p)$ be a tensegrity framework in $\mathbb R^d$. Then $(T, p)$ is infinitesimally rigid if and only if $(\overline T, p)$ is infinitesimally rigid and there exists a proper stress of $(T, p)$.

Figures (9)

  • Figure 1: A minimally rigid cable-strut framework in $\mathbb R^2$ with $4|V|-6$ members.
  • Figure 2: A rigid and a non-rigid generic realization of the same tensegrity graph in the plane.
  • Figure 3: Two rigid realizations of the same tensegrity graph in the plane. The first one is infinitesimally rigid, the second one is not.
  • Figure 4: A minimally infinitesimally rigid tensegrity framework in the plane with $2|B|+|C\cup S|=4|V|-6$.
  • Figure 5: A minimally infinitesimally rigid tensegrity framework in ${\mathbb R}^3$ with $2|B|+|C\cup S|=6|V|-12$.
  • ...and 4 more figures

Theorems & Definitions (50)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3: Steinitz Steinitz
  • Lemma 3.4
  • ...and 40 more