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Stress Matrices and Global Rigidity of Frameworks on Surfaces

Abstract

In 2005, Bob Connelly showed that a generic framework in \({\mathbb {R}}^d\) is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in \({\mathbb {R}}^3\). For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum-rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.

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Notes

  1. We can consider \(\det M\) as a polynomial in the coordinates of \((p(v_1),p(v_2),p(v_3))\). If \(\det M=0\), then genericness would imply that this polynomial evaluates to 0 at all points in \(S(r_1)\times S(r_2)\times S(r_3)\). It is straightforward to show that this is not the case by finding points \((p_1,p_2,p_3)\in S(r_1)\times S(r_2)\times S(r_3)\) at which the polynomial is non-zero. When \({S}={\mathcal {Y}}(r)\), we can take \(p_1=(\sqrt{r_1},0,0)\), \(p_2=(0,\sqrt{r_2},0)\) and \(p_3=(\sqrt{r_3},0,1)\); when \({S}={\mathcal {C}}(r)\), we can take \(p_1=(\sqrt{r_1},0,1)\), \(p_2=(\sqrt{r_2},0,-1)\) and \(p_3=(0,\sqrt{r_3},1)\); and when \({S}={\mathcal {E}}(r)\), we can take \(p_1=(\sqrt{r_1},0,0)\), \(p_2=(0,\frac{\sqrt{r_2}}{\sqrt{\alpha }},0)\) and \(p_3=(0,0,\frac{\sqrt{r_3}}{\sqrt{\beta }})\).

  2. Partial results are known for particular surfaces: there exists a framework (Gq) with \({\text {rank}}\,R_{{S}^q}(G',q)={\text {rank}}\,R_{S}(G,p)+3\) and \({S}^q={S}\) when \({S}={\mathcal {Y}}\) [15], and when \({S}={\mathcal {C}}(1)\) or \({S}={\mathcal {E}}(1)\) [16].

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Acknowledgments

We would like to thank the School of Mathematics, University of Bristol, for providing partial financial support for this research. We would also like to thank Lee Butler for helpful discussions concerning semi-algebraic sets and Bob Connelly for many helpful discussions.

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Correspondence to Anthony Nixon.

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Jackson, B., Nixon, A. Stress Matrices and Global Rigidity of Frameworks on Surfaces. Discrete Comput Geom 54, 586–609 (2015). https://doi.org/10.1007/s00454-015-9724-8

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Keywords

  • Rigidity
  • Global rigidity
  • Stress matrix
  • Framework on a surface

Mathematics Subject Classification

  • 52C25
  • 05C10
  • 53A05