### Abstract

A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edge-labeled graph T=(V;C,S), in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization of T as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative on the struts, or equivalently, if every convex realization of T is infinitesimally rigid. We characterize the robust abstract tensegrity polygons on n vertices with n-2 struts, answering a question of Roth and Whiteley (Trans Am Math Soc 265:419-446, 1981) and solving an open problem of Connelly (Recent progress in rigidity theory, 2008).

Original language | English |
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Pages (from-to) | 537-551 |

Number of pages | 15 |

Journal | Discrete & Computational Geometry |

Volume | 50 |

Issue number | 3 |

DOIs | |

Publication status | Published - Oct 2013 |

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### Keywords

- Rigid framework
- Tensegrity framework
- Tensegrity polygon

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Discrete & Computational Geometry*,

*50*(3), 537-551. https://doi.org/10.1007/s00454-013-9539-4

**Robust Tensegrity Polygons.** / Geleji, János; Jordán, T.

Research output: Contribution to journal › Article

*Discrete & Computational Geometry*, vol. 50, no. 3, pp. 537-551. https://doi.org/10.1007/s00454-013-9539-4

}

TY - JOUR

T1 - Robust Tensegrity Polygons

AU - Geleji, János

AU - Jordán, T.

PY - 2013/10

Y1 - 2013/10

N2 - A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edge-labeled graph T=(V;C,S), in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization of T as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative on the struts, or equivalently, if every convex realization of T is infinitesimally rigid. We characterize the robust abstract tensegrity polygons on n vertices with n-2 struts, answering a question of Roth and Whiteley (Trans Am Math Soc 265:419-446, 1981) and solving an open problem of Connelly (Recent progress in rigidity theory, 2008).

AB - A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edge-labeled graph T=(V;C,S), in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization of T as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative on the struts, or equivalently, if every convex realization of T is infinitesimally rigid. We characterize the robust abstract tensegrity polygons on n vertices with n-2 struts, answering a question of Roth and Whiteley (Trans Am Math Soc 265:419-446, 1981) and solving an open problem of Connelly (Recent progress in rigidity theory, 2008).

KW - Rigid framework

KW - Tensegrity framework

KW - Tensegrity polygon

UR - http://www.scopus.com/inward/record.url?scp=84884416904&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884416904&partnerID=8YFLogxK

U2 - 10.1007/s00454-013-9539-4

DO - 10.1007/s00454-013-9539-4

M3 - Article

AN - SCOPUS:84884416904

VL - 50

SP - 537

EP - 551

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -