### Abstract

A d-dimensional body-hinge framework is a collection of d-dimensional rigid bodies connected by hinges, where a hinge is a (d- 2)-dimensional affine subspace, i.e., pin-joints in 2-space, line-hinges in 3-space, plane-hinges in 4-space and etc. Bodies are allowed to move continuously in Rd so that the relative motion of any two bodies connected by a hinge is a rotation around it and the framework is called rigid if every motion provides a framework isometric to the original one. A body-hinge framework is expressed as a pair (G, p) of a multigraph G = (V, E) and a mapping p from e∈. E to a (d- 2)-dimensional affine subspace p(e) in Rd. Namely, v∈V corresponds to a body and uv∈E corresponds to a hinge p(uv) which joins the two bodies corresponding to u and v. Then, G is said to be realized as a body-hinge framework (G, p) in Rd, and is called a body-hinge graph. It is known [9,12] that the infinitesimal rigidity of a generic body-hinge framework (G, p) is determined only by its underlying graph G. So, a graph G is called (minimally) rigid if G can be realized as a (minimally) infinitesimally rigid body-hinge framework in d-dimension. In this paper, we shall present an inductive construction for minimally rigid body-hinge simple graphs in d-dimension with d≥ 3.

Original language | English |
---|---|

Pages (from-to) | 2-12 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 556 |

Issue number | C |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*556*(C), 2-12. https://doi.org/10.1016/j.tcs.2014.08.007

**An inductive construction of minimally rigid body-hinge simple graphs.** / Kobayashi, Yuki; Higashikawa, Yuya; Katoh, Naoki; Kamiyama, Naoyuki.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 556, no. C, pp. 2-12. https://doi.org/10.1016/j.tcs.2014.08.007

}

TY - JOUR

T1 - An inductive construction of minimally rigid body-hinge simple graphs

AU - Kobayashi, Yuki

AU - Higashikawa, Yuya

AU - Katoh, Naoki

AU - Kamiyama, Naoyuki

PY - 2014/1/1

Y1 - 2014/1/1

N2 - A d-dimensional body-hinge framework is a collection of d-dimensional rigid bodies connected by hinges, where a hinge is a (d- 2)-dimensional affine subspace, i.e., pin-joints in 2-space, line-hinges in 3-space, plane-hinges in 4-space and etc. Bodies are allowed to move continuously in Rd so that the relative motion of any two bodies connected by a hinge is a rotation around it and the framework is called rigid if every motion provides a framework isometric to the original one. A body-hinge framework is expressed as a pair (G, p) of a multigraph G = (V, E) and a mapping p from e∈. E to a (d- 2)-dimensional affine subspace p(e) in Rd. Namely, v∈V corresponds to a body and uv∈E corresponds to a hinge p(uv) which joins the two bodies corresponding to u and v. Then, G is said to be realized as a body-hinge framework (G, p) in Rd, and is called a body-hinge graph. It is known [9,12] that the infinitesimal rigidity of a generic body-hinge framework (G, p) is determined only by its underlying graph G. So, a graph G is called (minimally) rigid if G can be realized as a (minimally) infinitesimally rigid body-hinge framework in d-dimension. In this paper, we shall present an inductive construction for minimally rigid body-hinge simple graphs in d-dimension with d≥ 3.

AB - A d-dimensional body-hinge framework is a collection of d-dimensional rigid bodies connected by hinges, where a hinge is a (d- 2)-dimensional affine subspace, i.e., pin-joints in 2-space, line-hinges in 3-space, plane-hinges in 4-space and etc. Bodies are allowed to move continuously in Rd so that the relative motion of any two bodies connected by a hinge is a rotation around it and the framework is called rigid if every motion provides a framework isometric to the original one. A body-hinge framework is expressed as a pair (G, p) of a multigraph G = (V, E) and a mapping p from e∈. E to a (d- 2)-dimensional affine subspace p(e) in Rd. Namely, v∈V corresponds to a body and uv∈E corresponds to a hinge p(uv) which joins the two bodies corresponding to u and v. Then, G is said to be realized as a body-hinge framework (G, p) in Rd, and is called a body-hinge graph. It is known [9,12] that the infinitesimal rigidity of a generic body-hinge framework (G, p) is determined only by its underlying graph G. So, a graph G is called (minimally) rigid if G can be realized as a (minimally) infinitesimally rigid body-hinge framework in d-dimension. In this paper, we shall present an inductive construction for minimally rigid body-hinge simple graphs in d-dimension with d≥ 3.

UR - http://www.scopus.com/inward/record.url?scp=84925114554&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925114554&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2014.08.007

DO - 10.1016/j.tcs.2014.08.007

M3 - Article

AN - SCOPUS:84925114554

VL - 556

SP - 2

EP - 12

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - C

ER -