An inductive construction of minimally rigid body-hinge simple graphs

Yuki Kobayashi, Yuya Higashikawa, Naoki Katoh, Naoyuki Kamiyama

Research output: Contribution to journalArticle

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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 languageEnglish
Pages (from-to)2-12
Number of pages11
JournalTheoretical Computer Science
Volume556
Issue numberC
DOIs
Publication statusPublished - Jan 1 2014

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Hinges
Simple Graph
Rigid Body
Graph in graph theory
Subspace
Motion
Multigraph
Framework
Isometric
Rigidity
Join
Line

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

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

In: Theoretical Computer Science, Vol. 556, No. C, 01.01.2014, p. 2-12.

Research output: Contribution to journalArticle

Kobayashi, Yuki ; Higashikawa, Yuya ; Katoh, Naoki ; Kamiyama, Naoyuki. / An inductive construction of minimally rigid body-hinge simple graphs. In: Theoretical Computer Science. 2014 ; Vol. 556, No. C. pp. 2-12.
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