An inductive construction of minimally rigid body-hinge simple graphs

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 R-d 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 is an element of E to a (d - 2)-dimensional affine subspace p(e) in R-d. Namely, v is an element of V corresponds to a body and uv is an element of 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 R-d, 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. (C) 2014 Elsevier B.V. All rights reserved.