Point-hyperplane frameworks, slider joints, and rigidity preserving transformations

Abstract

A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in R d and those in S d is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis.Among other results, we derive a combinatorial characterization of graphs that can be realized as infinitesimally rigid frameworks in the plane with a given set of points collinear. This extends a result by Jackson and Jordán, which deals with the case when three points are collinear.

Pairing Symmetries for Euclidean and Spherical Frameworks

“…Remark 2.3. As discussed in [9], translating a hyperplane in a point-hyperplane framework does not affect its infinitesimal rigidity properties. We may therefore assume without loss of generality that every hyperplane contains the origin.…”

“…Given a point-hyperplane framework (G, p, ) in R d , we may construct a corresponding spherical framework (G, q) with all points in the upper hemisphere by setting q(i) =p i p i , wherê p i = (p i , 1), for all i ∈ V P , and q(j) = (a j , 0) for all j ∈ V H . It was shown in [9,34] that (G, p, ) is infinitesimally rigid in R d if and only if (G, q) is infinitesimally rigid in S d with all points in the upper hemisphere. We show that this operation also preserves the Γ symmetry.…”

“…It remains to show that (b) and (c) are equivalent. It was shown in [9] that (G, p, ) is infinitesimally rigid as a point-hyperplane framework in…”

“…Next we will extend the transfer results of Section 3 to the context of forced Γ-symmetric rigidity, where the action is free on the vertices, by adapting the approach in [9].…”

“…In [9] this line of research was extended to include point-hyperplane frameworks which consist of points and hyperplanes combined with point-point distance constraints, point-hyperplane distance constraints and hyperplane-hyperplane angle constraints (see Section 2.5 for a rigorous definition). These types of frameworks have practical applications in areas such as mechanical and civil engineering as well as CAD, since point-hyperplane distance constraints may be used to model slider-joints in engineering structures [9,16]. In particular the following result showed that the (infinitesimal) rigidity of such frameworks is equivalent to the (infinitesimal) rigidity of Euclidean and spherical frameworks with a certain special subset of vertices (that correspond to the hyperplanes).…”