Abstract
We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in {mathbb {R}}^d. In particular, for a graph G=(V,E) and a framework (G, p), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset Xsubset V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.
Highlights
Given a collection of primitive geometric objects in a space satisfying particular geometric constraints, a fundamental question is whether the given constraints uniquely determine the whole configuration up to congruence
The following are equivalent: (a) G can be realised as an infinitesimally rigid bar-joint framework on Sd such that the points assigned to X lie on the equator. (b) G can be realised as an infinitesimally rigid point-hyperplane framework in Rd such that each vertex in X is realised as a hyperplane and each vertex in V \X is realised as a point. (c) G can be realised as an infinitesimally rigid bar-joint framework in Rd such that the points assigned to X lie on a hyperplane
Given a framework that admits some point group symmetry we show, in Sects. 3 and 4, that both forced symmetric and incidentally symmetric infinitesimal rigidity can be transferred between spherical frameworks with a given set X of vertices realised on the equator and point-hyperplane frameworks, where the vertices of X are exactly the vertices realised as hyperplanes
Summary
Given a collection of primitive geometric objects in a space satisfying particular geometric constraints, a fundamental question is whether the given constraints uniquely determine the whole configuration up to congruence. A standard approach to study the rigidity of bar-joint frameworks is to linearise the problem by differentiating the length constraints on the corresponding pairs of points. This leads to the notion of infinitesimal (or equivalently, static) rigidity. A fundamental example is that half-turn rotation and mirror symmetry on the 2-sphere have geometrically equivalent infinitesimal rigidity properties, in both the incidental and forced contexts. 7, we consider the corresponding results when the action of the symmetry group is not free on the vertices of the symmetric graph In this context we present some examples and again discuss some combinatorial consequences.
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