Pairing Symmetries for Euclidean and Spherical Frameworks

Abstract

In this paper we consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in R d . In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in X 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.2010 Mathematics Subject Classification. 52C25, 05C10, 51E15

States of self-stress in symmetric frameworks and applications

“…(i) If v c = 0, then we obtain the same count for Γ(m) − Γ(s) as we did for reflection symmetry, with e σ being replaced by e 2 . (This is not surprising, given the transfer results for infinitesimal rigidity between C s and C 2 established in [5].) So the same observations we made for the reflection symmetry also apply to the half-turn symmetry in the case when v c = 0.…”

Symmetric contact systems of segments, pseudotriangulations and inductive constructions for corresponding surface graphs

“…For the case when τ (C) is generated by a rotation of order 2, we conjecture that a result analogous to Corollary 7.5 may be established by allowing the action θ to be nonfree on the edges of G, and hence allowing an edge of G to go through the cone point of R 2 /τ (Γ). By the transfer results for C-rigidity established in [5], this result would then immediately also extend to the case of reflection symmetry.…”

“…Then B is also balanced and so f (B) ≥ 3. Together with (5) We refer to the graph B whose existence is asserted by Lemma 9.9 as type 1/2/3 blocker according to whichever case of the lemma applies. Note that for a given blocker B exactly one of (1)-( 3) is true.…”

Rigidity through a Projective Lens

“…This identification of antipodal points creates the elliptical model of the projective space. Any switching of a point and its antipode on the sphere preserves infinitesimal and static rigidity in this elliptical metric, as is explored in [160]. This is again a thoroughly projective perspective that offers insights into the rigidity behaviour of projections into Euclidean space.…”