On the existence of paradoxical motions of generically rigid graphs on the sphere

Abstract

We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with $3+3$ vertices where no two vertices coincide or are antipodal.

42 pages. This is the accepted version of the manuscript; the final version of this work is https://doi.org/10.1137/19M1289467

References in corpus (4)

• Graphs with Flexible Labelings
• Graphs with Flexible Labelings allowing Injective Realizations
• Counting realizations of Laman graphs on the sphere
• A Survey on the Theory of Bonds

Cited by in corpus (6)

• On the Classification of Motions of Paradoxically Movable Graphs
• Flexing infinite frameworks with applications to braced Penrose tilings
• Zero-sum cycles in flexible polyhedra
• Complete bipartite graphs flexible in the plane
• Flexible placements of graphs with rotational symmetry
• Flexible placements of periodic graphs in the plane