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.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 DOI:
 10.48550/arXiv.1908.00467
 arXiv:
 arXiv:1908.00467
 Bibcode:
 2019arXiv190800467G
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Robotics;
 Mathematics  Algebraic Geometry;
 Mathematics  Metric Geometry
 EPrint:
 42 pages. This is the accepted version of the manuscript