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Rigidity of frameworks on expanding spheres

Research output: Contribution to journalJournal article

<mark>Journal publication date</mark>15/11/2018
<mark>Journal</mark>SIAM Journal on Discrete Mathematics
Issue number4
Number of pages21
Pages (from-to)2591-2611
Publication statusPublished
Original languageEnglish


A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterizations are established for the rigidity of generic frameworks for d=1 with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterization of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in one and two dimensions and to some extent between rigidity in two and three dimensions. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.

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© 2018, Society for Industrial and Applied Mathematics