Rights statement: © 2018, Society for Industrial and Applied Mathematics
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Rigidity of frameworks on expanding spheres
AU - Schulze, Bernd
AU - Nixon, Anthony
AU - Tanigawa, Shin-ichi
AU - Whiteley, Walter
N1 - © 2018, Society for Industrial and Applied Mathematics
PY - 2018/11/15
Y1 - 2018/11/15
N2 - 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.
AB - 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.
U2 - 10.1137/17M1116088
DO - 10.1137/17M1116088
M3 - Journal article
VL - 32
SP - 2591
EP - 2611
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
SN - 0895-4801
IS - 4
ER -