A characterization of generically rigid frameworks on surfaces of revolution

Nixon, Anthony and Owen, John and Power, Stephen (2014) A characterization of generically rigid frameworks on surfaces of revolution. SIAM Journal on Discrete Mathematics, 28 (4). pp. 2008-2028. ISSN 0895-4801

[thumbnail of NOPsurfaces_revised]
PDF (NOPsurfaces_revised)
NOPsurfaces_revised.pdf - Accepted Version
Available under License Creative Commons Attribution.

Download (359kB)


A foundational theorem of Laman provides a counting characterization of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterization was obtained for frameworks in three dimensions whose vertices are constrained to concentric spheres or to concentric cylinders. Noting that the plane and the sphere have 3 independent locally tangential infinitesimal motions while the cylinder has 2, we obtain here a Laman-type theorem for frameworks on algebraic surfaces with a 1-dimensional space of tangential motions. Such surfaces include the torus, helicoids, and surfaces of revolution. The relevant class of graphs are the (2,1)-tight graphs, in contrast to (2,3)-tightness for the plane/sphere and (2,2)-tightness for the cylinder. The proof uses a new characterization of simple (2,1)-tight graphs and an inductive construction requiring generic rigidity preservation for 5 graph moves, including the two Henneberg moves, an edge joining move, and various vertex surgery moves. Read More: http://epubs.siam.org/doi/abs/10.1137/130913195

Item Type:
Journal Article
Journal or Publication Title:
SIAM Journal on Discrete Mathematics
Uncontrolled Keywords:
?? mathematics(all) ??
ID Code:
Deposited By:
Deposited On:
15 Dec 2014 10:04
Last Modified:
27 Jan 2024 00:27