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

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Abstract

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:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
72141
Deposited By:
Deposited On:
15 Dec 2014 10:04
Refereed?:
Yes
Published?:
Published
Last Modified:
11 Jul 2020 05:03