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A characterization of generically rigid frameworks on surfaces of revolution

Research output: Contribution to journalJournal articlepeer-review

Published
<mark>Journal publication date</mark>2014
<mark>Journal</mark>SIAM Journal on Discrete Mathematics
Issue number4
Volume28
Number of pages21
Pages (from-to)2008-2028
Publication StatusPublished
<mark>Original language</mark>English

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