From infinitesimal to global rigidity
refreshments at 3:45pm
Suppose that a finite set of labeled points in the plane, a
configuration, is given. Some pairs of those points, determined by a finite
simple graph G, are to be kept at their same fixed distance apart. If the
coordinates of the configuration are generic, then the question of whether the
configuration is rigid is a purely combinatorial property of G, for which there isa polynomial-time algorithm. (A configuration is generic if there is no
non-trivial integral polynomial satisfied by the coordinates.)
But rigidity is a
local property equivalent to there being only trivial infinitesimal motions
satisfying the distance constraints. Is it possible to determine the
more basic question as to whether there is any other non-congruent configuration in the planethat satisfies the distance constraints? If the distance constraints determine
the configuration up to congruence, the configuration is called globally rigid,
and if the configuration is assumed to be generic, the graph G is called
generically globally rigid (in the plane). Necessary conditions for generic
global rigidity in the plane are generic rigidity even when any edge is removed
from G (generic redundant rigidity), It
turns out that these conditions are also sufficient,
which were conjectured by B.
Hendrickson some years ago. The proof of this involves the calculation of the
rank of a matrix (what I call a stress matrix), and a construction, due to T.
Jordan and W. Sullivan, that builds G starting from the complete graph K(4) by
operations that preserve the properties of being vertex 3-connected and generic
and that G should be vertex
Speaker's Contact Info: connelly(at-sign)math.cornell.edu
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