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If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid components span one, two, or three vertices, and when c > c_2, w.h.p. there is a giant rigid component. 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