  We provide a way of determining the infinitesimal rigidity of rod
configurations realizing a rank two incidence geometry in the Euclidean plane.
We model each rod with a cone over its point set and prove that the resulting
geometric realization of the incidence geometry is infinitesimally rigid in
regular position if and only if the resulting cone graph is infinitesimally
rigid in generic position. This is a generalization of the Molecular
conjecture.
