There is a well known correspondence between nontrivial infinitesimal motions of a planar framework and parallel redrawings preserving the incidence structure of the graph. We define weak projective matroid representations, which generalize the notion of a parallel redrawing. These matroid representations can be interpreted as line arrangements, and our motivation is to use examples from rigidity theory to study algebraic structures associated to line arrangements. This is joint work with Michael DiPasquale and Will Traves.