For generic placements of points of a graph G in the plane and straight line segments as edges, we expect any two edges to cross at most once and no three line segments to have a common intersection point. Introducing vertices of degree~4 at the edge intersection points yields a planar graph G+ with the same rigidity properties as the original graph . We now use the planarity of G+ to get a geometric dual. We show that the trucated union of the cycle matroid of G+ and the cocycle matroid of its embedded dual is the 2-d rigidity matroid We generalize this construction to surfaces of any genus. While 3-connected planar graphs have essentially unique embeddings in the plane, this is not the case for embeddings on surfaces of higher genus. We give examples of how the rigidity properties of a graph depend on the embedding.