We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the C d−2 d−1 -cofactor rigidity of points along any (non-degenerate) conic in the plane. For hyperconnectivity in dimension two, having the points in the moment curve is no loss of generality.We also show that, restricted to bipartite graphs, the bar-and-joint rigidity matroid is freer than the hyperconnectivity matroid.