We present new results for formation control in the plane. We work with the usual assumptions stipulating that each agent in the formation has only access to the position of its leader(s) and the target edge length it has to maintain, i.e. only has partial knowledge of the global objective and global state of the formation. In this paper, we show how local stability of the formation around a target equilibrium is affected by this partial knowledge. In particular, we show why many natural control laws work for a formation of three agents but fail to work for slightly more complex, yet minimally persistent, formations.