Which point configurations are determined by the distribution of their pairwise distances?

In a previous paper we showed that, for any n ≥ m + 2, most sets of n points in ℝ^m are determined (up to rotations, reflections, translations and relabeling of the points) by the distribution of their pairwise distances. But there are some exceptional point configurations which are not reconstructible from the distribution of distances in the above sense. In this paper, we concentrate on the planar case m = 2 and present a reconstructibility test with running time O(n¹¹). The cases of orientation preserving rigid motions (rotations and translations) and scalings are also discussed.