Boutin and Kemper have shown that the set of unlabeled pairwise distances between the points of a generic point-set in Rnmiddot is a lossless representation of the shape of the point-set. In this paper, we extend this result to the case where each of the points observed is drawn from a similar spherical Gaussian distribution in R2. More precisely, we consider the distribution of the (squared) distance between two points independently drawn from a mixture of spherical Gaussians, each Gaussian having the same variance sigma2. We then show that two generic such mixtures of spherical Gaussians have the same shape (i.e., they are related by a rigid motion) if and only if their distribution of distances are the same.