A Computer Vision Problem in Flatland

When is it possible to project two sets of labeled points of equal cardinality lying in a pair of projective planes to the same image on a projective line? We give a complete answer to this question, obtaining the following results. We first show that such a pair of projections exist if and only if the two point sets are themselves images of a common point set in projective space. Moreover, we find that for generic pairs of point sets, a common projection exists if and only if their cardinality is at most seven. In these cases, we give an explicit description of the loci of projection centers that enable a common image.