We show that we can optimally represent the set of 2D images produced by the point features of a rigid 3D model as two lines in two high-dimensional spaces. We then decribe a working recognition system in which we represent these spaces discretely in a hash table. We can access this table at run time to find all the groups of model features that could match a group of image features, accounting for the effects of sensing error. We also use this representation of a model's images to demonstrate significant new limitations of two other approaches to recognition: invariants, and non- accidental properties.