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