This paper studies the geometry of perspective projection into multiple images and the matching constraints that this induces between the images. The combined projections produce a 3D subspace of the space of combined image coordinates called the joint image. This is a complete projective replica of the 3D world defined entirely in terms of image coordinates, up to an arbitrary choice of certain scale factors. Projective reconstruction is a canonical process in the joint image requiring only the rescaling of image coordinates. The matching constraints tell whether a set of image points is the projection of a single world point. In 3D there are only three types of matching constraint: the fundamental matrix, Shashua's trilinear tensor, and a new quadrilinear 4 image tensor. All of these fit into a single geometric object, the joint image Grassmannian tensor. This encodes exactly the information needed for reconstruction: the location of the joint image in the space of combined image coordinates.