Affine hybrid systems are hybrid systems in which the discrete domains are affine sets and the transition maps between discrete domains are affine transformations. The simple structure of these systems results in interesting geometric properties; one of these is the notion of spatial equivalence. In this paper, a formal framework for describing affine hybrid systems is introduced. As an application, it is proven that every compact hybrid system H is spatially equivalent to a hybrid system H id in which all the transition maps are the identity. An explicit and computable construction for H id is given.