  We show that a unified and maximally generalized approach to spatial
transformation design is possible, one that encompasses all second order waves,
rays, and diffusion processes in anisotropic media. Until the final step, it is
unnecessary to specify the physical process for which a specific transformation
design is to be implemented. The principal approximation is the neglect of wave
impedance, an attribute that plays no role in ray propagation, and is therefore
irrelevant for pure ray devices; another constraint is that for waves the
spatial variation in material parameters needs to be sufficiently small
compared with the wavelength. The key link between our general formulation and
a specific implementation is how the spatial metric relates to the speed of
disturbance in a given medium, whether it is electromagnetic, acoustic, or
diffusive. Notably, we show that our generalised ray theory, in allowing for
anisotropic indexes (speeds), generates the same predictions as does a wave
theory, and the results are closely related to those for diffusion processes.
