A family of orthogonal polynomials on the disk (which we call scattering polynomials) serves to formulate a remarkable Fourier expansion of the composition of a sequence of Poincare disk automorphisms.  Scattering polynomials are tied to an exotic riemannian structure on the disk that is hybrid between hyperbolic and euclidean geometries, and the expansion therefore links this exotic structure to the usual hyperbolic one.  The resulting identity is intimately connected with the scattering of plane waves in piecewise constant layered media.  Indeed, a recently established combinatorial analysis of scattering sequences provides a key ingredient of the proof.  At the same time, the polynomial obtained by truncation of the Fourier expansion elegantly encodes the structure of the nonlinear measurement operator associated with the finite time duration scattering experiment.