  We present a self-contained discussion of the use of the transfer-matrix
formalism to study one-dimensional scattering. We elaborate on the geometrical
interpretation of this transfer matrix as a conformal mapping on the unit disk.
By generalizing to the unit disk the idea of turns, introduced by Hamilton to
represent rotations on the sphere, we develop a method to represent transfer
matrices by hyperbolic turns, which can be composed by a simple
parallelogramlike rule.
