  We present a comprehensive and self-contained discussion of the use of the
transfer matrix to study propagation in one-dimensional lossless systems,
including a variety of examples, such as superlattices, photonic crystals, and
optical resonators. In all these cases, the transfer matrix has the same
algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as
well as the group of unimodular real matrices underlying the structure of the
abcd law, which explains many subtle details. We elaborate on the geometrical
interpretation of the transfer-matrix action as a mapping on the unit disk and
apply a simple trace criterion to classify the systems into three types with
very different geometrical and physical properties. This approach is applied to
some practical examples and, in particular, an alternative framework to deal
with periodic (and quasiperiodic) systems is proposed.
