  We reelaborate on the basic properties of lossless multilayers by using
bilinear transformations. We study some interesting properties of the
multilayer transfer function in the unit disk, showing that hyperbolic geometry
turns out to be an essential tool for understanding multilayer action. We use a
simple trace criterion to classify multilayers into three classes that
represent rotations, translations, or parallel displacements. Moreover, we show
that these three actions can be decomposed as a product of two reflections in
hyperbolic lines. Therefore, we conclude that hyperbolic reflections can be
considered as the basic pieces for a deeper understanding of multilayer optics.
