I am doing a presentation for my history of math class over the development of non-euclidean geometry. Basically I am attempting to find examples of applications of both spherical and hyperbolic geometries. So far I have only found uses in navigation and the mapping of the celestial sphere for spherical geometry, however, I am trying to find a few more relevant applications perhaps in physics (maybe electric fields?). As for hyperbolic geometry, I only have the application used in the general theory of relativity. Perhaps this question might be suited better to a physics forum..but really I am looking for any interesting applications that I cannot think of. Any help is greatly appreciated!
Not as useful an application, but quite neat: tesselations of the hyperbolic plane in Poincaré disc model are at the core of plenty of M.C.Escher's beautiful artworks, such as the following one.
On a more serious side, and as far as mathematical applications are concerned, hyperbolic geometry has proved itself very important in the last 20 years in Gromov's geometric group theory (Gromov was recently awarded an Abel prize for his work). The rough idea of this theory is to transpose geometric concepts to groups (in particular through their Cayley graph) and derive connections between their algebraic and geometric properties; this geometrical insight enables a better understanding of certain properties. And "hyperbolic groups", i.e. groups whose geometry somehow shares deep properties with standard hyperbolic geometry, play a prominent role in this theory. You'll easily find further information on the web.