We study the Casimir energy of a massless scalar field that obeys Dirichlet
boundary conditions on a hyperboloid facing a plate. We use the optical
approximation including the first six reflections and compare the results with
the predictions of the proximity force approximation and the semi-classical
method. We also consider finite size effects by contrasting the infinite with a
finite plate. We find sizable and qualitative differences between the new
optical method and the more traditional approaches.Comment: v2: 14 pages, 11 eps figures; typo in eq. (21) removed, clarification
added, fig. 10 improved; version published in Phys. Rev.