Lens Stars and Platonic Lenses: Connecting Transformation Optics and Kepler Problem

: Lens stars comprise identical ideal thin lenses arranged in a regular star shape centred on the common principal point. They satisfy the edge-imaging condition of transformation optics (TO) and are thus suitable as building blocks of ideal-lens TO devices [J. Opt. Soc. Am. A 37 , 305 (2020)]. Here we show that the ray trajectories in lens stars are piecewise straight approximations of conic sections. We also generalise lens stars to Platonic lenses, highly symmetric combinations of lens stars based on Platonic solids, and find that ray trajectories in Platonic lenses are closed and planar; we design a more general ideal-lens cloak; and we clarify the process of designing ideal-lens TO devices. Throughout, we illustrate our results with ray-tracing simulations. Our results add to the knowledge of TO with ideal lenses.