A Natural Classification of Curves and Surfaces With Reflection Properties

Abstract

Parabolic mirrors and elliptic domes (whispering galleries) are familiar examples of focally reflective surfaces. In this paper we show that only conic curves and their corresponding surfaces of revolution (and in degenerate cases, lines and planes) are focally reflective. In [1], curves and surfaces with reflection properties were classified by solving differential equations in polar and spherical coordinates. Here we use a coordinate-free method to achieve the same classification by relating reflection properties to the defining focal properties of conics. The well-known orthogonality property of confocal conics comes as a bonus.