A Problem on the Geodesics of a Surface of Revolution which is of Interest in Microwave Optics

摘要

A general discussion is given of the following problem:Given a circle A and a function F(ψ), to find a surface of revolution S of which A is a parallel, such that a geodesic leaving A under an angle ψ should intersect A a second time after describing a longitude ψ = F(ψ). This problem depends on an integral equation of Abel’s type, it is shown how the singularities of F (ψ) and F(ψ) influence the shape of S. As an example a new surface is described.