Lightcurve inversion problem for objects with negative Gaussian curvature

Abstract

In this paper we consider a second order nonlinear hyperbolic equation of Monge-Ampère type arizing in the Light Curve Inversion problem [7,8,4] when dealing with objects with regions with negative gaussian curvature. The above works imply that such regions are of practical importance in the Light Curve Inversion problem. It is shown here that in such regions existence and uniqueness of solutions to the original equation can be obtained by solving a system of four first order quasi-linear differential equations supplemented by suitable initial and boundary conditions. Furthermore, these solutions can be found by integrating a system of ordinary differential equations. The numerical methods in [14] can be applied to actually find these solutions. We present here the completed part of the overall project, which establishes the necessary basis for efficient numerical solution. It is expected that the part of the project dealing with numerics will be addressed in a separate paper. This paper is written in a form accessible to physicists, engineers and applied mathematicians working on the Light Curve Inversion problem and design of reflectors.