Caustics of a Paraboloid and Apollonius Problem

We study the caustics of an elliptical paraboloid. In the paper two ways of generating the surface, one with cartesian coordinates using the formula for the principal curvatures, and the other one with the parabolic coordinates using Seidel's formula were demonstrated. By finding the intersection curves of these caustics with the paraboloid we extend the solution of F. Caspari for the classical Apollonius problem about the number of concurrent normals to the points of the paraboloid itself. A complete classification of all possible cases of intersections of these caustics with their paraboloid is given.