Caustics of a Paraboloid and Apollonius Problem

TL;DR

This paper explores the caustics of elliptical paraboloids, compares historical and modern representations, and extends classical geometric problems by classifying intersection cases with detailed mathematical analysis.

Contribution

It introduces two methods for generating paraboloid caustics and extends the classical Apollonius problem to include intersection classifications.

Findings

Two methods for generating paraboloid caustics demonstrated

Extended Apollonius problem to paraboloid caustic intersections

Complete classification of intersection cases provided

Abstract

We study caustics of an elliptical paraboloid and the history of their various representations from 3D models in XIX century to the recent computer graphics. In the paper two ways of generating the surface, one with cartesian coordinates using formula for principal curvatures, and the other one with 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 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.