Caustics of a Paraboloid and Apollonius Problem

TL;DR

This paper studies caustics of an elliptical paraboloid and their role in solving the Apollonius problem for concurrent normals. It develops two representations of the paraboloid, derives two caustic surfaces as centers of curvature, and introduces a nodal curve as their intersection. It provides explicit parametrizations for the caustics and their intersections with the paraboloid and coordinate planes, enabling a complete four-regime classification of configurations. The work extends Caspari's classical analysis to the paraboloid and connects to optics and computer graphics through Seidel's formula and locus analysis.

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.

Figures (6)

• Figure 1: Parabola $y=\frac{ax^2}{2}$ (green), hyperbola $y=m-\frac{x-l}{ax}$ (blue), semicubical parabola $(ay-1)^3=\frac{27}{8}a^2x^2$ (red), and the normal lines (black): https://www.geogebra.org/calculator/qpvn9gzj
• Figure 2: Elliptical paraboloid $z=\frac{ax^2+by^2}{2}$ (green), parametric curve $\left(\frac{l}{1+at},\frac{m}{1+bt},n+t\right)$ (blue), its asymptotes (grey), and the normal line (black): https://www.geogebra.org/3d/rbp3mxnc
• Figure 3: Elliptical paraboloid $z=\frac{ax^2+by^2}{2}$ (green), the intersection curves (yellow) of the paraboloid with its caustics (violet and blue), the self-intersection points (red) of these curves: https://www.geogebra.org/3d/p7vrgv6w
• Figure 4: Caustics (purple and blue) of paraboloid, the nodal curve (orange), and its intersection points $E_i$ (red) with the paraboloid (not shown): https://www.geogebra.org/3d/d66y3gjw
• Figure 5: Only the first quadrant: the caustics (gray and yellow) of paraboloid (cyan), curves (14) (orange), (15) (red), (16) (magenta), (17) (purple), (18) (green), (19) (blue), the nodal curve (brown), and their intersection points (black): https://www.geogebra.org/3d/gt5pj7jn

Theorems & Definitions (3)

• Lemma 6.1
• Theorem 7.1
• proof