Relative position of a parabola or a hyperbola and an ellipse without computing intersection points

TL;DR

This work develops a discriminant-based, root-free framework to decide the relative position of conics without computing intersection points. By reducing to canonical forms via affine transformations and analyzing the characteristic cubic of the conic pencil, the authors derive sign-based criteria expressed in terms of discriminants and transformed coefficients. They establish nine distinct configurations for parabola-ellipse and eleven for hyperbola-ellipse, all determined from polynomial signs rather than root extraction, and extend to parameter-dependent setups. The approach is practical for robotics, CAD/CAM, and dynamic scenes where moving conics must be evaluated efficiently.

Abstract

Efficient methods to determine the relative position of two conics are of great interest for applications in robotics, computer animation, CAGD, computational physics, and other areas. We present a method to obtain the relative position of a parabola or a hyperbola, and a coplanar ellipse, directly from the coefficients of their implicit equations, even if they are not given in canonical form, and avoiding the computation of the corresponding intersection points (and their characteristics).

Figures (3)

• Figure 1: Ellipse and Parabola - Theorem \ref{['ep']}
• Figure 2: Circle and Hyperbola. Proposition \ref{['ch']}
• Figure 3: Ellipse and Hyperbola - Theorem \ref{['he']}

Theorems & Definitions (3)

• proof
• proof
• proof