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

TL;DR

This paper introduces a direct algebraic method to determine the relative position of a parabola or hyperbola with an ellipse, avoiding intersection point calculations, which benefits various computational fields.

Contribution

It provides a novel approach to assess conic positions solely from their coefficients, bypassing the need for intersection point computation.

Findings

Method accurately determines relative positions without intersection points

Applicable to conics in arbitrary implicit form

Reduces computational complexity in conic analysis

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).