Contact detection between an ellipsoid and a combination of quadrics

TL;DR

This paper presents a novel method for contact detection between an ellipsoid and a quadric by analyzing the characteristic polynomial and its discriminants, enabling efficient and parameter-based contact assessment.

Contribution

It introduces a necessary and sufficient condition for contact detection using polynomial discriminants, directly derived from the quadrics' parameters, improving efficiency and classification.

Findings

Contact detection can be performed efficiently using polynomial discriminants.

The method classifies relative positions based on polynomial coefficient signs.

The approach provides a direct parameter-based criterion for contact detection.

Abstract

We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of discriminants of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial.