Classification of the relative positions between a hyperboloid and a sphere

TL;DR

This paper presents a method to classify the relative positions between a hyperboloid and a sphere using polynomial roots, enabling contact detection in practical applications.

Contribution

It introduces a polynomial-based classification method for hyperboloid-sphere positions, extending to hyperboloid-ellipsoid cases, facilitating contact detection.

Findings

Classification based on polynomial roots effectively determines relative positions.

Method applicable to real-world contact detection scenarios.

Extends to hyperboloid-ellipsoid configurations in certain cases.

Abstract

We characterize all possible relative positions between a hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. The classification is also suitable for a hyperboloid and a ellipsoid in some situations. As an application, this provides a method to detect contact between the two surfaces by a simple calculation in many real world applications.