Classification of the relative positions between an ellipsoid and an elliptic paraboloid

TL;DR

This paper provides a classification of the relative positions between an ellipsoid and an elliptic paraboloid, enabling easy contact detection through polynomial coefficient analysis, especially when the ellipsoid is small relative to the paraboloid.

Contribution

It introduces a comprehensive classification method for ellipsoid-paraboloid positions based on polynomial analysis, focusing on small ellipsoids.

Findings

Classification scheme for all relative positions

Polynomial coefficient analysis for contact detection

Applicable when ellipsoid is small compared to paraboloid

Abstract

We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid ({\it small} meaning that the ellipsoid cannot be tangent to the paraboloid at two points simultaneously). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.