On the Interference Problem for Ellipsoids: Experiments and Applications

Abstract

The problem of detecting when two moving ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc. By analysing symbolically the sign of the real roots of the characteristic polynomial of the pencil defined by two ellipsoids $\mathcal {A}$ and $\mathcal {B}$ we use and analyse the new closed formulae introduced in [9] characterising when $\mathcal {A}$ and $\mathcal {B}$ overlap, are separate and touch each other externally for determining the interference of two moving ellipsoids. These formulae involves a minimal set of polynomial inequalities depending only on the entries of the matrices A and B (defining the ellipsoids $\mathcal {A}$ and $\mathcal {B}$), need only to compute the characteristic polynomial of the pencil defined by A and B and do not require the computation of the intersection points between them. This characterisation provides a new approach for exact collision detection of two moving ellipsoids since the analysis of the univariate polynomials (depending on the time) in the previously mentioned formulae provides the collision events between them.

Both authors are partially supported by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2017-88796-P.