A new approach to characterizing the relative position of two ellipses depending on one parameter

By using several tools coming from Real Algebraic Geometry, Computer Algebra and Projective Geometry (Sturm–Habicht sequences and the classification of pencils of conics in P2(R)), a new approach for characterizing the ten relative positions of two ellipses is introduced.Each relative position is exclusively characterized by a set of equalities and inequalities depending only on the matrices defining the two considered ellipses and does not require in advance the computation or knowledge of the intersection points between them.Moreover, this characterization is specially well adapted for computionally treating the case where the considered ellipses depend on one or several parameters.