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

New algebraic conditions for the identification of the relative position of two coplanar ellipses

“…This necessary and sufficient condition readily permits to discern between contacts of type 10 and type 11, despite they have the same root pattern for P (λ) = 0, because the latter type contains a double line (repeated three times) in its pencil. Since the triple root in root pattern 2, according to (19), is λ t = −l 2 /l 3 , the contact between the two ellipses is of type 11 if Q(λ t ) = R(λ t ) = 0, and type 10, otherwise.…”

“…A further refinement in the approach based on the analysis of the pencil consists in using Sturm-Habicht sequences, as explained in [19], where ten positional relationships which do not exactly match those in Fig. 1 are considered (for example, the osculating and hyperosculating contact are considered as a single case).…”

Equations, inequations and inequalities characterizing the configurations of two real projective conics

“…For the sign sequence +00−, the Sturm query is computed as for ++: one permanence, no change, this gives 1. 9 The discriminant of the characteristic form, Disc(Φ), is called the Tact invariant by the classics, because it vanishes exactly when the two conics are tangent [5].…”

“…They are considered for instance in the articles [9,20] (and [21] for the similar problem for ellipsoids). We consider this paper as the systematization of their main ideas.…”

“…We consider this paper as the systematization of their main ideas. Specially, in [9], an algorithm was proposed to determine the configuration of a pair of ellipses, by means of calculations of Sturm-Habicht sequences. Our approach is different: when there, computations were performed for each particular case, we perform the computations once for all in the most general case.…”