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

Abstract

Abstract. Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well-adapted to the study of the relative position of two conics defined by equations depending on parameters.MSC2000: 13A50 (invariant theory), 13J30 (real algebra).

A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form

“…It is well known that the conics A 1 , A 2 have four common points (which need not all be distinct). Further, the relative position of the conics A 1 , A 2 has nine cases according to the nature (generic or nongeneric, real or complex) of their common points [10,11]. ey are presented in Table 1.…”

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

“…As a consequence, λ 2 and λ 3 are both positive or negative. If λ 2 and λ 3 are negative then, according to (14), l 2 < 0, and according to (15), l 1 < 0. A sharper result than the converse is also true: if λ 2 and λ 3 are positive then, l 2 < 0 or l 1 < 0.…”

“…One important feature of this approach is that it permits the manipulation of the derived formulae for the cases where the considered ellipses depend on one parameter. More recently, a complete systematization of the problem is finally provided in [15], where it is also generalized to find the relative position of two arbitrary real conics, not necessary ellipses. The approach is based on the characterization of the orbits of pencils of conics using classical invariant theory, and the characterization of the rigid isotropy class for each orbit.…”

“…This requires the use of Descartes' law of signs and subresultant sequences. Building upon these results, the invariant theory of pencils of conics is further explored in [16] to show that some of the conditions obtained in [15] have unnecessary high degrees.…”

Computing the intersections of three conics according to their Jacobian curve

“…Previous studies have focused on two aspects: determining the collision (e.g., Wang et al, 2001;Liu and Chen, 2004;Briand, 2007) and finding the intersection (e.g., Wang, 2002). Some studies have also focused on both of these two aspects (e.g., Chen et al, 2011).…”