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