An Explicit Criterion for the Positional Relationship of an Ellipse and a Parabola

Detecting the positional relationship between two planar conics is a common problem in fields of computer animation, computer graphics, machine vision, robotics, virtual reality and CAD/CAM. Based on the generalized characteristic polynomial and the affine invariants of two conics, explicit conditions are derived for classifying the various positional relationships of an ellipse and a parabola by considering all the cases: separation, exterior tangency, intersection, interior tangency and inclusion. The explicit conditions can be expressed in terms of the coefficients of the quadratic forms representing the ellipse and the parabola.