An Algebraic Condition for the Positional Relationship of an Ellipse and a Parabola (Hyperbola)

Abstract

In many scientific areas such as computer animation, computer graphics, computer aided geometry design and robot, it is very common to detect the positional relationship of several entities. We derive an algebraic condition (based on the generalized characteristic polynomial) for detecting the positional relationship of a planar ellipse and a planar parabola (hyperbola) by considering all the cases: separation, exterior contact, intersection, interior contact and inclusion. The algebraic condition can be represented by the distribution of the roots of the generalized characteristic equation of the ellipse and parabola (hyperbola). The criterion is simple and effective, and easier than old methods.