An Explicit Criterion of the Positional Relationship of Two Planar Ellipses

Detecting the positional relationship between two planar ellipses 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 ellipses, explicit conditions are derived for classifying the various positional relationships of two planar ellipses 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 two ellipses