Abstract: In many scientific fields such as computer animation,computer graphics,computer aided geometry design and robotics,it is very common to need detecting the positional relationship of several entities.Based on the generalized characteristic polynomial,we give an algebraic condition for detecting the positional relationship of two planar ellipses,namely,separation,exterior contact,intersection,interior contact or inclusion.The algebraic condition can be classified by the distribution of roots of the generalized characteristic equation. The criterion is simple,effective,and easier than other known methods.