In many fields of computer science such as computer animation, computer graphics, computer aided geometric design and robotics, it is a common problem to detect the positional relationships of several entities. Based on generalized characteristic polynomials and projective transformations, we derive algebraic conditions for detecting the various positional relationships of two planar conics, namely, outer separation, exterior contact, intersection, interior contact and inclusion. We then apply the results to detecting the positional relationships of a cylinder (or a cone) and a quadrics. The criteria is very effective and easier to use than other known methods.