[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient
condition for an efficient method to detect contact. This condition, named smallness
condition, is a feature on the size and the shape of the quadrics and can be checked
directly from their parameters. Under this hypothesis, contact can be noticed by means
of the expressions in a discriminant system of the characteristic polynomial. Furthermore,
relative positions can be classified through the sign of the coefficients of this polynomial.
As an application of these results, a method to detect contact between a small ellipsoid
and a combination of quadrics is given
