Contact detection between a small ellipsoid and another quadric

[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.