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We analyze the characteristic polynomial associated to an ellipsoid andanother quadric in the context of the contact detection problem. We obtain anecessary and sufficient condition for an efficient method to detect contact.This condition is a feature on the size and the shape of the quadrics and canbe checked directly from their parameters. Under this hypothesis, contact canbe noticed by means of discriminants of the characteristic polynomial.Furthermore, relative positions can be classified through the sign of thecoefficients of this polynomial.

Contact detection between an ellipsoid and a combination of quadrics