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 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 discriminants of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial.

翻译：我们分析接触检测问题中与一个环球体和另一个四等相联的特性。我们获得一个有效检测接触方法的必要和充分条件。这是四等体大小和形状的特点,可以直接从参数中检查。在这种假设下,通过对特性多等体的辨别,可以注意到接触。此外,相对位置可以通过该多等体系数的标记进行分类。