Classification of the relative positions between a circular hyperboloid of one sheet and a sphere

Abstract

We characterize all possible relative positions between a circular hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. As an application, this provides a method to detect contact between the 2 surfaces by a simple calculation in many real world applications.

Classification of the relative positions between a small ellipsoid and an elliptic paraboloid

“…For example, for a general S and P, the sum of the paths 1], ensures that there is not contact between the surfaces and the center at the end of the path is located in the negative part of the OZ-axis (see Figure 3 (a)). Now, a similar argument to that given in assertion (1) applies to prove assertion (2). Proof.…”

“…In this section we are going to carefully analyze the relation between the multiple roots and the existence of a tangent point between S and P. First note that if the quadrics are tangent then there exists a multiple root. The proof of the following two results is analogous to those given in [2] (see Lemma 26 and Lemma 25, respectively), so we omit them in the interest of brevity. Lemma 8.…”

“…The roots −a 2 and −b 2 are indeed special in Lemma 9. The following example shows how −a 2 can be a double root and there is no tangency between S and P: P : x 2 1,2 2 + y 2 1,5 2 − z = 0, S : x 2 + (y − 0, 5) 2 + (z − 0, 712045) 2 = 0, 25 2 .…”

“…If a = b the paraboloid is circular and is an exceptional case in the analysis developed in Sections 3 and 5. Situations like this, where the quadric is a surface of revolution, were considered previously in the literature (see [2]). When we work with a circular paraboloid and a sphere, −a 2 is always a root of the characteristic polynomial.…”

“…Thus, since 0 is never a root by Lemma 5, a position where S is interior to P and S is tangent to P from inside has roots λ 3 ≤ λ 4 < 0 for any -paraboloid, so in the limit, when → 0, also has λ 3 ≤ λ 4 < 0. The fact that a position of contact between S and P provides complex roots for f is obtained by redoing the argument of Lemma 14 for the circular paraboloid, taking into account that −a 2 is always a root and working with a third degree polynomial (see also [2] for an analogous situation with a circular hyperboloid). For any -paraboloid, that S is tangent from outside to P is characterized by a positive double root λ 3 = λ 4 > 0, so when → 0 this will happen too, because 0 is not a root.…”