A Simple Mathematical Approach for Determining Intersection of Quadratic Surfaces

Abstract

This paper is primarily concerned with the mathematical formulation of the conditions for intersection of two surfaces described by general second degree polynomial (quadratic) equations. The term quadric surface is used to denote implicitly a surface described by a quadratic equation in three variables. Of special interest is the case of two ellipsoids in the three dimensional space for which the determination of intersection has practical applications. Even the simplest of traditional approaches to this intersection determination has been based on a constrained numerical optimization formulation in which a requisite combined rotational, translational and dilational transformation reduces one of the ellipsoids to a sphere, and then a numerical search procedure is performed to obtain the point on the other ellipsoid closest to the sphere’s center. Intersection is then determined according to whether this shortest distance exceeds the radius of the sphere. An alternative novel technique, used by Alfano and Greer [AG01] is based on formulating the problem in four dimensions and then determining the eigenvalues which yield a degenerate quadric surface. This method has strictly relied on many numerical observations of the eigenvalues to arrive at the conclusion whether these ellipsoids intersect. A rigorous mathematical formulation and solution was provided by Chan [Cha01] to explain the myriads of numerical observations obtained through trial and error using eigenvalues. Moreover, it turns out that this mathematical analysis may also be extended in two ways. First, it is also valid for quadric surfaces in general: ellipsoids, hyperboloids of one or two sheets, elliptic paraboloids, hyperbolic paraboloids, cylinders of the elliptic, hyperbolic and parabolic types, and double elliptic cones. (The term ellipsoids includes spheres, and elliptic includes circular.) The general problem of analytically determining the intersection of any pair of these surfaces is not simple. This formulation provides a much desired simple solution. The second way of generalization is to extend it to n dimensions in which we determine the intersection of higher dimensional surfaces described by quadratic equations in n variables. The analysis using direct substitution and voluminous algebraic simplification turns out to be very laborious and troublesome, if at all possible in the general case. However, by using abstract symbolism and invariant properties of the extended (n+1) by (n+1) matrix, the analysis is greatly simplified and its overall structure made comprehensive and comprehensible. These results are also included in this paper. They also serve as a starting point for further theoretical investigations in higher dimensional analytical geometry.