||The Area of the Solid of Intersection of a Sphere and an Ellipsoid, a Parametric Approach
||Vladimir SHELOMOVSKII, Wei-Chi YANG and Skip THOMPSON
In a previous paper we considered the intersection of a sphere and an ellipsoid using rectangular coordinates. In this paper we use a different approach based on using parametric coordinates and the use of a graphics program GInMA in order to gain further insight into this problem. As in our previous paper, we determine the surface area of the respective portions of the ellipsoid and the sphere that are inside each other. We provide examples to illustrate the various possibilities that arise and we provide Maple worksheets that can be used to deal with the calculations that must be performed. The task of the present paper is the derivation of the equations that allow us to represent graphically the solid of intersection and to calculate its surface area accurately and efficiently. We choose a system of coordinates with the origin at the center of the sphere and its axis directed toward the center of a spherical piece of the solid of intersection. We examine a variable step-size integration method. For an accuracy of approximately 10-6, 100 to 600 calculation points are typically sufficient and a typical calculation time is less than a minute.