Linear and geometric algebra approaches for sphere and spherical shell intersections in [formula omitted

Abstract

Many practical problems involve sphere intersections. Examples include but are not limited to estimations using the Global Positioning System (GPS), data science applications and 3D protein structure determination. Motivated by practical situations, where radii of spheres are not known precisely, we consider what happens when a spherical shell must be included in the intersection. We present and compare two approaches for this problem: one uses linear algebra and the other is based on conformal geometric algebra (CGA). The theoretical development is illustrated with some numerical examples, where it is possible to note the main advantage of CGA compared to the linear algebra approach: even in dimensions higher than three, CGA naturally preserves the geometric intuition of the problem.