Computing sphere and spherical shell intersections with conformal geometric algebra

Abstract

This paper studies the intersection of multiple spherical shells and presents a dimension-independent algorithm based on conformal geometric algebra (CGA), providing both a decision test and a characterization of the feasible region. At a high level, the method proceeds in two stages. First, it tightens each shell's radius interval by comparing it with the result of the intersection of spheres. Second, it reconciles shells pairwise to ensure mutual compatibility, updating bounds until either emptiness is certified or a consistent family of intervals is obtained. The output is a compact, interpretable description of the solution set that integrates directly with standard CGA pipelines. This formulation is especially useful when multiple shells must be enforced simultaneously. Examples in R3 and R6 illustrate how the approach captures interval-based constraints without resorting to dimension-specific constructions or ad hoc geometric casework.

This article is part of the theme issue ‘Modern applications of geometric algebra’.