Computing sphere and spherical shell intersections with conformal geometric algebra

Area of Science:

• Computational geometry
• Geometric algebra
• Robotics and computer vision

Background:

• Intersections of spherical shells are fundamental in various geometric computations.
• Existing methods often lack dimension independence or require complex case analysis.

Purpose of the Study:

• To develop a unified, dimension-independent algorithm for analyzing multiple spherical shell intersections.
• To provide a robust method for characterizing the feasible region of such intersections.

Main Methods:

• The algorithm employs conformal geometric algebra (CGA) for a unified representation.
• It involves a two-stage process: tightening radius intervals and pairwise shell reconciliation.
• The method iteratively updates bounds to certify emptiness or find a consistent solution.

Main Results:

• A dimension-independent algorithm for spherical shell intersection is presented.
• The algorithm provides a decision test and characterizes the feasible region.
• The output is a compact representation compatible with standard CGA pipelines.

Conclusions:

• The proposed CGA-based method efficiently handles simultaneous constraints from multiple shells.
• It avoids dimension-specific constructions and ad hoc geometric casework.
• The approach is effective in various dimensions, as demonstrated in R3 and R6.