Paraxial Geometric Optics in 3D Through Point-Based Geometric Algebra

The versors of a homogeneous-point-based geometric algebra \(\mathbb {R}_{d,0,1}\) (dubbed HGA) are related to the basic operations in geometric paraxial optics. Odd versors represent reflections in spherical mirrors (be they concave or convex) and even versors implement the lens equation. We extend the results to arbitrarily positioned optical elements by embedding \(\mathbb {R}_{d,0,1}\) into CGA \(\mathbb {R}_{d+1,1}\) . The total transformation through a paraxial optical system now consists of successive teleportation (by CGA dot and outer product) to the next optical center, and then applying its local HGA versors. The result is a straightforward sequence of operations which implements a total system of arbitrarily placed paraxial lenses and mirrors in 3D (or any dimension), parameterized by their CGA tangent vectors (from each optical center to the corresponding focal point) for each optical component. This can be used to compile the homogeneous transformation matrices of a total paraxial system in terms of those geometric parameters.