Beyond the ABCDs: A projective geometry treatment of paraxial ray tracing using homogeneous coordinates

Homogeneous coordinates are a projective geometry tool particularly well suited to paraxial geometric optics. They are useful because they allow the expression of rotations, translations, affine transformations, and projective transformations as linear operators (matrices). While these techniques are common in the computer graphics community, they are not well-known to physicists. Here we apply them to paraxial ray tracing. Geometric optics is often implemented by tracing the paths of non-diffracting rays through an optical system. In the paraxial limit rays can be represented through their height and slope and traced through the optical system using ray transfer (colloquially, "ABCD") matrices. The homogeneous representation allows us to add translations and rotations to our set of operations, allowing us to consider decentered and rotated optical elements. The homogeneous representation also distinguishes the orientation of rays (forwards or backwards). Using projective duality, we also show how to image points through an optical system, given the homogeneous ray transfer matrix. We demonstrate the usefulness of these methods with several examples, and discuss future directions to expand this technique.