The usefulness of homogeneous coordinates in paraxial optics

Homogeneous coordinates are a well known tool in the computer graphics community because they allow the expression of rotations, translations, perspective transforms, and affine transformations as linear operators in homogeneous space, but these methods are rarely used in the physics community. Homogeneous coordinates are particularly well suited to paraxial geometric optics because they are the natural setting for projective geometry (specifically oriented projective geometry) which itself underlies optics in the paraxial limit.

Most optics work in this regard has been on the tracing of non-diffracting rays through the optical system.The key result we derive is that given the ray transfer matrix for an optical system, we can use homogenous coordinates to also easily define a point transfer matrix that maps points in the object space of the optical system onto their respective image points, including points infinitely far away. We demonstrate the usefulness of this method with several examples, and discuss future directions to expand this technique.