Beyond the ABCDs: A better matrix method for geometric optics by using homogeneous coordinates

Geometric optics is often described as tracing the paths of non-diffracting rays through an optical system. In paraxial limit, ray traces can be calculated using transfer matrices (colloquially, ABCD matrices), which are 2x2 acting on height and slope rays. A known limitation that they only work for elements centered normal to axis. this article, we provide improved 3x3 matrix method calculating systems applicable how these actually arranged table: lenses mirrors in any orientation or position (e.g.~in laboratory coordinates), with path zig-zagging along table. Using projective duality, also show directly image points system a point from system's matrix. We demonstrate usefulness methods several examples discuss future directions expand applications technique.