Explicit Cartesian formulas y = f(x) for caustic curves in spherical reflection and refraction

The caustic curves produced by reflection and refraction at spherical surfaces have conventionally been described by polar, parametric, or implicit quartic equations. Deriving Explicit Cartesian formulas y = f(x) for the caustic curves from these equations is a hard task and requires solving high-degree equations. Despite its practical and theoretical importance, explicit formulas remained almost absent in geometrical optics. In this work, we bridge this long-standing gap by introducing a new analytical technique to derive these Cartesian formulas. The technique is based on new non-trigonometric forms of the Coddington equations. This reformulation could be achieved using a linear alternative to Snell's law. Hence, we could easily derive the explicit Cartesian formulas y = f(x) for the caustic curve equations in cases of reflection and refraction at spherical surfaces and for a plano-convex lens. The derivation is simple, and the obtained formulas are closed single-valued functions. The results of this paper provide a powerful new tool for theoretical research, optical design, and education.