The Fermat Algorithm: A Unified Calculus Approach to Geometric Optics

Standard geometric methods in optics often rely on laborious angle-chasing and fail when the refractive index varies continuously. This paper proposes replacing these geometric constructions with a unified calculus-based protocol: extremizing the optical-path integral using the Beltrami Identity. By exploiting translational symmetry, we derive Snell’s Law, the exact trajectory of light in gradient-index media (mirages), and the invariants of ray optics within a single framework. This "Fermat Algorithm" recasts Feynman’s variational arguments into an elementary routine suitable for advanced secondary school students, effectively bridging the gap between high school geometry and university analytical mechanics.