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Ray Tracing

The method of approximating the path of light by drawing straight lines (rays) through the optical system, simplifying calculations.
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The statement of the theorem

Let the optical medium be defined by a spatially varying refractive index n(r)n(\mathbf{r}). The propagation of the wavefront Φ(r,t)\Phi(\mathbf{r}, t) is governed by the eikonal equation, which is the scalar form of the wave equation in the high-frequency limit:\n(∂Φ∂r)2=n2(r)\left(\frac{\partial \Phi}{\partial \mathbf{r}}\right)^2 = n^2(\mathbf{r})\nwhere r=(x,y,z)\mathbf{r} = (x, y, z) and Φ\Phi is the eikonal function. The ray path r(s)\mathbf{r}(s) is parameterized by the arc length ss and follows the direction of the wave vector k=∇Φ\mathbf{k} = \nabla \Phi. The trajectory r(s)\mathbf{r}(s) must satisfy the differential equation:\ndds(n(r)drds)=∇n(r)\frac{d}{ds}\left(n(\mathbf{r}) \frac{d\mathbf{r}}{ds}\right) = \nabla n(\mathbf{r})\nThis equation, derived from the Hamiltonian formulation of the optical path length minimization, describes the continuous path of the ray through the inhomogeneous medium.
Source: Wikipedia