Ray tracing

Remarks

Example: in a homogeneous medium, $n$ is constant and $\nabla n = 0$. The ray equation becomes $n \, x'' = 0$, so rays are straight lines, and ray tracing reduces to intersecting lines with surfaces.

Non-example: in the medium $n(x, y, z) = 1 + a y$ with $a > 0$, the straight line $x(s) = (s, 0, 0)$ is not a ray. Along it $\frac{d}{ds}(n x') = 0$ while $\nabla n = (0, a, 0) \ne 0$, so clause 1 fails; the true ray starting in this direction bends towards larger $y$, where the index is larger.

In Hamiltonian form, with $p = n \, dx/ds$ and the parameter $\sigma$ such that $d\sigma = ds / n$, the ray equation becomes the first-order system $dx/d\sigma = p$, $dp/d\sigma = n \nabla n$, with $\abs{p} = n$. This form is convenient for numerical integration.