<p>We derive a generalized Snell’s Law for refraction and reflection in the presence of a phase discontinuity using the calculus of variations. We then apply this to several specific situations, especially when there is a surface discontinuity between air and a nematic liquid crystal. In this case, we take the refractive index of the liquid crystal to be the effective refractive index for the extraordinary ray. We also apply a variational framework of wavefronts and Eikonal functions to give an alternate derivation of Snell’s Law in the presence of a phase discontinuity. In particular, a precise interpretation of light rays as orthogonal trajectories of wavefronts is provided for general inhomogeneous, anisotropic media. Finally, the Eikonal equation for the extraordinary ray in a nematic liquid crystal is derived by determining the correct Hamiltonian function for the relevant Hamilton-Jacobi equation.</p>