Abstract: When propagating light\|ray passes through two fixed points in the gradient index (GRIN) mediums and a cuspidal variable\|point （at this point the trace equation of propagating light\|ray is non\|differentiable） at the borderline of two different GRIN mediums, the mathematical relationship, which the trace equation of propagating light\|ray and borderline equation of two different GRIN mediums should satisfy, is derived by the Fermat’s principle and calculus of variation. Some applied examples show that the differential relationship not only contains fundamental content in geometrical optics, such as the three laws of geometrical optics, imaging formula of single spheric face, but is another form of differential equation of the light ray.

Key words: geometrical optics, Fermat principle, variational calculus, equation of the light rays