The caustic structure near a grazing point in three-dimensional space

By definition, a grazing point is a point of the edge of a concave mirror where the incident ray is tangent to the mirror. In its neighborhood, the rays are reflected a great number of times and an ordered series of caustics is formed, a caustic of order n corresponding to n reflections by the mirror. We study, in the framework of geometrical optics in three dimensions, the structure of this remarkable infinite set of caustic surfaces passing through the grazing point. We calculate the geometrical characteristics of the caustics at the grazing point, namely, their Gaussian curvatures and their mean curvatures. We show that these characteristics tend to those of the mirror when the caustic's order tends to infinity We also show that the boundary of every caustic surface forms a corner at the grazing point. We calculate the angle of the corner and we show that this angle tends to zero when the caustic's order tends to infinity We show that, unlike the case of a grazing point in the plane, a grazing point in three dimensional space possesses the fundamental property of being stable. It thus constitutes a new generic type of optical singularity, in addition to the well known five Lagrangian singularities of space, i.e. the fold, the cusp, the swallowtail, the elliptic umbilic, and the hyperbolic umbilic.