Ray tracing in monocentric and ball lenses by a general exact formula

In this work, we derive a general ray-tracing formula for monocentric lenses based on a linear form of Snell's law. From this formula, we can directly get the meridional formula for a single symmetric (or asymmetric) monocentric lens. All of these formulas can be reduced to their corresponding paraxial forms, and their spherical aberration exact formulas can be easily derived. Also, we easily apply this general formula to get the position of the final image formed by a system of monocentric lenses having a common optical axis. The successive lenses of this system can be a mixture of different monocentric lenses of any kind having known separation distances between their centers. We have no need to apply Snell's law in its trigonometric form, no need to determine the angles of incidence or refraction, and no need to know the height of the ray at any point along its path. If the ray between any two lenses crosses the optical axis, the procedure will continue without any change and without adding any special equations. We also provide an exact procedure to trace a skew ray through a system of monocentric lenses that can be centered or decentered. In this procedure, we have no need to determine the points of incidence or the normal to the surface. The contribution of the procedure in this work is that we can apply our equations lens by lens, and not surface by surface, as in classical ray-tracing methods. This greatly reduces effort and mathematical complexity.