Lens design uses a calculation of the lens' surfaces that permits us to obtain an image from a given object. A set of general rules and laws permits us to calculate the essential points of the optical system, such as distances, thickness, pupils, and focal distances, among others. Now, the theory on which classical lens design is based has changed radically, as our theoretical foundations do not rely on the classical ray-tracing rules. We show that with the rules expressed in a reduced vector analytical solution set of equations, we can take into account all optical elements, i.e., refractive, reflective, and catadioptric. These foundations permit us to keep under control the system aberration budget in every surface. It reduces the computation time dramatically. The examples presented here were possible because of the versatility of this theoretical approach.