  Lens design uses a calculation of the lens' surfaces that permit to obtain an
image from a given object. A set of general rules and laws permits to calculate
the essential points of the optical system such as distances, thickness,
pupils, and focal distance among others. Now, the theory on which the classical
lens design is based changes 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, catadioptric. These
foundations permit 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.
