The theorem of Malus and Dupin

The vast majority of lenses are used for image formation. Camera lenses, eye glasses, binoculars, and all sorts of other lens systems are useful because they form, each in their own way, an image of an object. To begin a discussion of the image forming process we choose a fixed source point P0 in the object space of a lens that we wish to study, and follow the rays emerging from this point all the way through the lens to the image space. One might hope that the rays emerging into the image space pass through a single point again, but this is rarely the case. Usually each of the emerging rays intersects the nominal image plane at a slightly different point. It is the task of the lens designer to bring, for a specified set of conditions, these intersection points as close together as possible.

Although the emerging rays do not usually pass through one point, the ray patterns in the image space created by a single source point in the object space are not completely arbitrary. Fermat's principle imposes an important restriction. To analyze this restriction we choose in the image space any convenient Cartesian coordinate system (x, y, z) and introduce the path function E(x, y) from the source point P0 to points (x, y, 0) in the z = 0 coordinate plane.