In the late 1830’s, Gauss discovered that the image positions and sizes in the paraxial region (i.e.,
for rays that are always “nearly ”parallel to the optical axis) may be determined from knowledge of
only four specific locations: the two focal points (F1, F2) and the two principal points (sometimes
labeled H1 and H2, though other symbols are also used). We have already defined the focal points
to be the locations on the optical axis of the images created from objects located at +∞ for F1 and
−∞ for F2. The principal points are pair of points (one each in object space and image space) that
are images of each other with magnification MT = +1. Note that this is not the pair of points with
so = si = 2f that we often think of with unit magnification; these are images at “equal conjugates”
with MT = −1.

In 1845, Listing developed the additional concept of nodal points, which are the points of equal
angular magnification. In other words, a ray at an angle θ from the optical axis and directed at the
front nodal point N1 will emerge from the back nodal point N2 at angle θ . Note that for a lens in
“air ”(actually, in vacuum), where both the object and image spaces have unit refractive index, the
nodal points and principal points are coincident. Figure 2 shows the concept of nodal points within
a thick lens.
For nonparaxial rays, the “surfaces ” of unit magnification are called the principal planes. The
intersection of the optical axis with the front and rear surface of the lens are called the front and rear
vertices respectively. Though not cardinal points, the vertices are used to specify the focal distances,
which are the distances F1V1 and V2F2 in Figure 1. By contrast, the object- and image-space focal
lengths are the distances F1H1 and H2F2.
Note that the “back” or “rear” focal distance V2F2 is a good figure to keep in mind for the
clearance required between the lens and the sensor in a real imaging system.

The locations of the six cardinal points of an existing lens may be determined by four means:
the nodal slide, the foco-collimator (or swinging collimator), the two-magnifications method, and the
use of the Newtonian lens formula. Four methods are briefly described below, and the last two will
be used in this experiment.