
Cardinal points
The paraxial properties of an axisymmetric lens can be described in terms of its cardinal points. The cardinal points of a lens are:
- first nodal point
- Nodal point for reverse beam. A reverse ray aimed at this point will exit the lens travelling parallel to its initial direction.
- first principal focus
- Focal point for reverse beam. Reverse rays parallel to the axis will converge on this point.
- first principal point
- Principal point for reverse beam. The point containing the principal plane, where a thin lens can replace the lens, for reverse rays.
- second nodal point
- Nodal point for forward beam. A forward ray aimed at this point will exit the lens travelling parallel to its initial direction.
- second principal focus
- Focal point for forward beam. Forward rays parallel to the axis will converge on this point.
- second principal point
- Principal point for forward beam. The point containing the principal plane, where a thin lens can replace the lens, for forward rays.
The definition of the principal and focal points is given in the figure below. For convenience we also show the vertex V1 of the first surface of the lens (the first surface after the object surface that is not a coordinate break) and the vertex V2 of the last surface of the lens (the surface before the image). The first princpal focus is the point F1 where a reverse beam parallel to the axis will cross the axis after being deflected by the lens. and the first principal point P1is the interssection of the axis and the plane where a thin lens would be positioned to mimic this focusing of a reverse ray. The second princpal focus is the point F2 where a forward beam parallel to the axis will after being deflected by the lens., and the second principal point P2is the interssection of the axis and the plane where a thin lens would be positioned to mimic this focusing of a forward ray.

Numerically these points are specified by their z-coordinate relative to the first glass surface for "first" points and relative to the last glass surface for "second" points.
The convention is that the first cardinal points are given by their distance to the right of
vertex V1 and the second cardinal points are given by their distance to the right of
vertex V2.
With this convention, the firstPrincipalFocus
(F1) is equal to the negative of the front focal length
and the secondPrincipalFocus
(F2) is equal to the back focal length.
In the figure above, the value of
firstPrincipalPoint
(position of P1 relative to V1) is postive and
the value of secondPrincipalPoint
(position of P2 relative to V2) is negative.
Closely related to the cardinal points are the focal lengths:
- first focal length
- Distance between first focal point and first principal point
- second focal length
- Distance between second focal point and second principal point
The sign convention for the focal lengths is that the focal length is positive if the ray reaches the focal point after crossing the principal plane. For example, in the figure below for a biconvex lens the first and second focal lengths are f1 and f2 which are both positive:

The nodal points are where the rays have to cross the axis to exit the lens at the same angle they entered it,
as shown in the figure below.
The sign convention for nodal points is the same as for principal points.
In the figure,
firstNodalPoint
is positive and
secondNodalPoint
is negative. If the material on the input side of the lens has the same index of
refraction as the material on the output side, then the nodal points coincide with the principal points.

The cardinal points are paraxial properties and are all determined by the three (!) independent quantities in the ABCD matrix.
The cardinal points are displayed on the prescription listing.