In Gaussian optics, the **cardinal points** consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the **focal points**, the **principal points**, and the **nodal points**.^{[1]} For *ideal* systems, the basic imaging properties such as image size, location, and orientation are completely determined by the locations of the cardinal points; in fact only four points are necessary: the focal points and either the principal or nodal points. The only ideal system that has been achieved in practice is the plane mirror,^{[2]} however the cardinal points are widely used to *approximate* the behavior of real optical systems. Cardinal points provide a way to analytically simplify a system with many components, allowing the imaging characteristics of the system to be approximately determined with simple calculations.

The cardinal points lie on the optical axis of the optical system. Each point is defined by the effect the optical system has on rays that pass through that point, in the paraxial approximation. The paraxial approximation assumes that rays travel at shallow angles with respect to the optical axis, so that **cardinal points** consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the **focal points**, the **principal points**, and the **nodal points**.^{[1]} For *ideal* systems, the basic imaging properties such as image size, location, and orientation are completely determined by the locations of the cardinal points; in fact only four points are necessary: the focal points and either the principal or nodal points. The only ideal system that has been achieved in practice is the plane mirror,^{[2]} however the cardinal points are widely used to *approximate* the behavior of real optical systems. Cardinal points provide a way to analytically simplify a system with many components, allowing the imaging characteristics of the system to be approximately determined with simple calculations.
Gaussian optics, the

The cardinal points lie on the optical axis of the optical system. Each point is defined by the effect the optical system has on rays that pass through that point, in the paraxial approximation. The paraxial approximation assumes that rays travel at shallow angles with respect to the optical axis, so that and .^{[3]} Aperture effects are ignored: rays that do not pass through the aperture stop of the system are not considered in the discussion below.

The front focal point of an optical system, by definition, has the property that any ray that passes through it will emerge from the system parallel to the optical axis. The rear (or back) focal point of the system has the reverse property: rays that enter the system parallel to the optical axis are focused such that they pass through the rear focal point.

The front and rear (or back) focal *planes* are defined as the planes, perpendicular to the optic axis, which pass through the front and rear focal points. An object infinitely far from the optical system forms an image at the rear focal plane. For objects a finite distance away, the image is formed at a different location, but rays that leave the object parallel to one another cross at the rear focal plane.

A diaphragm or "stop" at the rear focal plane can be used to filter rays by angle, since:

- It only allows rays to pass that are emitted at an angle (relative to the optical axis) that is sufficiently small. (An infinitely small aperture would only allow rays that are emitted along the optical axis to pass.)
- No matter where on the object the ray comes from, the ray will pass through the aperture as long as the angle at which it is emitted from the object is small enough.

Note that the aperture must be centered on the optical axis for this to work as indicated. Using a sufficiently small aperture in the focal plane will make the lens telecentric.

Similarly, the allowed range of angles on the output side of the lens can

The front focal point of an optical system, by definition, has the property that any ray that passes through it will emerge from the system parallel to the optical axis. The rear (or back) focal point of the system has the reverse property: rays that enter the system parallel to the optical axis are focused such that they pass through the rear focal point.

The front and rear (or back) focal *planes* are defined as the planes, perpendicular to the optic axis, which pass through the front and rear focal points. An object infinitely far from the optical system forms an image at the rear focal plane. For objects a finite distance away, the image is formed at a different location, but rays that leave the object parallel to one another cross at the rear focal plane.