The cardinal points and the associated cardinal planes are a set of special points and planes in an optical system, which help in the analysis of its paraxial properties. The analysis of an optical system using cardinal points is known as gaussian optics, named after Carl Friedrich Gauss. A spatial point is an entity with a location in space but no extent (volume, area or length). ... Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ... See also: List of optical topics Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... In geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system (such as a lens). ... (help· info) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...

The cardinal points and planes of an optical system include:

• The focal points and focal planes
• The principal planes and principal points
• The surface vertices (or vertexes)
• The nodal points

For a lens, there will be two of each of these, identified by "front" and "rear" depending on whether they are on the input or the output side of the lens, respectively. The focus or image point is the point where light rays, originating from a point in the object, converge [1]. The principal focus or focal point of a lens or parabolic mirror is the point onto which collimated light parallel to the axis is focused. ... A lens. ...

These points and planes, together with the aperture stop, and the chief and marginal rays of the system, define the locations and sizes of the entrance and exit pupils of the system, as well as its other image-forming properties, such as the focal length and magnification. In optics, an aperture is something which restricts the diameter of the light path through one plane in an optical system. ... In an optical system, the entrance pupil is a virtual aperture that defines the area at the entrance of the system that can accept light. ... The exit pupil is a concept in optics, and is defined as the minimum diameter of the light beam leaving an eyepiece though which all of the light from the eyepiece passes. ... For images in Wikipedia, see Wikipedia:Images. ... The focal point F and focal length f of a positive lens, a negative lens, a concave mirror, and a convex mirror. ... Magnification is the process of enlarging something only in appearance, not physical size. ...

More detailed and accurate analysis of an optical system's performance can be achieved by raytracing, either within the paraxial approximation or using "real rays", i.e. rays that refract and reflect according to Snell's law and the law of reflection, without approximation. A ray traced scene. ... Snells law is the simple formula used to calculate the refraction of light when travelling between two media of differing refractive index. ... The reflection of sunlight on water Reflection is the abrupt change in direction of a wave front at an interface between two dissimilar media so that the wave front returns into the medium from which it originated. ...

Definitions

The cardinal points of a thick lens in air.
F, F' front and rear focal points,
P, P' front and rear principal points,
V, V' front and rear surface vertices.

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. Aperture effects are ignored—rays that do not pass through the aperture stop of the system are ignored in the discussion below. In telecommunication, the term optical axis has the following meanings: 1. ... In optics, a ray is an idealized narrow beam of light. ...

Focal points and planes

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 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 focus or image point is the point where light rays, originating from a point in the object, converge [1]. The principal focus or focal point of a lens or parabolic mirror is the point onto which collimated light parallel to the axis is focused. ... The focal point F and focal length f of a positive lens, a negative lens, a concave mirror, and a convex mirror. ...

The front and rear focal planes are defined as the planes, perpendicular to the optic axis, which pass through the front and rear focal points. An object an infinite distance away from the optical system forms an image at the rear focal plane. For images in Wikipedia, see Wikipedia:Images. ...

Principal planes and points

The two principal planes have the property that a ray emerging from the lens appears to have crossed the rear principal plane at the same distance from the axis that that ray appeared to cross the front principal plane, as viewed from the front of the lens. The principal planes are crucial in defining the optical properties of the system, since it is the distance of the object and image from the front and rear principal planes that determines the magnification of the system. The principal points are the points where the principal planes cross the optical axis. Magnification is the process of enlarging something only in appearance, not physical size. ...

If the medium surrounding the optical system has a refractive index of 1 (e.g. air), then the distance from the principal planes to their corresponding focal points is just the focal length of the system. If the medium is not air or vacuum, the distance to the foci is multiplied by the index of refraction of the medium. The refractive index of a material is the factor by which the phase velocity of electromagnetic radiation is slowed relative to vacuum. ... The focal point F and focal length f of a positive lens, a negative lens, a concave mirror, and a convex mirror. ... Look up Vacuum in Wiktionary, the free dictionary For other uses, see vacuum (disambiguation) A vacuum is a volume of space that is empty of matter, including air, so that gaseous pressure is much less than standard atmospheric pressure. ...

For a thin lens in air, the principal planes both lie at the location of the lens. The point where they cross the optical axis is sometimes misleadingly called the optical centre of the lens. Note, however, that for a real lens the principal planes do not necessarily pass through the centre of the lens, and in general may not lie inside the lens at all. A lens can be considered a thin lens if d << f. ...

Surface vertices

The surface vertices are the points where each surface crosses the optical axis. They are important primarily because they are the physically measurable parameters for the position of the optical elements, and so the positions of the other cardinal points must be known with respect to the vertices to describe the physical system.

In anatomy, the surface vertices of the eye's lens are called the anterior and posterior poles of the lens[1]. Anatomical drawing of the human muscles from the Encyclopédie. ... Light from a single point of a distant object and light from a single point of a near object being brought to a focus by changing the curvature of the lens. ...

Nodal points

N, N' The front and rear nodal points of a thick lens.

The front and rear nodal points have the property that a ray that passes through one of them will also pass through the other, and with the same angle with respect to the optical axis. The nodal points therefore do for angles what the principal planes do for transverse distance. If medium on both sides of the optical system is the same (e.g. air), then the front and rear nodal points coincide with the front and rear principal planes, respectively.

The nodal points are widely misunderstood in photography, where it is commonly asserted that the light rays "intersect" at "the nodal point", that the iris diaphragm of the lens is located there, and that this is the correct pivot point for panoramic photography, so as to avoid parallax error. These claims are all false, and generally arise from confusion about the optics of camera lenses, as well as confusion between the nodal points and the other cardinal points of the system. The correct pivot point for panoramic photography can be shown to be the centre of the system's entrance pupil.^{[1] [2]} Photography is the process of making pictures by means of the action of light. ... Parallax (Greek: παραλλαγή (parallagé) = alteration) is the change of angular position of two stationary points relative to each other as seen by an observer, due to the motion of an observer. ... In an optical system, the entrance pupil is a virtual aperture that defines the area at the entrance of the system that can accept light. ...

Notes and References

1. ↑ Kerr, Douglas A. (2005). The Proper Pivot Point for Panoramic Photography. (PDF) The Pumpkin. URL accessed on 2006-03-05.
2. ↑ van Walree, Paul. Misconceptions in photographic optics. URL accessed on 2006-03-05. Item #6.

• Greivenkamp, John E. (2004). Field Guide to Geometrical Optics, SPIE Field Guides vol. FG01, SPIE. ISBN 0-8194-5294-7.
• Hecht, Eugene (1987). Optics, 2nd ed., Addison Wesley. ISBN 0-201-11609-X.
• Lambda Research Corporation (2001). OSLO Optics Reference (PDF), Version 6.1. URL accessed 2006-03-05. Pages 74–76 define the cardinal points.