Fisheye lens

In photography, a fisheye lens is a wide-angle lens that takes in an extremely wide, hemispherical image. Originally developed for use in astronomy and called "whole-sky lenses", fisheye lenses quickly became popular in general photography for their unique, distorted appearance. They are often used by photographers shooting broad landscapes to suggest the curve of the Earth.

All ultra-wide angle lenses suffer from some amount of distortion. While this can easily be corrected for moderately wide angles of view, rectilinear ultra-wide angle lenses with angles of view greater than 90 degrees are difficult to design. Fisheye lenses achieve extremely wide angles of view by foregoing a rectilinear image, opting instead for a special mapping (for example: equisolid angle), which gives images a characteristic convex appearance. A panorama by rotating lens or stitching images (cylindrical perspective) is not a fisheye photo.

Types of fisheye lenses

Circular

The first types of fisheye lenses to be developed were "circular fisheyes" - lenses which took in a 180-degree hemisphere and projected this as a circle within the film frame. Some circular fisheyes were available in orthographic projection models for scientific applications.

Full-frame

As fisheye lenses gained popularity in general photography, camera companies began manufacturing fisheye lenses that enlarged the image circle to cover the entire 35 mm film frame. Because of this, the picture angle produced by these lenses only measures 180 degrees when measured from corner to corner. The first full-frame fisheye lens to be mass-produced was a 16 mm lens made by Nikon in the late 1960s. This is the fisheye most commonly used by photographers.

Focal length

The focal lengths of fisheye lenses depend on the film format. For the popular 35 mm film format, typical focal lengths of fisheye lenses are between 8 mm and 10 mm for circular lenses, and 15-16 mm for full-frame lenses.

The widest lens ever produced was a 6 mm circular fisheye made by Nikon. Initially designed for an expedition to Antarctica, it featured a 220-degree field of view, designed to capture the entire sky and surrounding ground when pointed straight up. This lens is still manufactured by Nikon upon special orderand is used nowadays to produce interactive virtual-reality images such as QuickTime VR and IPIX. Having an f-ratio of , it is very large and cumbersome - weighing 5.2 Kg (11.5 lb) and having a diameter of 236 mm (9.3 in). It dwarfs a regular 35mm SLR camera[http://www.mir.com.my/rb/photography/companies/nikon/nikkoresources/late70nikkor/fisheyes/6mmf28.htm and has its own tripod mounting point, a feature normally seen in large long-focus or telephoto lenses to reduce strain on the lens mount because the lens is heavier than the camera.

Other uses

The peepholes in most doors contain a fisheye lens.
Most planetariums use a form of fisheye lens to project a two-dimensional film image of the night sky onto the interior of a dome.
Similarly, the Omnimax motion-picture format involves photography through a circular fisheye lens, and projection through the same onto a hemispherical screen.

Mapping function

The mapping of a sideways object leads to a picture position displacement from the image centre. The manner of this conversion is the mapping function. The distance of a point from the image centre 'r' is dependent on the focal length of the optical system 'f', and the angle from the optical axis 'θ'.

Normal (non-fisheye) lens:

Gnomonical: $r = f * tan(\theta)$ . Works like the pinhole camera. Straight lines remain straight (distortion free). "θ" has to be smaller than 90°. The aperture angle is gauged symmetrically to the optical axis and has to be smaller than 180°. Large aperture angles are difficult to design and lead to high prices.

Fisheyes can have many different mapping functions:

Linear scaled (equidistant): $r = f \cdot \theta$ , where θ is in radians. Practical for angle measurement e.g star maps. PanoTools uses this type.

Orthographic: $r = f * sin(\theta)$ . Looks like an orb with the surroundings lying on < max. 180° aperture angle.

Equal area (equisolid angle): $r = 2 f \sin(\theta / 2)$ . Looks like a mirror image on a ball, best special effect (unsophisticated distances), suitable for area comparison (clouds grade determination). This type is popular but it compresses marginal objects. The prices of these lenses are high, but not extreme.

Stereographic (conform): $r = 2 f \tan(\theta / 2)$ . This mapping would be ideal for photographers because it doesn't compress marginal objects. Although no lens has yet been developed for this type, this mapping is easily implemented by software.

All types of fisheye lens bend straight lines. Aperture angles of 180° or more are possible only with large amounts of barrel distortion.