Fisheye lens

A fisheye lens is an ultra wide-angle lens that produces strong visual distortion intended to create a wide panoramic or hemispherical image.[4][5]^:145 Fisheye lenses achieve extremely wide angles of view. Instead of producing images with straight lines of perspective (rectilinear images), fisheye lenses use a special mapping (for example: equisolid angle), which gives images a characteristic convex non-rectilinear appearance.

Circular fisheye photograph of Oude Kerk Amsterdam

Fisheye

Introduced in	1924
Author	Wood (1905),[1] Bond (1922),[2] and Hill (1924)[3]
Construction	Var. elements in Var. groups

The term fisheye was coined in 1906 by American physicist and inventor Robert W. Wood based on how a fish would see an ultrawide hemispherical view from beneath the water (a phenomenon known as Snell's window).[1][5]^:145 Their first practical use was in the 1920s for use in meteorology[3][6] to study cloud formation giving them the name "whole-sky lenses". The angle of view of a fisheye lens is usually between 100 and 180 degrees[4] while the focal lengths depend on the film format they are designed for.

Mass-produced fisheye lenses for photography first appeared in the early 1960s[7] and are generally used for their unique, distorted appearance. For the popular 35 mm film format, typical focal lengths of fisheye lenses are between 8 mm and 10 mm for circular images, and 15–16 mm for full-frame images. For digital cameras using smaller electronic imagers such as ​¹⁄₄" and ​¹⁄₃" format CCD or CMOS sensors, the focal length of "miniature" fisheye lenses can be as short as 1 to 2 mm.

These types of lenses also have other applications such as re-projecting images that were originally filmed through a fisheye lens, or created via computer generated graphics, onto hemispherical screens. Fisheye lenses are also used for scientific photography such as recording of aurora and meteors, and to study plant canopy geometry and to calculate near-ground solar radiation. They are perhaps most commonly encountered as peephole door viewers to give the user a wide field of view.

History and development

"Vue circulaire des montagnes qu'on découvre du sommet du Glacier de Buet", Horace-Benedict de Saussure, Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve. Neuchatel, 1779–96, pl. 8.

Panoramas with fisheye distortion predate photography and the fisheye lens. In 1779, Horace Bénédict de Saussure published his downward-facing fisheye view of the Alps: "All the objects are drawn in perspective from the centre".[8]

Pail (top) and improved (bottom) camera

First known fisheye image recorded in 1905 using Wood's pail apparatus

Figures 1 and 2 from Wood's 1906 paper[1]

In 1906, Wood published a paper detailing an experiment in which he built a camera in a water-filled pail starting with a photographic plate at the bottom, a short focus lens with a pinhole diaphragm located approximately halfway up the pail, and a sheet of glass at the rim to suppress ripples in the water. The experiment was Wood's attempt "to ascertain how the external world appears to the fish" and hence the title of the paper was "Fish-Eye Views, and Vision under Water".[1] Wood subsequently built an improved "horizontal" version of the camera omitting the lens, instead using a pinhole pierced in the side of a tank, which was filled with water and a photographic plate. In the text, he described a third "Fish-Eye" camera built using sheet brass, the primary advantages being that this one was more portable than the other two cameras, and was "absolutely leaktight".[1] In his conclusion, Wood thought that "the device will photograph the entire sky [so] a sunshine recorder could be made on this principle, which would require no adjustment for latitude or month" but also wryly noted "the views used for the illustration of this paper savour somewhat of the 'freak' pictures of the magazines."[1]

Bond's hemispherical lens (1922)

W.N. Bond described an improvement to Wood's apparatus in 1922 which replaced the tank of water with a simple hemispheric glass lens, making the camera significantly more portable. The focal length depended on the refractive index and radius of the hemispherical lens, and the maximum aperture was approximately f/50; it was not corrected for chromatic aberration and projected a curved field onto a flat plate. Bond noted the new lens could be used to record cloud cover or lightning strikes at a given location.[2] Bond's hemispheric lens also reduced the need for a pinhole aperture to ensure sharp focus, so exposure times were also reduced.[9]

Hill Sky Lens

Hill/Beck "Sky Lens" (1923, GB 225,398)[10]

In 1924, Robin Hill first described a lens with 180° coverage that had been used for a cloud survey in September 1923[3] The lens, designed by Hill and R. & J. Beck, Ltd., was patented in December 1923.[10] The Hill Sky Lens is now credited as the first fisheye lens.[5]^:146 Hill also described three different mapping functions of a lens designed to capture an entire hemisphere (stereographic, equidistant, and orthographic).[3][11] Distortion is unavoidable in a lens that encompasses an angle of view exceeding 125°, but Hill and Beck claimed in the patent that stereographic or equidistant projection were the preferred mapping functions.[10] The three-element, three-group lens design uses a highly divergent meniscus lens as the first element to bring in light over a wide view followed by a converging lens system to project the view onto a flat photographic plate.[10]

The Hill Sky Lens was fitted to a whole sky camera, typically used in a pair separated by 500 metres (1,600 ft) for stereo imaging, and equipped with a red filter for contrast; in its original form, the lens had a focal length of 0.84 in (21 mm) and cast an image 2.5 in (64 mm) in diameter at f/8.[12] Conrad Beck described the camera system in an article published in 1925.[13] At least one has been reconstructed.[14]

German and Japanese development

Schulz/AEG Weitwinkelobjektiv (1932, DE 620538)[15]

In 1932, the German firm Allgemeine Elektricitäts-Gesellschaft AG (AEG) filed for a patent on the Weitwinkelobjektiv (wide-angle lens), a 5-element, 4-group development of the Hill Sky Lens.[5]^:148 [15] Compared to the 1923 Hill Sky Lens, the 1932 Weitwinkelobjektiv featured two diverging meniscus elements ahead of the stop and used a cemented achromatic group in the converging section.[15] Miyamoto credits Dr Hans Schulz with the design of the Weitwinkelobjektiv.[11] The basic patented design was produced for cloud recording as a 17 mm f/6.3 lens,[16] and the artist known as Umbo used the AEG lens for artistic purposes, with photographs published in a 1937 issue of Volk und Welt.[17]

The AEG Weitwinkelobjektiv formed the basis of the later Fish-eye-Nikkor 16 mm f/8 lens of 1938, which was used for military and scientific (cloud cover) purposes.[16][18] Nikon, which had a contract to supply optics to the Imperial Japanese Navy, possibly gained access to the AEG design under the Pact of Steel.[18] After the war, the lens was mated to a medium format camera and was produced in slightly modified form (focal length increased slightly to 16.3 mm) as the "Sky-image Recording Camera" in March 1957 for the Japanese government,[19] followed by a commercial release as the Nikon Fisheye Camera (also known as the "Nikon Sky Camera" or "Nikon Cloud Camera") in September 1960, which had a retail price of ¥200,000 (equivalent to ¥1,130,000 in 2019).[20] The revised lens created a circular image 50 mm (2.0 in) in diameter and covered a complete hemispherical field of 180°.[21] Only 30 examples of the Nikon Fisheye Camera were manufactured, and of those, 18 were sold to customers, mainly in the United States; Nikon likely destroyed the remaining stock to avoid tax penalties.[22] A photograph of pole vaulter Bob Gutowski taken by the Fisheye Camera was published in Life in 1957.[23]

Richter/Zeiss Pleon (1938, US 2,247,068)[24]

Also in 1938, Robert Richter of Carl Zeiss AG patented the 6-element, 5-group Pleon lens,[24] which was used for aerial surveillance during World War II. The converging rear group of the Pleon was symmetrical, reminiscent of the 4-element Topogon design, also designed by Richter for Zeiss in 1933. Testing on a captured lens after the war showed the Pleon provided an equidistant projection to cover a field of approximately 130°, and negatives were printed using a special rectifying enlarger to eliminate distortion.[5]^:149 [25] The Pleon had a focal length of approximately 72.5 mm with a maximum aperture of f/8 and used a plano-concave front element 300 mm (12 in) in diameter; the image on the negative was approximately 85 mm (3.3 in) in diameter.[25]

Merté/Zeiss Sphaerogon (1935, DE 672 393[26] and US 2,126,126)[27]

35 mm development

At approximately the same time that Schulz was developing the Weitwinkelobjektiv at AEG, Willy Merté at Zeiss was developing the Sphaerogon, which was also designed to encompass a 180° field of view.[26][27] Unlike the Weitwinkelobjektiv, Merté's Sphaerogon was not limited to medium format cameras; prototype versions of the Sphaerogon were constructed for the Contax I miniature format camera. The first prototype Sphaerogon lenses constructed had a maximum aperture of f/8, but later examples were computed half a stop faster, to f/6.8.[28] Several prototype examples of Sphaerogon lenses were recovered as part of the Zeiss Lens Collection seized by the Army Signal Corps as war reparations in 1945;[29] the collection, which the Zeiss firm had retained as a record of their designs, was later documented by Merté, the former head of optical computation for CZJ, working under Signal Corps officer Edward Kaprelian.[30][31]

The Nikon Fisheye Camera was discontinued in September 1961,[19] and Nikon subsequently introduced the first regular production fisheye lens for miniature cameras in 1962,[11] the Fish-eye-Nikkor 8 mm f/8,[32] which required the reflex mirror on its Nikon F and Nikkormat cameras to be locked up prior to mounting the lens. Prior to the early 1960s, fisheye lenses were used primarily by professional and scientific photographers, but the advent of the fisheye for the 135 format increased its popular use.[33] The Nikkor 8 mm f/8 has a field of view of 180° and uses 9 elements in 5 groups; it is fixed focus and has built-in filters mainly intended for black-and-white photography. Research indicates that fewer than 1,400 lenses were built.[34]

Nikon subsequently released several more milestone circular fisheye lenses in Nikon F mount through the 1960s and 70s:

• 10 mm f/5.6 OP (1968), the first fisheye to feature orthographic projection, which was also the first lens to feature an aspherical element[35]
• 6 mm f/5.6 (1969), the first fisheye to feature a 220° field of view;[7] interestingly, the patent accompanying this lens includes a design for a lens with a 270° field of view.[36] A 6.2 mm f/5.6 SAP fisheye was later produced in limited numbers with an aspherical surface, encompassing a 230° field of view.[37]
• 8 mm f/2.8 (1970), the first circular fisheye with variable focus, automatic aperture, and reflex viewing (mirror lock-up no longer required).[7]

Fish-eye Takumar 11/18mm on a modern Pentax K-1 DSLR

Meanwhile, other Japanese manufacturers were developing the so-called full-frame or diagonal fisheyes, which captured approximately a 180° field of view across the diagonal of the film frame. The first such diagonal fisheye was the Fish-eye Takumar 18 mm f/11, released by Pentax (Asahi Optical) in 1962,[37][38][39] followed by the slightly faster UW Rokkor-PG 18 mm f/9.5 from Minolta in 1966.[40] Both of these were reflex viewing and fixed-focus, and Pentax and Minolta followed up with faster lenses with variable focus in 1967 (Super Fish-eye-Takumar 17 mm f/4)[41] and 1969 (Rokkor-OK 16 mm f/2.8),[42][43] respectively. The 16 mm Rokkor was later adopted by Leica as the Fisheye-Elmarit-R (1974) and then converted to autofocus (1986) for the Alpha system. As of 2018, the same basic optical design is still sold as the Sony SAL16F28.

Design

Types of using format

circular		cropped circle	full-frame
3:2	52 % sensor	78 % FOV, 92 % sensor	59 % FOV
4:3	59 % sensor	86 % FOV, 90 % sensor	61 % FOV
Circular fisheye for 35 mm			Full-frame fisheye with rudimentary lens hood
ESO's VLT image taken with a circular fisheye lens.		35 mm circular fisheye with DX-format-camera	Full-frame fisheye used in a closed space (Nikkor 10.5mm)

In a circular fisheye lens, the image circle is inscribed in the film or sensor area; in a full-frame fisheye lens the image circle is circumscribed around the film or sensor area.

Further, different fisheye lenses distort images differently, and the manner of distortion is referred to as their mapping function. A common type for consumer use is equisolid angle.

Although there are digital fisheye effects available both in-camera and as computer software they can't extend the angle of view of the original images to the very large one of a true fisheye lens.

Focal length

The focal length is determined by the angular coverage, the specific mapping function used, and the required dimensions of the final image. Focal lengths for popular amateur camera sizes are computed as:

Calculated fisheye focal lengths[lower-alpha 1]

		Stereographic	Equidistant	Equisolid Angle	Orthographic
Inverse Mapping Function[44]		${\displaystyle f={\frac {r}{2\tan(\theta /2)}}}$	${\displaystyle f={\frac {r}{\theta }}}$	${\displaystyle f={\frac {r}{2\sin(\theta /2)}}}$	${\displaystyle f={\frac {r}{\sin(\theta )}}}$
Circular[lower-alpha 2]	APS-C (${\displaystyle r}$ = 8.4 mm)	4.2	5.3	5.9	8.4
	135 (${\displaystyle r}$ = 12 mm)	6.0	7.6	8.5	12.0
	6×6 (${\displaystyle r}$ = 28 mm)	14.0	17.8	19.8	28.0
Full-frame[lower-alpha 3]	APS-C (${\displaystyle r}$ = 15.1 mm)	7.5	9.6	10.6	15.1
	135 (${\displaystyle r}$ = 21.7 mm)	10.8	13.8	15.3	21.7
	6×6 (${\displaystyle r}$ = 39.6 mm)	19.8	25.2	28.0	39.6

Notes

1. Assumes 180° maximum angle of view for the mapping function, ${\displaystyle \theta ={\frac {\pi }{2}}}$
2. For circular fisheyes, the maximum dimension ${\displaystyle r}$ is half the length of the shortest side.
3. For full-frame fisheyes, the maximum dimension ${\displaystyle r}$ is half the length of the diagonal.

Circular fisheye

The first types of fisheye lenses to be developed were "circular fisheye"—lenses which took in a 180° hemisphere and projected this as a circle within the film frame. Some circular fisheyes were available in orthographic projection models for scientific applications. These have a 180° vertical angle of view, and the horizontal and diagonal angle of view are also 180°. By design, most circular fisheye lenses cover a smaller image circle than rectilinear lenses, so the corners of the frame will be completely dark.

Sigma currently makes a 4.5mm fisheye lens that captures a 180-degree field of view on a crop body.[45] Sunex also makes a 5.6mm fisheye lens that captures a circular 185-degree field of view on a 1.5x Nikon and 1.6x Canon DSLR cameras.

Fisheye-Nikkor 6mm f/2.8 mounted on a Nikon F2 in the Nikon Museum.

Nikon produced a 6 mm circular fisheye lens that was 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 no longer manufactured by Nikon,[46] and is used nowadays to produce interactive virtual-reality images such as QuickTime VR and IPIX. Because of its very wide field of view, it is very large and cumbersome—weighing 5.2 kilograms (11 lb), having a diameter of 236 millimetres (9.3 in), a length of 171 millimetres (6.7 in) and an angle of view of 220 degrees. It dwarfs a regular 35 mm SLR camera[47] 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. The lens is extremely rare.[48]

The fish eye lens Laowa 4 mm f/2,8 of the manufacturer Venus Optics

However, there are new developments by the Japanese manufacturer Entaniya for the Micro Four Thirds standard, which offer an angle of view of 250 degrees with lenses that have a focal length of 2.3 millimetres (0.091 in) to 3.6 millimetres (0.14 in), an aperture of f/2.8 to f/4.0, a weight of 1.6 kilograms (3.5 lb), a diameter of 120 millimetres (4.7 in) and a length below 100 millimetres (3.9 in).[49] In 2018 Venus Optics introduced a 210° fisheye lens for the Micro Four Thirds system.[50]

An 8 mm fisheye lens, also made by Nikon, has proven useful for scientific purposes because of its equidistant (equiangular) projection, in which distance along the radius of the circular image is proportional to zenith angle.

Full-frame fisheye

As fisheye lenses gained popularity in general photography, camera companies began manufacturing fisheye lenses that enlarged the image circle to cover the entire rectangular frame, called a "full-frame fisheye".[51]

The picture angle produced by these lenses only measures 180 degrees when measured from corner to corner: these have a 180° diagonal angle of view, while the horizontal and vertical angles of view will be smaller; for an equisolid angle-type 15 mm full-frame fisheye, the horizontal AOV will be 147°, and the vertical AOV will be 94°.[52]

One of the first full-frame fisheye lens to be mass-produced was the Fisheye-Nikkor 16mm f/3.5, made by Nikon in the early 1970s. Digital cameras with APS-C sized sensors require a 10.5 mm lens (or, for Canon APS-C cameras, a 10 mm lens) to get the same effect as a 16 mm lens on a camera with full-frame sensor.[53]

Miniature fisheye lenses

Miniature digital cameras, especially when used as security cameras, often tend to have fisheye lenses to maximize coverage. Miniature fisheye lenses are designed for small-format CCD/CMOS imagers commonly used in consumer and security cameras.[54][55] Popular image sensor format sizes used include ​¹⁄₄", ​¹⁄₃", and ​¹⁄₂". Depending on the active area of the image sensor, the same lens can form a circular image on a larger image sensor (e.g. ​¹⁄₂"), and a full frame on a smaller one (e.g. ​¹⁄₄").

Examples and specific models

Fisheye lenses for APS-C cameras

The APS-C image sensor used in Canon cameras is 22.3 mm × 14.9 mm (0.88 in × 0.59 in) or 26.82 mm (1.056 in) on the diagonal, which is slightly smaller than the sensor size used by other popular manufacturers of cameras with APS-C sensors, such as Fuji, Minolta, Nikon, Pentax, and Sony. The other common APS-C sensors range from 23.6 to 23.7 mm (0.93 to 0.93 in) on the long dimension and 15.6 mm (0.61 in) on the shorter side, for a diagonal measurement between 28.2 to 28.4 mm (1.11 to 1.12 in).

Circular APS-C fisheye lenses

• Sigma 4.5 mm f/2.8 for APS-C sensors
• Lensbaby 5.8 mm f/3.5 circular fisheye

Full-frame APS-C fisheye lenses

• Nikon 10.5 mm f/2.8
• Samyang 8 mm f/3.5. Notable for its stereographic projection
• Samyang 8 mm f/2.8. For various mirrorless cameras. Notable for its stereographic projection
• Sigma 10 mm f/2.8

Zoom APS-C fisheye lenses

• Pentax 10–17 mm f/3.5–4.5 / Tokina 10–17 mm f/3.5–4.5 (jointly developed)

Circular fisheye

A Peleng 8mm circular fisheye lens.

• Peleng 8 mm f/3.5
• Sigma 8 mm f/4.0
• Sigma 8 mm f/3.5 - replaces the Sigma 8 mm f/4.0 EX DG

Full-frame fisheye

• Canon EF 15 mm f/2.8 (discontinued)
• Canon Fisheye FD 15 mm f/2.8 (old lens, does not work on EF mount)
• Fuji Photo Film Co. EBC Fujinon Fish Eye 16mm F2.8 (m42 & X-Fujinon mounts)(discontinued)
• Minolta AF 16 mm f/2.8
• Sigma 15 mm f/2.8 EX DG Diagonal Fisheye
• Fisheye-Nikkor AF 16mm f/2.8 D
• Samyang 12 mm f/2.8 ED AS NCS Diagonal Fish-eye
• Zenitar 16 mm f/2.8 Fisheye lens

Zoom fisheye

• Canon EF 8–15 mm f/4L Fisheye USM – lens can be used as both a full-frame fisheye and a circular fisheye on a 35 mm full-frame film or DSLR such as the 5D (Mark I – IV) cameras; it can only be used as a cropped circular or as a full-frame fisheye on EOS DSLRs with APS-C/H size sensors (a zoom lock is included).
• Nikon AF-S Fisheye Nikkor 8–15 mm f/3.5–4.5E ED – designed for Nikon's full-frame FX DSLRs, this lens is a circular fisheye at the short end of the zoom range and becomes a full-frame fisheye at longer focal lengths.
• Tokina AT-X 10–17mm f3.4-4.5 AF DX – a fisheye zoom lens designed for APS-C sensor cameras. It is also sold as a NH version that comes without integrated lens hood, then the fisheye lens is usable on full frame cameras. The lens is also sold under Pentax brand.
• Pentax F 17–28 mm 1:3.5–4.5 Fisheye – lens was born for full-frame film cameras, to take the place of the 16 mm f/2.8 in the AF era. It starts from a 17 mm full-frame fisheye and reaches the end of the excursion as an overdistorted 28 mm. Was intended as a "special effect" lens and never had big sales.[56]
• Pentax DA 3,5-4,5/10-17 ED IF Fisheye is designed for APS-C cameras, but can be used on a full-frame after a small modification of the hood.

Sample images

• 
  An image of the Louvre museum entry taken with the 7.5 mm f/5.6 circular fisheye Nikkor lens
• 
  Fisheye used to capture entire Wells Cathedral Chapter House room
• 
  Canon 8–15 mm zoom at 8 mm of BMW M3
• 
  Image shot with a 16 mm full-frame fisheye lens before and after remapping to rectilinear perspective.[n 1]
• 
  Comparison of conventional (rectilinear) mapping function with four different fisheye mapping functions, given a constant focal length.

Other applications

The Curves of ESO’s Headquarters through a fish-eye lens.[57]

• Many planetariums now use fisheye projection lenses to project the night sky or other digital content onto the interior of a dome.
• Fish-eye lenses are used in POV pornography to make things right in front of the camera look bigger.
• Flight simulators and visual combat simulators use fisheye projection lenses in order to create an immersive environment for pilots, air traffic controllers, or military personnel to train in.
• Similarly, the IMAX Dome (previously 'OMNIMAX') motion-picture format involves photography through a circular fisheye lens, and projection through the same onto a hemispherical screen.
• Scientists and resource managers (e.g., biologists, foresters, and meteorologists) use fisheye lenses for hemispherical photography to calculate plant canopy indices and near-ground solar radiation. Applications include evaluation of forest health, characterization of monarch butterfly winter roosting sites, and management of vineyards.
• Astronomers use fisheye lenses to capture cloud cover and light pollution data.
• Photographers and videographers use fisheye lenses so they can get the camera as close as possible for action shots whilst also capturing context, for example in skateboarding to focus on the board and still retain an image of the skater.
• The "eye" of the HAL 9000 computer from 2001: A Space Odyssey was constructed using a Fisheye-Nikkor 8 mm f/8 lens.[58] HAL's point-of-view was filmed using a Fairchild-Curtis 'bug-eye' lens originally designed for films in the Cinerama 360 dome format.[59]
• The first music video to be shot completely with fisheye lens was for the Beastie Boys song "Hold It Now, Hit It" in 1987.
• In Computer Graphics, circular fisheye images can be used to create environment maps from the physical world. One complete 180-degree wide angle fisheye image will fit to half of cubic mapping space using the proper algorithm. Environment maps can be used to render 3D objects and virtual panoramic scenes.
• Many personal weather station online cameras around the world upload fisheye images of the current local sky conditions as well as a previous day time-lapse sequence with climate conditions such as temperature, humidity, wind and rainfall amounts.[60]

Mapping function

The subject is placed in the image by the lens according to the mapping function of the lens. The mapping function gives ${\displaystyle r}$, the position of the object from the center of the image, as a function of ${\displaystyle f}$, the focal length, and ${\displaystyle \theta }$, the angle from the optical axis. ${\displaystyle \theta }$ is measured in radians.

Comparison of mapping functions

Subject	Original tunnel to be photographed, with camera looking from inside center to left wall.
	Normal	Fisheye[61][44]
	Rectilinear	Stereographic[62]	Equidistant	Equisolid angle	Orthographic
Other names	gnomonic, perspective, conventional	panoramic, conform, planisphere	linear, linear-scaled	equal-area	orthogonal
Image
Mapping function[44]	${\displaystyle r=f\tan(\theta )}$	${\displaystyle r=2f\tan(\theta /2)}$	${\displaystyle r=f\cdot \theta }$	${\displaystyle r=2f\sin(\theta /2)}$[lower-alpha 1]	${\displaystyle r=f\sin(\theta )}$
Notes	Works like the pinhole camera. Straight lines remain straight (distortion free). ${\displaystyle \theta }$ has to be smaller than 90°. The aperture angle is gaged symmetrically to the optical axis and has to be smaller than 180°. Large aperture angles are difficult to design and lead to high prices.	Maintains angles. This mapping would be ideal for photographers because it doesn't compress marginal objects as much. Samyang is the only manufacturer to produce this kind of fisheye lens, but it is available under different brand names. This mapping is easily implemented by software.	Maintains angular distances. Practical for angle measurement (e.g., star maps). PanoTools uses this type of mapping.	Maintains surface relations. Every pixel subtends an equal solid angle, or an equal area on the unit sphere. 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.	Maintains planar illuminance. Looks like an orb with the surroundings lying on < max. 180° aperture angle. Highly distorted near the edge of the image, but image in center is less compressed.
Examples[63][64][65]	(numerous)	Samyang f = 7.5 mm f/2.8 Samyang f = 8 mm f/2.8 Samyang f = 12 mm f/2.8	Canon f = 7.5 mm f/5.6 Coastal Optical f = 7.45 mm f/5.6 Nikkor f = 6 mm f/2.8 Nikkor f = 7.5 mm f/5.6 Nikkor f = 8 mm f/2.8 Nikkor f = 8 mm f/8.0 Peleng f = 8 mm f/3.5 Rokkor f = 7.5 mm f/4.0 Sigma f = 8 mm f/3.5	Canon f = 15 mm f/2.8 (1988) Minolta f = 16 mm f/2.8 (1971) Nikkor f = 10.5 mm f/2.8[lower-alpha 2] Nikkor f = 16 mm f/2.8 (1995) Sigma f = 4.5 mm f/2.8 Sigma f = 8 mm f/4.0[lower-alpha 3] Sigma f = 15 mm f/2.8 (1990) Zuiko f = 8 mm f/2.8	Nikkor f = 10 mm f/5.6 OP Yasuhara Madoka180 f = 7.3 mm f/4

Details

1. Special case of ${\displaystyle r=k_{1}\cdot f\sin {\frac {\theta }{k_{2}}}}$, where ${\displaystyle k_{1}=k_{2}=2}$. Some fisheyes, such as the AF Nikkor DX 10.5 mm f/2.8 have slightly different values for ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$.
2. For this lens, ${\displaystyle k_{1}=1.47}$ and ${\displaystyle 1/k_{2}=0.713}$, determined empirically.[66]
3. In this case, ${\displaystyle k_{1}=1.88}$ and ${\displaystyle 1/k_{2}=0.54}$.[66]

Other mapping functions (for example Panomorph Lenses) are also possible for enhancing the off-axis resolution of fisheye lenses.

With appropriate software, the curvilinear images produced by a fisheye lens can be remapped to a conventional rectilinear projection. Although this entails some loss of detail at the edges of the frame, the technique can produce an image with a field of view greater than that of a conventional rectilinear lens. This is particularly useful for creating panoramic images.

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

See also

• Azimuthal equidistant projection
• Dashcam
• Little planet effect
• Miniature faking
• Stereographic projection
• de:Fischaugenobjektiv Fisheye lens with more information in German

Notes

1. Camera: a 35 mm format digital SLR, editing tool: Panorama Tools

References

1. Wood, R. W. (August 1906). "Fish-Eye Views, and Vision under Water". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 6. XII (LXVIII): 159–161. doi:10.1080/14786440609463529.
2. Bond, W. N. (November 1922). "A Wide Angle Lens for Cloud Recording". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 6. XLIV (CCLXIII): 999–1001. doi:10.1080/14786441208562576. Retrieved 6 November 2018.
3. Hill, Robin (July 1924). "A lens for whole sky photographs". Quarterly Journal of the Royal Meteorological Society. 50 (211): 227–235. Bibcode:1924QJRMS..50..227H. doi:10.1002/qj.49705021110.
4. Henry Horenstein (2005-04-20). Black and White Photography: A Basic Manual. Little, Brown. p. 55. ISBN 9780316373050.
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External links

• Fisheye projection theory
• Vascon, Luca (2007). "the fish list". Archived from the original on 7 December 2016.
• Various fisheye projections
• Kumler, James "Jay"; Bauer, Martin (2000). Fish-eye lens designs and their relative performance. International Symposium of Optical Science and Technology. San Diego, California: SPIE. doi:10.1117/12.405226. Alternative archived URL