View source for Engineering:Fisheye lens

{{Short description|Wide-angle photographic lens with strong barrel distortion}}
{{Infobox lens design
| name     =Fisheye
| scheme   =Fisheye-Nikkor Auto 8mm 1-2.8.jpg
| year     =1924
| author   =Wood (1905),<ref name=Wood-1906 /> Bond (1922),<ref name=Bond-1922 /> and Hill (1924)<ref name=Hill-1924 />
| elements =Var.
| groups   =Var.
| aperture =
}}
A '''fisheye lens''' is an [[Engineering:Ultra wide angle lens|ultra wide-angle lens]] that produces strong [[Physics:Distortion (optics)|visual distortion]] intended to create a wide panoramic or [[Sphere#Hemisphere|hemispherical]] image.<ref name=Horenstein>{{cite book|url=https://archive.org/details/blackwhitephotog00hore_0 |url-access=registration |author=Henry Horenstein|title=Black and White Photography: A Basic Manual|publisher=Little, Brown |page=[https://archive.org/details/blackwhitephotog00hore_0/page/55 55]|isbn=9780316373050|date=2005-04-20}}</ref><ref name=Kingslake>{{Kingslake-lens-history |chapter=10}}</ref>{{rp|145}} Fisheye lenses achieve extremely wide [[Angle of view|angles of view]], well beyond any rectilinear lens. Instead of producing images with straight lines of perspective ([[Engineering:Rectilinear lens|rectilinear images]]), fisheye lenses use a special mapping ("distortion"; for example: [[Lambert azimuthal equal-area projection|equisolid angle]], see below), which gives images a characteristic convex non-rectilinear appearance.

[[File:Circular fisheye view of Oude Kerk Amsterdam Daniel D. Teoli Jr..jpg|thumb|Circular fisheye photograph of Oude Kerk Amsterdam. [[Physics:Chromatic aberration|Chromatic aberration]] can clearly be seen toward the outer edges.]]
The term ''fisheye'' was coined in 1906 by American physicist and inventor [[Biography:Robert W. Wood|Robert W. Wood]] based on how a fish would see an ultrawide hemispherical view from beneath the water (a phenomenon known as [[Physics:Snell's window|Snell's window]]).<ref name="Wood-1906">{{cite journal|author=Wood, R. W.|date=August 1906|title=Fish-Eye Views, and Vision under Water|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|series=6|volume=XII|pages=159–161|number=LXVIII|doi=10.1080/14786440609463529|url=https://zenodo.org/record/1725410}}</ref><ref name=Kingslake />{{rp|145}} Their first practical use was in the 1920s for use in [[Physics:Meteorology|meteorology]]<ref name=Hill-1924>{{cite journal |last=Hill |first=Robin |date=July 1924 |title=A lens for whole sky photographs |journal=Quarterly Journal of the Royal Meteorological Society |volume=50 |issue=211 |pages=227–235 |doi=10.1002/qj.49705021110 |bibcode=1924QJRMS..50..227H}}</ref><ref name=Brooks>{{cite book |url=https://books.google.com/books?id=_gqwwMWZm84C&q=first+practical+fisheye+lens|author=David Brooks|title=Lenses and lens accessories: a photographer's guide|page=29|isbn=9780930764340|year=1982}}</ref> 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,<ref name=Horenstein /> although lenses covering up to 280 degrees exist (see below). Their [[Focal length|focal length]]s depend on the [[Engineering:Film format|film format]] they are designed for.

[[Finance:Mass production|Mass-produced]] fisheye lenses for [[Photography|photography]] first appeared in the early 1960s<ref name=Compendium>{{cite book|url=https://books.google.com/books?id=u4rXBqSmjPAC&pg=PA209 |title=The New Nikon Compendium |author1=Stafford, Simon |author2=Hillebrand, Rudi |author3=Hauschild, Hans-Joachim |publisher=Lark Books |isbn=1-57990-592-7 |pages=209–210 |date=2004}}</ref> and are generally used for their unique, distorted appearance. For the popular [[135 film|35{{nbsp}}mm film]] format, typical focal lengths of fisheye lenses are 8–10&nbsp;mm for circular images, and 12–18&nbsp;mm for diagonal images filling the entire frame. For digital cameras using smaller imagers such as {{frac|4}}″ and {{frac|3}}″ format [[Engineering:Charge-coupled device|CCD]] or [[Engineering:CMOS|CMOS]] sensors, the focal length of "miniature" fisheye lenses can be as short as 1–2&nbsp;mm.

Fisheye lenses also have other applications, such as re-projecting images originally filmed through a fisheye lens, or created via computer-generated graphics, onto hemispherical screens. They are also used for scientific photography, such as recordings of aurora and meteors, and to study plant canopy geometry, and to calculate near-ground [[Physics:Solar radiation|solar radiation]]. In everyday life, they are perhaps most commonly encountered as [[Engineering:Peephole|peephole]] door viewers to give a wide field of view.

==History and development==
Panoramas with fisheye distortion predate photography and the fisheye lens. In 1779, [[Biography:Horace Bénédict de Saussure|Horace Bénédict de Saussure]] published his downward-facing fisheye view of the Alps: "All the objects are drawn in perspective from the centre".<ref>Veronica della Dora, "Mountains as a Way of Seeing: From Mount of Temptation to Mont Blanc", in Emily Goetsch, ''Mountains, Mobilities and Movement'', 2017, {{isbn|1137586354}}, p. 205</ref>
<gallery mode="packed" heights="200px">
File:Vue circulaire des montagnes qu ‘on decouvre du sommet du Glacier de Buet, from Horace-Benedict de Saussure, Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve. Neuchatel, l779-96, pl. 8.jpg|"Vue circulaire des montagnes qu'on découvre du sommet du Glacier de Buet", Horace-Benedict de Saussure<ref>''Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve''. Neuchatel, 1779–96, pl. 8.</ref>
File:Wood-1906 - Fig 1.jpg|Wood's pail (top) and improved (bottom) camera (1906)
File:Wood-1906 - Fig 2.jpg|First known fisheye image recorded in 1905 using Wood's pail apparatus (1906)
File:Bond-1922 - Fig 1.png|Bond's hemispherical lens (1922)
</gallery>

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".<ref name=Wood-1906 /> 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".<ref name=Wood-1906 /> 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."<ref name=Wood-1906 />

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 [[Physics:Chromatic aberration|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.<ref name=Bond-1922>{{cite journal |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015077756651;view=1up;seq=1033 |title=A Wide Angle Lens for Cloud Recording |author=Bond, W. N. |date=November 1922 |pages=999–1001 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |series=6 |volume=XLIV |number=CCLXIII |access-date=6 November 2018 |doi=10.1080/14786441208562576}}</ref> Bond's hemispheric lens also reduced the need for a pinhole aperture to ensure sharp focus, so exposure times were also reduced.<ref>{{cite journal |doi=10.1175/1520-0477(1989)070<1243:AACOCO>2.0.CO;2 |title=Almost a Century of "Imaging" Clouds Over the Whole-Sky Dome |author1=McGuffie, K. |author2=Henderson-Sellers, A. |date=October 1989 |pages=1243–1253 |journal=Bulletin of the American Meteorological Society |volume=70 |number=10|bibcode=1989BAMS...70.1243M |doi-access=free }}</ref>

===Hill Sky Lens===
[[File:Hill Sky Lens (1924).svg|thumb|right|Hill/Beck ''Sky Lens'' (1923, GB&nbsp;225,398)<ref name=GB225398/>]]
In 1924, Robin Hill first described a lens with 180° coverage that had been used for a cloud survey in September 1923<ref name=Hill-1924 /> The lens, designed by Hill and R. & J. Beck, Ltd., was patented in December 1923.<ref name=GB225398 /> The Hill ''Sky Lens'' is now credited as the first fisheye lens.<ref name=Kingslake />{{rp|146}} Hill also described three different mapping functions of a lens designed to capture an entire hemisphere (stereographic, equidistant, and orthographic).<ref name=Hill-1924 /><ref name=Miyamoto-1964>{{cite journal |doi=10.1364/JOSA.54.001060 |author=Miyamoto, Kenro |title=Fish Eye Lens |date=1964 |journal=Journal of the Optical Society of America |volume=54 |issue=8 |pages=1060–1061}}</ref> 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.<ref name=GB225398 /> 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.<ref name=GB225398>{{cite patent |country=GB |number=225398 |status=Application |title=Improvements in photographic lenses |inventor=Robin Hill |pridate=20 December 1923 |assign=R & J Beck Ltd. |pubdate=4 December 1924 |url=https://patents.google.com/patent/GB225398A/}}</ref>

The Hill ''Sky Lens'' was fitted to a [[Engineering:Whole sky camera|whole sky camera]], typically used in a pair separated by {{convert|500|m}} for [[Stereoscopy|stereo imaging]], and equipped with a red filter for contrast; in its original form, the lens had a focal length of {{convert|0.84|in|0|abbr=on}} and cast an image {{convert|2.5|in|abbr=on}} in diameter at {{f/}}8.<ref>{{cite book |chapter-url=https://books.google.com/books?id=h6zMCgAAQBAJ&pg=PA535 |title=Scientific Photography and Applied Imaging |author=Ray, Sidney |chapter=Fisheye lens photography |page=535 |date=1999 |publisher=Focal Press |location=Burlington, Massachusetts |isbn=978-0-240-51323-2 |access-date=7 November 2018}}</ref> Conrad Beck described the camera system in an article published in 1925.<ref>{{cite journal |doi=10.1088/0950-7671/2/4/305 |title=Apparatus to photograph the whole sky |author=Beck, Conrad |journal=Journal of Scientific Instruments |volume=2 |number=4 |date=1925 |pages=135–139|bibcode=1925JScI....2..135B }}</ref> At least one has been reconstructed.<ref>{{cite web |url=http://www.novacon.com.br/odditycameras/hillscloud.htm |title=1920s Hill's Cloud Camera with Hill Sky lens made by Beck London |publisher=Novacon |access-date=7 November 2018}}</ref>

===German and Japanese development===
[[File:AEG Weitwinkelobjectiv (1932).svg|thumb|right|Schulz/AEG ''Weitwinkelobjektiv'' (1932, DE&nbsp;620538)<ref name=DE620538 />]]
In 1932, the German firm [[Company:AEG (German company)|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''.<ref name=Kingslake />{{rp|148}}<ref name=DE620538>{{cite patent |country=DE |number=620538 |status=patennt |title=Abänderung eines Weitwinkelobjektivs (Modification of a wide-angle lens) |inventor=Hans Schulz |pridate=6 August 1932 |assign=Allgemeine Elektricitäts-Gesellschaft |gdate=23 October 1935 |url=https://patents.google.com/patent/DE620538C/}}</ref> 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.<ref name=DE620538 /> Miyamoto credits Hans Schulz with the design of the ''Weitwinkelobjektiv''.<ref name=Miyamoto-1964 /> The basic patented design was produced for cloud recording as a 17&nbsp;mm {{f/}}6.3 lens,<ref name=AEG-patent>{{cite web |url=http://aeg-weitwinkelobjektiv.de/Patent-eng.htm |title=AEG Weitwinkelobjektiv patent information |access-date=7 November 2018}}</ref> and Umbo used the AEG lens for artistic purposes, with photographs published in a 1937 issue of ''Volk und Welt''.<ref>{{cite web |url=http://aeg-weitwinkelobjektiv.de/umbo1.htm |title=Umbo (Otto Maximilian Umbehr) |access-date=7 November 2018}}</ref>

The AEG ''Weitwinkelobjektiv'' formed the basis of the later Nippon Kogaku (Nikon) ''Fisheye-Nikkor'' 16&nbsp;mm {{f/}}8 lens of 1938, which was used for military and scientific (cloud cover) purposes.<ref name=AEG-patent /><ref name=Cavina-35>{{cite web |url=http://www.marcocavina.com/articoli_fotografici/AEG_fisheye_1935/00_pag.htm |title=AEG Fisheye 17mm f/6,3 del 1935 e Nikkor Fisheye 16,3mm f/8 del 1938: Le prove del travaso tecnologico nel campo dell'ottica fra Germania e Giappone negli anni del patto d'acciaio |author=Cavina, Marco |access-date=8 November 2018 |trans-title=AEG Fisheye 17mm f/6,3 of 1935 and Nikkor Fisheye 16,3mm f/8 of 1938: Proof of optical field technology transfer between Germany and Japan during the Pact of Steel era}}</ref> 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.<ref name=Cavina-35 />

[[File:Richter-Zeiss Pleon (1938).svg|thumb|right|Richter/Zeiss ''Pleon'' (1938, US&nbsp;2,247,068)<ref name=US2247068 />]]
Also in 1938, Robert Richter of Carl Zeiss AG patented the 6-element, 5-group ''Pleon'' lens,<ref name=US2247068>{{cite patent |country=US |number=2247068 |status=patent |title=Anastigmatic photographic objective |inventor=Robert Richter |pridate=18 July 1938 |assign=Carl Zeiss, Jena |gdate=24 June 1941 |fdate=7 July 1939 |url=https://patents.google.com/patent/US2247068A/}}</ref> which was used for aerial surveillance during World War II. The converging rear group of the Pleon was symmetrical, derived from the 4-element ''[[Engineering:Topogon|Topogon]]'' lens 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.<ref name=Kingslake />{{rp|149}}<ref name=Pleon-47 /> The ''Pleon'' had a focal length of approximately 72.5&nbsp;mm with a maximum aperture of {{f/}}8 and used a plano-concave front element {{convert|300|mm|abbr=on}} in diameter; the image on the negative was approximately {{convert|85|mm|abbr=on}} in diameter.<ref name=Pleon-47>{{cite journal |doi=10.1364/JOSA.38.000421 |title=Lenses of Extremely Wide Angle for Airplane Mapping |author1=Gardner, Irvine C. |author2=Washer, Francis E. |date=1948 |journal=Journal of the Optical Society of America |volume=38 |issue=5 |pages=421–431|s2cid=122928391 }}{{pad|1.0em}}[https://nvlpubs.nist.gov/nistpubs/jres/40/jresv40n2p93_A1b.pdf Direct URL from NIST]</ref>


===35&nbsp;mm development===
[[File:Merté-Zeiss Sphaerogon (1935, fig 1).svg|thumb|right|upright=0.7|Merté/Zeiss ''Sphaerogon'' (1935, DE&nbsp;672&nbsp;393<ref name=DE672393/> and US&nbsp;2,126,126)<ref name=US2126126/>]]
At approximately the same time that Schulz was developing the ''Weitwinkelobjektiv'' at AEG, {{ill|Willy Merté|de}} at Zeiss was developing the ''Sphaerogon'', which was also designed to encompass a 180° field of view.<ref name=DE672393>{{cite patent |country=DE |number=672393 |status=Grant |title=Lichtbildlinse (Photo lens) |inventor=Willi Merté |pridate=6 August 1932 |assign=Carl Zeiss SMT GmbH |gdate=3 October 1935 |url=https://patents.google.com/patent/DE672393C/}}</ref><ref name=US2126126>{{cite patent |country=US |number=2126126 |status=patent |title=Photographic objective |inventor=Willi Merté |pridate=26 November 1935 |assign=Carl Zeiss SMT GmbH |gdate=9 August 1938 |fdate=27 November 1936 |url=https://patents.google.com/patent/US2126126A/}}</ref> 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.<ref>{{cite web |url=http://www.marcocavina.com/articoli_fotografici/czj_perimetar_sphaerogon_pleon/00_pag.htm |title=Perimetar, Sphaerogon, Pleon: The definitive compendium about these super-wide and fisheye lenses of the '30s conceived by the firm Carl Zeiss Jena |author=Cavina, Marco |date=10 March 2010 |access-date=9 November 2018}}</ref> 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;<ref>{{cite web |url=http://www.photographyhistory.com/zeisslenscollection.html |title=Highlights from the Carl Zeiss Jena Lens Collection |author=Silver, David |date=2012 |access-date=9 November 2018}}</ref> 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.<ref>{{cite web |url=http://zeisshistoricasociety.org/edward-k-kaprelian/ |title=Edward K. Kaprelian, 1913-1997 |publisher=Zeiss Historica Society |access-date=9 November 2018}}</ref><ref>{{cite magazine |url=https://books.google.com/books?id=Dl0zAQAAMAAJ&pg=PA82 |title=Lens Oddities |author1=Steinhard, Walter (photographs) |date=April 1947 |volume=20 |number=5 |magazine=Popular Photography |publisher=Ziff-Davis |location=Chicago |pages=82–83 |access-date=9 November 2018}}</ref>

After the war, the ''Fisheye-Nikkor'' lens was mated to a medium format camera and was produced in slightly modified form (focal length increased slightly to 16.3&nbsp;mm) as the "Sky-image Recording Camera" in March 1957 for the Japanese government,<ref name=Nikon-FEC>{{cite web |url=http://www.nikon.co.jp/main/eng/d-archives/camera/fisheye_camera/fisheye_camera.htm |title=Fisheye Camera |publisher=Nikon |access-date=8 November 2018 |archive-url=https://web.archive.org/web/20010211181813/http://www.nikon.co.jp/main/eng/d-archives/camera/fisheye_camera/fisheye_camera.htm |archive-date=11 February 2001 |url-status=dead}}</ref> 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 {{JPY|200000|1960|round=-4}}.<ref name=Nikon-history>{{cite web |url=http://www.nikon.co.jp:80/main/eng/d-archives/camera/history_e.htm#camera8.0 |title=History of Nikon Cameras: Release of Fisheye Camera |publisher=Nikon |access-date=8 November 2018 |archive-url=https://web.archive.org/web/20010206204641/http://www.nikon.co.jp/main/eng/d-archives/camera/history_e.htm#camera8.0 |archive-date=6 February 2001 |url-status=dead }}</ref> The revised lens created a circular image {{convert|50|mm|abbr=on}} in diameter and covered a complete hemispherical field of 180°.<ref>{{cite web |url=https://imaging.nikon.com/lineup/filmcamera/120format/fisheye/index.htm |title=Fisheye Camera |publisher=Nikon Imaging Products |access-date=8 November 2018}}</ref> 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.<ref name=Grays>{{cite magazine |url=https://www.graysofwestminster.co.uk/pdf/gazette/gaz52.pdf |title=The Nikon Legend - Part LI |author=Levett, Gray |date=1998 |magazine=The Grays of Westminster Gazette |issue=52 |page=8 |publisher=Grays of Westminster |location=London, England |access-date=8 November 2018}}</ref> Photographs taken with the Fisheye Camera that were published in ''Life'' in 1957 marked the first wide exposure of the public to fisheye distortion;<ref name=PopPhoto2019>{{cite news |url= https://www.popphoto.com/story/culture/how-fisheye-lens-became-staple-of-music-industry/  |title=How the fisheye lens became a staple of the music industry |author=Moses, Jeanette D. |date=December 19, 2019 |work=Popular Photography |access-date=1 April 2023}}</ref> including a photograph of the United States Senate caucus room, taken by Ed Clark and published in an April 1957 issue,<ref>{{cite magazine |url=https://books.google.com/books?id=8EwEAAAAMBAJ&pg=PA42 |title=Fisheye views as Dave Beck gets the hook |author=Clark, Ed |date=April 8, 1957 |magazine=LIFE |pages=42–43 |volume=42 |number=14 |publisher=Time Inc. |location=Chicago, Illinois |access-date=28 March 2023}}</ref> and a photograph of pole vaulter Bob Gutowski taken by Ralph Crane, published in July 1957.<ref>{{cite magazine |url=https://books.google.com/books?id=ID8EAAAAMBAJ&pg=PA16 |title=Views along a pole |author=Crane, Ralph (photographer) |date=July 1, 1957 |magazine=Life |publisher=Time-Life |volume=43 |number=1 |page=16 |access-date=9 November 2018}}</ref>

[[File:Are You Experienced - US cover-edit.jpg|thumb|left|Album cover of ''Are You Experienced'' (1967) by The Jimi Hendrix Experience, featuring the trio photographed using a fisheye lens]]
The Nikon Fisheye Camera was discontinued in September 1961,<ref name=Nikon-FEC /> and Nikon subsequently introduced the first regular production fisheye lens for 35&nbsp;mm cameras in 1962,<ref name=Miyamoto-1964 /> the ''Fisheye-Nikkor'' 8&nbsp;mm {{f/}}8,<ref>{{cite book |url=https://books.google.com/books?id=0GhqDwAAQBAJ&pg=PT232 |title=Nikon: A Celebration |author=Long, Brian |edition=Third |date=2018 |publisher=The Crowood Press Ltd |location=Ramsbury, Marlborough |isbn=978-1-78500-470-4 |access-date=9 November 2018}}</ref> 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 35&nbsp;mm format increased its popular use.<ref>{{cite web |url=http://www.naturfotograf.com/lens_fish.html |title=Fisheye Lenses for Nikon 'F' Mount |first=Bjørn |last=Rørslett |date=27 November 2007 |access-date=9 November 2018}}</ref> The ''Fisheye-Nikkor'' 8&nbsp;mm {{f/}}8 has a field of view of 180° and uses 9 elements in 5 groups; it has a fixed focus and built-in filters intended for black-and-white photography. Research indicates that fewer than 1,400 lenses were built.<ref>{{cite web |url=https://blog.graysofwestminster.co.uk/2012/09/20/8mm-f8-fisheye-nikkor-lens-advert-preview/ |title=8mm f/8 Fisheye-Nikkor Lens |date=20 September 2012 |publisher=Grays of Westminster |access-date=9 November 2018}}</ref> As fisheye lenses became more widely available, the distinctive distortion grew in popularity, especially for album covers. For example, British fashion photographer Tim Walker used a fisheye lens to capture the cover of Harry Styles' 2019 pop/rock album, ''Fine Line''.<ref>{{Cite web |last=Greenwood |first=Douglas |title=Everything You Need to Know About the Cover for Harry Styles' Fine Line |url=https://i-d.vice.com/en/article/ywajwv/harry-styles-new-tim-walker-album-cover-fine-line |access-date=2023-04-08 |website=i-d.vice.com |language=en}}</ref> Other genres that have taken advantage of the fisheye lens look include punk rock, hip-hop, and skateboarding videos.<ref name=PopPhoto2019/>

[[File:Isshiki (1967).svg|thumb|right|Isshiki & Matsuki/Nikon ''Fisheye-Nikkor'' 6&nbsp;mm {{f/|5.6}} (US 3,542,697)<ref name=US3524697/>]]
Nikon subsequently released several more milestone circular fisheye lenses in Nikon F mount through the 1960s and 70s:
* 10&nbsp;mm {{f/}}5.6 OP (1968), the first fisheye to feature orthographic projection, which was also the first lens to feature an aspherical element<ref>{{cite web |url=https://imaging.nikon.com/history/story/0006/index.htm |title=NIKKOR - The Thousand and One Nights No.6: OP Fisheye-NIKKOR 10mm F5.6 |author=Ohshita, Kouichi |publisher=Nikon Imaging |access-date=9 November 2018}}</ref>
* 6&nbsp;mm {{f/}}5.6 (1969), the first fisheye to feature a 220° field of view;<ref name=Compendium/> interestingly, the patent accompanying this lens includes a design for a lens with a 270° field of view.<ref name=US3524697>{{cite patent |country=US |number=3524697 |status=Grant |title=Achromatic super wide-angle lens |invent1=Isshiki Masaki |invent2=Matsuki Keiji |pridate=27 April 1967 |assign=Nikon Corp. |gdate=18 August 1970 |url=https://patents.google.com/patent/US3524697A/}}</ref> A 6.2&nbsp;mm {{f/}}5.6 SAP fisheye was later produced in limited numbers with an aspherical surface, encompassing a 230° field of view.<ref name=Sato-53>{{cite web |url=https://imaging.nikon.com/history/story/0053/index.htm |title=NIKKOR: The Thousand and One Nights No. 53: Ai AF Fisheye-Nikkor 16mm F2.8D |author=Sato, Haruo |publisher=Nikon Imaging |access-date=13 November 2018}}</ref>
* 8&nbsp;mm {{f/}}2.8 (1970), the first circular fisheye with variable focus, automatic aperture, and reflex viewing (mirror lock-up no longer required).<ref name=Compendium />

[[File:Pentax K-1 with flat Fisheye (34927515900).jpg|thumb|left|Fish-eye Takumar 11/18mm on a modern Pentax K-1 DSLR]]
Contemporaneously, 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&nbsp;mm {{f/}}11, released by Pentax (Asahi Optical) in 1962,<ref name=Sato-53 /><ref>{{cite web |url=http://www.ricoh-imaging.co.jp/english/pentax/pentaxhistory/course/ |title=Course of History |publisher=Ricoh Imaging |access-date=10 November 2018}}</ref><ref>{{cite web |url=https://takumarguide.weebly.com/fish-eye-1--11--18-356.html |title=Fish-eye-Takumar 1:11/18mm |publisher=Takumar Field Guide |access-date=10 November 2018}}</ref> followed by the slightly faster ''UW Rokkor-PG'' 18&nbsp;mm {{f/}}9.5 from Minolta in 1966.<ref>{{cite web |url=http://artaphot.ch/minolta-sr/objektive/138-minolta-18mm-f95-fisheye |title=Minolta 18mm 1:95 Fisheye (MC) |publisher=artaphot |access-date=10 November 2018}}</ref> Both of these were reflex-viewing and fixed-focus. Both Pentax and Minolta followed up with faster lenses with variable focus in 1967 (''Super Fish-eye-Takumar'' 17&nbsp;mm {{f/}}4)<ref>{{cite web |url=https://takumarguide.weebly.com/fish-eye-1--4--17.html |title=Fish-eye-Takumar 1:4/17 |publisher=Takumar Field Guide |access-date=10 November 2018}}</ref> and 1969 (''Rokkor-OK'' 16&nbsp;mm {{f/}}2.8),<ref>{{cite web |url=http://artaphot.ch/minolta-sr/objektive/136-minolta-16mm-f28-fisheye |title=Minolta 16mm 1:2.8 Fisheye (MC/MD) |publisher=artaphot |access-date=10 November 2018}}</ref><ref name=US3589798>{{cite patent |country=US |number=3589798 |status=Grant |title=Wide-angle lens system with corrected lateral aberration |invent1=Ogura Toshinobu |pridate=25 March 1968 |assign=Minolta Camera Kabushiki Kaisha |gdate=29 June 1971 |url=https://patents.google.com/patent/US3589798A/}}</ref> respectively. The 16&nbsp;mm Rokkor was later adopted by Leica as the ''Fisheye-Elmarit-R'' (1974) for its SLRs 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.
{{clear}}

==Design==
Unlike rectilinear lenses, fisheye lenses are not fully characterised by focal length and aperture alone. Angle of view, image diameter, projection type, and sensor coverage all vary independently of these.

=== Image diameter and coverage ===
{| style="float:right; margin-left:1em; border:solid 1px #cccccc; text-align:center;"
|+ style="text-align:left; font-weight:bold;" | Types of using format
|-
! colspan="2" style="width:135px;" | Circular
! style="width:135px;" | Cropped circle
! style="width:135px;" | Full-frame
|- style="vertical-align:top;"
| colspan="2" | [[File:3-2-circular.png|134px]]
| [[File:3-2-circumcised.png|134px]] || [[File:3-2-fullFrame.png|134px]]
|- style="vertical-align:top; font-size:85%;"
| style="width:20px; font-weight:bold;" | 3:2
| style="width:104px;" | [[:File:3-2-circular.png|52%&nbsp;sensor]]
| [[:File:3-2-circumcised.png|78%&nbsp;FOV, 92%&nbsp;sensor]] || [[:File:3-2-fullFrame.png|59%&nbsp;FOV]]
|- style="vertical-align:top; font-size:85%;"
| style="font-weight:bold;" | 4:3
| [[:File:4-3-circular.png|59%&nbsp;sensor]] || [[:File:4-3-circumcised.png|86%&nbsp;FOV, 90%&nbsp;sensor]] || [[:File:4-3-fullFrame.png|61%&nbsp;FOV]]
|- style="vertical-align:middle; font-size:85%;"
| colspan="3" | [[File:Peleng 8mm Fisheye 8225.jpg|134px]]<br />Circular fisheye for 35&nbsp;mm<br />&nbsp;
| [[File:Sigma 10 mm F2,8 EX DC HSM Fisheye.jpg|134px]]<br />Full-frame fisheye with rudimentary lens hood<br />&nbsp;
|- style="vertical-align:top; font-size:85%;"
| colspan="2" | [[File:Imagine being a fly VLT.jpg|134px]]<br />ESO's [[Astronomy:Very Large Telescope|VLT]] image taken with a circular fisheye lens.
| [[File:The Squirrels 0048.jpg|134px]]<br />35&nbsp;mm circular fisheye with DX-format-camera<br />&nbsp; || [[File:Vlg shop.jpg|134px]]<br />Diagonal fisheye used in a closed space (Nikkor 10.5&nbsp;mm)
|}

In a '''circular fisheye lens''', the [[Engineering:Image circle|image circle]] is ''inscribed in'' the film or sensor area; in a '''diagonal ("full-frame") fisheye lens''', the image circle is ''circumscribed around'' the film or sensor area. This implies that using a fisheye lens for a different format than it was intended for is easy (as opposed to a rectilinear lens), and may change its characteristic.

Further, different fisheye lenses map ("distort") images differently, and the manner of distortion is referred to as their '''mapping function.''' A common type for consumer use is ''equi[[Solid angle|solid angle]].''

Although there are digital fisheye effects available both in-camera and as computer software, neither can 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:

{|class=wikitable style="text-align:center;font-size:85%;"
|+ Calculated fisheye focal lengths{{efn|Assumes 180° maximum [[Angle of view|angle of view]] for the mapping function, <math>\theta=\frac{\pi}{2}</math>}}
|-
! colspan=2 style="width:10.0%;" |&nbsp;
! style="width:22.5%;" | Stereographic
! style="width:22.5%;" | Equidistant
! style="width:22.5%;" | Equisolid angle
! style="width:22.5%;" | Orthographic
|-
! colspan=2 | Inverse mapping function<ref name=IMO-05 />
| <math>f = \dfrac r {2 \tan \dfrac \theta 2}</math>
| <math>f = \frac r \theta</math>
| <math>f = \dfrac r {2 \sin \dfrac \theta 2}</math>
| <math>f = \frac r {\sin \theta}</math>
|-
! rowspan=3 | Circular{{efn|For circular fisheyes, the maximum dimension <math>r</math> is half the length of the shortest side.}}
! [[Engineering:APS-C|APS-C]] ({{nowrap|<math>r</math> {{=}} 8.4 mm}})
| 4.2 || 5.3 || 5.9 || 8.4
|-
! 135 ({{nowrap|<math>r</math> {{=}} 12 mm}})
| 6.0 || 7.6 || 8.5 || 12.0
|-
! 6×6 ({{nowrap|<math>r</math> {{=}} 28 mm}})
| 14.0 || 17.8 || 19.8 || 28.0
|-
! rowspan=3 | Diagonal{{efn|For full-frame fisheyes, the maximum dimension <math>r</math> is half the length of the diagonal.}}
! APS-C ({{nowrap|<math>r</math> {{=}} 15.1 mm}})
| 7.5 || 9.6 || 10.6 || 15.1
|-
! 135 ({{nowrap|<math>r</math> {{=}} 21.7 mm}})
| 10.8 || 13.8 || 15.3 || 21.7
|-
! 6×6 ({{nowrap|<math>r</math> {{=}} 39.6 mm}})
| 19.8 || 25.2 || 28.0 || 39.6
|}

;Notes
{{notelist}}

===Circular fisheyes===
The first types of fisheye lenses developed were "circular" — lenses which took in a 180° hemisphere and projected it as a circle within the film frame. By design, circular fisheye lenses thus cover a smaller image circle than rectilinear lenses designed for the same sensor size. The corners of a circular fisheye image will be completely black. This blackness is different from the gradual vignetting of rectilinear lenses and sets on abruptly.

Some circular fisheyes were available in orthographic projection models for scientific applications. These have a 180° ''vertical,'' horizontal and diagonal [[Angle of view|angle of view]]. 

For APS and m43 cameras, several lenses have emerged that retain a 180° field of view on a crop body. The first of these was the Sigma 4.5{{nbsp}}mm.<ref>[https://www.sigmaphoto.com/lenses/fisheye-lenses/45mm-f28-ex-dc-hsm-circular-fisheye 4.5mm F2.8 EX DC Circular Fisheye]</ref> Sunex also makes a 5.6{{nbsp}}mm fisheye lens that captures a circular 185° field of view on a 1.5x Nikon and 1.6x Canon DSLR cameras.

[[File:Fisheye-Nikkor Auto 6mm f2.8 lens 2015 Nikon Museum.jpg|thumb|left|upright=1.5|Fisheye-Nikkor 6mm f/2.8 mounted on a Nikon F2 in the Nikon Museum.]]

[[Company:Nikon|Nikon]] produced a 6&nbsp;mm circular fisheye lens for 35&nbsp;mm film that was initially designed for an expedition to [[Earth:Antarctica|Antarctica]]. It featured a 220° field of view, designed to capture the entire sky and surrounding ground when pointed straight up. This lens is no longer manufactured,<ref>{{cite web|url=http://www.mir.com.my/rb/photography/companies/nikon/nikkoresources/fisheyes/6mmf28.htm|title=Additional Information on Fisheye-Nikkor 6mm f/2.8 lens|publisher=Malaysian Internet Resources|access-date=2008-11-11}}</ref> and is used nowadays to produce interactive virtual-reality images such as [[QuickTime VR]] and [[Software:IPIX|IPIX]]. Because of its very wide field of view, it is very large — weighing {{convert|5.2|kg|lb}}, having a diameter of {{convert|236|mm|in}}, a length of {{convert|171|mm|in}} and an angle of view of 220 degrees. It dwarfs a regular 35&nbsp;mm SLR camera<ref>{{cite web|url=http://www.mir.com.my/rb/photography/companies/nikon/nikkoresources/late70nikkor/fisheyes/6mmf28.htm|title=Additional Information on Fisheye-Nikkor 6mm f/2.8 lens: Late 70s|publisher=Malaysian Internet Resources|access-date=2008-11-11}}</ref> and has its own tripod mounting point, a feature normally seen in large long-focus or [[Engineering:Telephoto lens|telephoto lens]]es to reduce strain on the lens mount. The lens is extremely rare.<ref>{{cite journal|last=Laurent|first=Olivier Laurent|title=Rare extreme wide-angle Nikkor lens goes on sale|journal=British Journal of Photography|date=23 April 2012|url=http://www.bjp-online.com/british-journal-of-photography/news/2169615/rare-extreme-wide-angle-nikkor-lens-goes-sale|access-date=24 April 2012|archive-url=https://web.archive.org/web/20120425140604/http://www.bjp-online.com/british-journal-of-photography/news/2169615/rare-extreme-wide-angle-nikkor-lens-goes-sale|archive-date=25 April 2012|url-status=dead}}</ref>

[[File:Venus.Optics.Laowa.4mm.f2.8.jpg|thumb|The fish eye lens Laowa 4{{nbsp}}mm f/2,8 of the manufacturer Venus Optics]]

More recently, the Japanese manufacturer Entaniya offers several fisheye lenses with angles of view up to 250° on 35&nbsp;mm full frame, and up to 280° on smaller sensors (see list below). In 2018, [[Company:Venus Optics|Venus Optics]] introduced a 210° fisheye lens for the [[Engineering:Micro Four Thirds system|Micro Four Thirds]] system.<ref>[https://www.dpreview.com/products/venus/lenses/venus_laowa_4_2p8_fisheye_mft Venus Laowa 4mm F2.8 Fisheye MFT], dpreview, 20 April 2018, retrieved 30 September 2018</ref>

The 8&nbsp;mm and 7.5&nbsp;mm circular fisheye lenses made by [[Company:Nikon|Nikon]] have proven useful for scientific purposes because of their equidistant (equiangular) projection, in which distance along the radius of the circular image is proportional to the zenith angle.

=== Diagonal fisheyes (a.k.a. full-frame or rectangular) ===
As fisheye lenses gained popularity in general photography, camera companies began manufacturing fisheye lenses with an enlarged image circle to cover the entire rectangular film frame. They are called diagonal, or sometimes "rectangular" or "full-frame", fisheyes.  (This was well before digital photography, so the use of the term "full frame" with respect to fisheyes has nothing to do with the use of the term to designate a digital sensor measuring 36x24&nbsp;mm).<ref>{{cite book|url=https://books.google.com/books?id=P39ysk_UwZcC&pg=PA33 |title=The Magic of Digital Nature Photography|isbn=9781579907730|last1=Sheppard|first1=Rob|year=2006}}</ref>

The angle of view produced by diagonal fisheyes only measures 180° ''from corner to corner'': they have a 180° ''diagonal'' [[Angle of view|angle of view (AOV)]], while the horizontal and vertical angles of view will be smaller. For an equisolid angle 15&nbsp;mm full-frame fisheye, the horizontal AOV will be 147°, and the vertical AOV will be 94°.<ref name="FOV-formula">For an equisolid angle projection (typical of full-frame fisheyes), the angle of view is double <math>\theta</math>, the angle from the optical axis, and the resulting formula is <math>\text{AOV} = 4 \cdot \arcsin \left(\frac{\text{frame size}}{4\cdot \text{focal length}}\right)</math>, where <math>\text{frame size} = 2 \cdot r</math> which comes from [[Inverse function|solving]] the mapping function for <math>\theta</math>; [http://www.dyxum.com/columns/articles/lenses/Takumar-17f4/Takumar_17mmF4_fisheye_review.asp Dyxum, Gustavo Orensztajn]</ref>

One of the first diagonal fisheye lenses to be mass-produced was the Nikon Fisheye-Nikkor F 16mm {{f/}}3.5, made in the early 1970s. 

To obtain the same effect on digital cameras with smaller sensors, shorter focal lengths are required. Nikon makes a 10.5&nbsp;mm fisheye for their APS DX SLRs.<ref>{{cite web|url=http://www.nikonusa.com/template.php?productNr=2148&cat=1&grp=5|title=Cameras from Nikon - DSLR and Digital Cameras, Lenses, & More}}</ref> Several other companies make "full frame", i.e. diagonal, fisheyes for APS and m43 cameras, see next paragraph.

=== Portrait or cropped-circle fisheyes ===
An intermediate between a diagonal and a circular fisheye consists of a circular image optimised for the ''width'' of the film format rather than the ''height''. As a result, on any non-square film format, the circular image will be cropped at the top and bottom, but still show black edges on the left and right. This format is called a "portrait" fisheye;<ref>{{Cite web |last=ehopf |date=2017-03-16 |title=Neue Fisheyes für Panoramafotografie |url=https://pt4pano.com/blog/2017/neue-fisheyes-fuer-panoramafotografie |access-date=2022-04-21 |website=PT4Pano |language=de}}</ref> historically, it has been rather rare - only the 12 mm f/8 Accura lens (see list below) directly follows the portrait principle. Today however, a portrait fisheye effect is easily achieved by using a fisheye lens intended for full coverage of a smaller sensor format, like an APS diagonal fisheye on a 35 mm full frame camera, or an m43 diagonal fisheye on APS. 

===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.<ref>{{Cite web |url=http://www.omnitech.com/fisheye.htm |title=190 Degree FOV Miniature Fisheye Lens |access-date=2012-02-13 |archive-url=https://web.archive.org/web/20121030000611/http://www.omnitech.com/fisheye.htm |archive-date=2012-10-30 |url-status=dead }}</ref><ref>[http://www.optics-online.com/dsl_fisheye.asp Miniature fisheye lenses]</ref> Popular [[Engineering:Image sensor format#Table of sensor formats and sizes|image sensor format sizes used]] include {{frac|4}}", {{frac|3}}", and {{frac|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. {{frac|2}}"), and a full frame on a smaller one (e.g. {{frac|4}}").

==Examples and specific models==
For a comprehensive list of all current and all past fisheye lenses, see External Links below. 

===Noteworthy fisheye lenses for APS-C cameras===
The APS-C image sensor used in Canon cameras is {{convert|22.3|×|14.9|mm|abbr=on}}, or {{convert|26.82|mm|abbr=on}} 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 {{convert|23.6|to|23.7|mm|abbr=on}} on the long dimension and {{convert|15.6|mm|abbr=on}} on the shorter side, for a diagonal between {{convert|28.2|to|28.4|mm|abbr=on}}.

====Circular APS-C fisheye lenses====
* Sigma 4.5&nbsp;mm {{f/}}2.8 
* Lensbaby 5.8{{nbsp}}mm {{f/}}3.5 

====Diagonal APS-C fisheye lenses====
* Nikon 10.5&nbsp;mm {{f/}}2.8 for Nikon F DSLRs
* Samyang 8&nbsp;mm {{f/}}3.5 for various APS DSLRs. Notable for its [[Stereographic projection|stereographic projection]].
* Samyang 8&nbsp;mm {{f/}}2.8 for various mirrorless mounts. Notable for its [[Stereographic projection|stereographic projection]].
* Sigma 10&nbsp;mm {{f/}}2.8 for various APS DSLRs. 

====Zoom APS-C fisheye lenses====
* Pentax 10–17&nbsp;mm {{f/}}3.5–4.5 = Tokina 10–17&nbsp;mm {{f/}}3.5–4.5 (jointly developed) for various APS DSLRs. 

===Noteworthy fisheye lenses for 35&nbsp;mm full frame cameras===

====Circular fisheye lenses====
[[File:Peleng 8mm Fisheye 8225.jpg|thumb|The Peleng 8&nbsp;mm f/3.5 circular fisheye lens]]
* Accura 12&nbsp;mm {{f/}}8 (180° portrait fisheye lens, i.e. optimised for the height rather than the width of the frame, thus giving a circular image of larger diameter, i.e. cropped at the top and bottom. 1968. Sold as Beroflex, Berolina, Panomar, Sigma, Spiratone, Universa, Upsilon, Vemar etc. Very bad.)<ref>{{Cite web |title=The Strange Case of the 12mm Fisheye |url=http://thesybersite.com/general/12mm-fisheye/ |access-date=2022-04-19 |website=thesybersite.com}}</ref>
* C-4 Optics Hyperfisheye 4.9&nbsp;mm {{f/}}3.5 (270°, 2020, for mirrorless camera only, 13kg)<ref>{{Cite web |title=C-4 Precision Optics releases $39,000 4.9mm F3.5 Hyper Fisheye lens for Sony E mount |url=https://www.dpreview.com/news/4034995669/c-4-precision-optics-4-9mm-f3-5-hyper-fisheye-lens-for-sony-e-mount |access-date=2022-04-19 |website=DPReview}}</ref>
* Canon FD 7.5&nbsp;mm {{f/}}5.6 (180°, 1971, three versions: initial version with silver bayonet ring, 1973 S.S.C. version with silver bayonet ring, 1979 NewFD version with the same SSC coating, black bayonet ring; all have inbuilt, wheel-selectable colour filters)<ref name=":0">{{Cite web |title=Table of all Canon SLR Lenses ever made |url=https://docs.google.com/spreadsheets/d/1XHrXeijkKB_ULZxcgJK_v4iPBbxAfOuuo-VuP5ESDvA/edit?usp=sharing |access-date=19 April 2022 |website=Table of all Canon SLR Lenses ever made}}</ref>
* Entaniya HAL 200 6&nbsp;mm {{f/}}4 (200°, 19.9 mm image diameter, for mirrorless camera only)<ref>{{Cite web |title=HAL 200 6.0mm {{!}} Entaniya |url=https://products.entaniya.co.jp/en/products/fisheye-lens-hal-200-6-0mm/ |access-date=2022-04-19 |website=products.entaniya.co.jp|date=9 November 2021 }}</ref>
* Entaniya HAL 250 6&nbsp;mm {{f/}}5.6 (250°, 23.7 mm image diameter, for mirrorless camera only, fixed aperture, 2 kg (the company also makes a 280° model with however only 5mm image diameter))<ref>{{Cite web |title=HAL 250 6.0mm {{!}} Entaniya |url=https://products.entaniya.co.jp/en/products/fisheye-lens-hal-250-6-0mm/ |access-date=2022-04-19 |website=products.entaniya.co.jp|date=15 November 2021 }}</ref>
* Nikon F 6&nbsp;mm {{f/}}2.8 (220°, 1972)<ref name=":1">{{Cite web |title=Nikon Lenses |url=http://www.photosynthesis.co.nz/nikon/lenses.html#Fisheye |access-date=2022-04-19 |website=www.photosynthesis.co.nz}}</ref>
* Nikon F 6&nbsp;mm {{f/}}5.6 (220°, 1970)<ref name=":1" />
* Nikon F 6.2&nbsp;mm {{f/}}5.6 (230° and at its time the widest fisheye. Looks like the aforementioned 6 mm f/5.6, but different engraving: 6.2&nbsp;mm 230°. Reportedly the rarest Nikon lens in existence, only 3 produced)<ref>{{Cite web |title=The Fish List • Nikon Fisheye-Nikkor f5.6 6.2-mm SAP |url=https://www.thefisheyelist.com/lens/nikon-fisheye-nikkor-15-6-f6-2mm-230-sap/ |access-date=2022-04-22 |language=en-US}}</ref>
* Nikon F 7.5&nbsp;mm {{f/}}5.6 (220°, 1966)<ref name=":1" />
* Nikon F 8&nbsp;mm {{f/}}2.8 (180°, 1970)<ref name=":1" />
* Nikon F 8&nbsp;mm {{f/}}8 (180°, 1962)<ref name=":1" />
* Olympus OM Auto-Fisheye 8&nbsp;mm {{f/}}2.4 (180°, rare)
* Peleng 8&nbsp;mm {{f/}}3.5 (180°)
* Sigma 8&nbsp;mm {{f/}}4.0 EX DG (180°)
* Sigma 8&nbsp;mm {{f/}}3.5 EX DG (180°, succeccor of the above Sigma 8&nbsp;mm {{f/}}4.0)

====Full-frame (i.e. diagonal) fisheye lenses====
* Canon EF 15&nbsp;mm {{f/}}2.8 (optically simpler successor of the below FD model; since discontinued)<ref name=":0" />
* Canon Fisheye FD 15&nbsp;mm {{f/}}2.8 (predecessor of the above, incompatible with EF mount. Two versions: original with silver bayonet ring, 1973; NewFD with black bayonet ring, 1980. Both have inbuilt colour filters and S.S.C. coating)<ref name=":0" />
* Fuji Photo Film Co. EBC Fujinon Fish Eye 16&nbsp;mm f/2.8 (M42 and X-Fujinon mounts, discontinued)
* Minolta AF 16&nbsp;mm {{f/}}2.8, since continued as Sony A
* Nikon Fisheye-Nikkor 16&nbsp;mm {{f/}}2.8 AI-s and AF D (since 1979)
* Nikon Fisheye-Nikkor 16 mm {{f/}}3.5 (1973, predecessor of the above)
* Pentax SMC 17 mm {{f/}}4 Fish-Eye
* Pentax 18 mm f/11 [[Engineering:Pancake lens|Pancake]] Fisheye (160°)<ref>{{Cite web |title=The Fish List • Pentax Fish-eye-Takumar 1:11 / 18mm |url=https://www.thefisheyelist.com/lens/pentax-fish-eye-takumar-111-18mm/ |access-date=2022-04-22 |language=en-US}}</ref>
* Samyang 12&nbsp;mm {{f/}}2.8 ED AS NCS Diagonal Fisheye (famous for its stereographic projection; available in several SLR and mirrorless camera mounts)
* Sigma 15&nbsp;mm {{f/}}2.8 EX DG Diagonal Fisheye
* TTArtisan 11mm {{f/}}2.8 Fisheye (notable as the first fisheye lens marketed, among others, in Leica M rangefinder mount and for Fuji GFX (the lens does not cover the full GFX frame, though!). An expert found it's actually 15 mm in focal length and has an angle of view of only 176°<ref>{{Cite web |last=ehopf |date=2020-09-26 |title=TTArtisan 11mm Fullframe Fisheye |url=https://pt4pano.com/blog/2020/ttartisan-11mm-fullframe-fisheye |access-date=2022-04-21 |website=PT4Pano |language=de}}</ref>)
* Zenitar 16&nbsp;mm {{f/}}2.8 Fisheye lens

====Zoom fisheye lenses====
*Canon EF 8–15{{nbsp}}mm {{f/}}4L Fisheye USM{{spaced ndash}}180° at all focal lengths, but turning from a circular to a diagonal fisheye on a 35&nbsp;mm full-frame camera, i.e. changing in vertical angle of view. On a crop camera with APS-C/H size sensors, it only yields a cropped circular and full-frame image. A zoom lock is included which avoids leaving the focal length range of full coverage on crop sensor cameras.
* Nikon AF-S Fisheye Nikkor 8–15{{nbsp}}mm {{f/}}3.5–4.5E ED{{spaced ndash}}designed for full-frame and FX cameras, this lens behaves identically to the Canon.
* [[Company:Tokina|Tokina]] AT-X AF DX = Pentax DA ED IF 10–17&nbsp;mm {{f/}}3.4-4.5{{spaced ndash}}a fisheye zoom lens designed for APS-C sensor cameras, also sold as an NH version without integrated lens hood: then, it is usable on full frame cameras.
* Pentax F 17–28{{nbsp}}mm {{f/}}3.5–4.5 Fisheye{{spaced ndash}}This lens was born for 35&nbsp;mm full frame film cameras, to take the place of the 16{{nbsp}}mm f/2.8 in the AF era. It starts from a 17{{nbsp}}mm full-frame (diagonal) fisheye. When it reaches 28{{nbsp}}mm, the fisheye effect is almost gone, leaving an overdistorted wideangle image. It was intended as a "special effect" lens and is claimed not to have sold too many copies.<ref>{{cite web|url=http://panopticus.altervista.org/fishlist/pentax/17-28pkaf.htm|title=TheFishList - Pentax DA 10-17 -}}</ref>{{citation needed|date=January 2013}}

==== Curiosities ====

* Canon 5.2&nbsp;mm f/2.8 RF L<ref>{{Cite web |last=Europe |first=Canon |title=Canon RF 5.2mm F2.8L Dual Fisheye Lens |url=https://www.canon-europe.com/lenses/rf-5-2mm-f2-8l-dual-fisheye-lens/ |access-date=2022-04-20 |website=Canon Europe |language=en-EU}}</ref> (a 190° stereographic fisheye with ''two'' fisheye lens systems: for 3D virtual reality shooting onto a single 35&nbsp;mm full-frame image sensor, and less relevant for photography: fitting two image circles onto one frame of 36&nbsp;mm width means that each can only be 18&nbsp;mm in diameter, wasting quite a bit of resolution)
* Pentax K "Bird's eye" 8.4&nbsp;mm {{f/}}2.8 (prototype, 1982, rendition not quite the same as a fisheye)<ref>{{Cite web |title=The Fish List • PENTAX smc BIRDS-EYE 1:2.8 8.4 mm |url=https://www.thefisheyelist.com/lens/pentax-smc-birds-eye-12-8-8-4-mm/ |access-date=2022-04-22 |language=en-US}}</ref>

==Sample images==
<gallery widths="200" heights="190">
Museumfisheye.jpg|An image of the Louvre museum entry taken with the 7.5&nbsp;mm {{f/|5.6}} circular fisheye Nikkor lens
WellsCathedral-28F12wyrdlight.jpg| Fisheye used to capture entire Wells Cathedral Chapter House room
Car Fisheye.jpg|Canon 8–15{{nbsp}}mm zoom at 8{{nbsp}}mm of BMW M3
Panotools5618.jpg|Image shot with a 16{{nbsp}}mm full-frame fisheye lens before and after remapping to [[Engineering:Rectilinear lens|rectilinear]] perspective.{{refn|group=n|Camera: a 35&nbsp;mm format digital SLR, editing tool: [[Software:Panorama Tools|Panorama Tools]]}}
File:Fisheye 16mm vs Ultra Wide Angle 11mm.jpg|Comparison of rectilinear 11mm lens and fisheye 16mm lens on a full frame DSLR
File:Fisheye mapping function (animation).gif|Comparison of conventional (rectilinear) mapping function with four different fisheye mapping functions, given a constant focal length.
</gallery>

==Other applications==
[[File:The Curves of ESO’s Headquarters.jpg|thumb|The curves of ESO's headquarters through a fish-eye lens.<ref>{{cite news|title=The Curves of ESO's Headquarters|url=http://www.eso.org/public/images/potw1408a/|access-date=26 February 2014|newspaper=ESO Picture of the Week}}</ref> ]]

* Many [[Astronomy:Planetarium|planetarium]]s 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#IMAX Dome/OMNIMAX|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|hemispherical photography]] to calculate plant canopy indices and near-ground solar radiation. Applications include evaluation of forest health, characterization of [[Biology:Monarch butterfly|monarch butterfly]] winter roosting sites, and management of vineyards.
* Astronomers use fisheye lenses to capture cloud cover and [[Astronomy:Light pollution|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&nbsp;mm {{f/}}8 lens.<ref>{{cite web |url=http://oberlehrer.de/der-hal-9000-stammt-von-nikon-oder-so/ |title=Der HAL 9000 stammt von Nikon (oder so) |trans-title=The HAL 9000 is from Nikon (maybe) |author=Luhm, Thorsten |date=1 May 2017 |publisher=Oberlehrer<sup>2</sup> |access-date=7 November 2018}}</ref> HAL's point-of-view was filmed using a Fairchild-Curtis 'bug-eye' lens originally designed for films in the [[Engineering:Cinerama|Cinerama]] 360 dome format.<ref>{{cite news |url=http://articles.latimes.com/2003/nov/27/entertainment/et-hal27 |title=Tracking HAL's Odyssey |author=Welkos, Robert W. |date=November 27, 2003 |newspaper=Los Angeles Times |access-date=7 November 2018}}</ref>
* 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 [[Engineering:Weather station|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.<ref>url=http://www.bloomsky.com</ref>
* In door's [[Engineering:Peephole|peephole]]s, to allow the viewer inside to see a wider view from the outside.

==Mapping function==
The subject is placed in the image by the lens according to the ''mapping function'' of the lens. The mapping function gives <math>r</math>, the position of the object from the center of the image, as a function of <math>f</math>, the focal length, and <math>\theta</math>, the angle from the optical axis. <math>\theta</math> is measured in [[Radian|radian]]s.

{|class=wikitable style="text-align:center;"
|+ Comparison of mapping functions
!Subject
|colspan=5 | [[File:PeterW zt 1.png|300px]]<br/>Original tunnel to be photographed, with camera looking from inside center to left wall.<br/>
|-
! rowspan=2 style="width:10%;" |&nbsp;!! Normal !! colspan=4 | Fisheye<ref>{{cite web|url=http://www.lenstip.com/index.php?test=obiektywu&test_ob=160|title=Samyang 8 mm f/3.5 Aspherical IF MC Fish-eye review - Introduction - Lenstip.com}}</ref><ref name=IMO-05>{{cite journal |title=Imaging: Fisheye lenses |author=Bettonvil, Felix |date=6 March 2005 |journal=WGN |publisher=International Meteor Organization |volume=33 |number=1 |pages=9–14 |bibcode = 2005JIMO...33....9B}}</ref>
|-
! style="width:18%;" |Rectilinear
! style="width:18%;" |Stereographic<ref>{{cite web |url=http://www.versacorp.com/vlink/jcreview/sy8rv9jc.pdf |title=Review of Samyang 8 mm f/3.5 Proportional Projection Ultra-wide Angle Lens. |author=Charles, Jefrey R. |date=4 December 2009 |publisher=Versacorp |access-date=6 November 2018}}</ref>
! style="width:18%;" |Equidistant
! style="width:18%;" |Equisolid angle
! style="width:18%;" |Orthographic
|-
! Other names
| gnomonic, perspective, conventional
| panoramic, conform, planisphere
| linear, linear-scaled
| equal-area
| orthogonal
|-
!Image
|[[File:PeterW zt 2.png|130px]]
|[[File:PeterW zt 4.png|130px]]
|[[File:PeterW zt 5.png|130px]]
|[[File:PeterW zt 6.png|130px]]
|[[File:PeterW zt 7.png|130px]]
|-
!Mapping function<ref name=IMO-05 />
|<math>r = f \tan \theta</math>
|<math>r = 2 f \tan \frac \theta 2</math>
|<math>r = f \, \theta</math>
|<math>r = 2 f \sin \frac \theta 2</math>{{efn|name=NotESA|Special case of <math>r = k_1 f \sin{\frac{\theta}{k_2}}</math>, where <math>k_1 = k_2 = 2</math>. Some fisheyes, such as the {{nowrap|AF Nikkor DX}} 10.5&nbsp;mm {{f/}}2.8 have slightly different values for <math>k_1</math> and <math>k_2</math>. }}
|<math>r = f \sin \theta</math>
|- style="vertical-align:top; text-align:left; font-size:85%;"
! Notes
| Works like the pinhole camera. Straight lines remain straight (distortion free). <math>\theta</math> 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. [[Company:Samyang Optics|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|solid angle]], or an equal area on the [[Unit sphere|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.
|- style="vertical-align:top; font-size:85%;"
! Examples<ref>{{cite web |url=http://michel.thoby.free.fr/Fisheye_history_short/Projections/Various_lens_projection.html |title=About the various projections of the photographic objective lenses |author=Thoby, Michel |date=6 November 2012 |access-date=6 November 2018}}</ref><ref>{{cite web |url=http://www.pierretoscani.com/fisheyes-(in-english).html |title=Fisheyes |author=Toscani, Pierre |date=20 December 2010 |access-date=6 November 2018}}</ref><ref>{{cite web |url=http://www.kurazumi.net/home/fisheye |title=Fish-eye lenses |publisher=Kurazumi Office |access-date=14 November 2018}}</ref>
| {{n/a|(Numerous)}}
|
* Samyang ''f''&nbsp;=&nbsp;8&nbsp;mm {{f/}}2.8
* Samyang ''f''&nbsp;=&nbsp;12&nbsp;mm {{f/}}2.8
|
* Canon FD ''f''&nbsp;=&nbsp;7.5&nbsp;mm {{f/}}5.6
* Coastal Optical ''f''&nbsp;=&nbsp;7.45&nbsp;mm {{f/}}5.6
* Nikkor ''f''&nbsp;=&nbsp;6&nbsp;mm {{f/}}2.8
* Nikkor ''f''&nbsp;=&nbsp;7.5&nbsp;mm {{f/}}5.6
* Nikkor ''f''&nbsp;=&nbsp;8&nbsp;mm {{f/}}2.8
* Nikkor ''f''&nbsp;=&nbsp;8&nbsp;mm {{f/}}8.0
* Peleng ''f''&nbsp;=&nbsp;8&nbsp;mm {{f/}}3.5
* Rokkor ''f''&nbsp;=&nbsp;7.5&nbsp;mm {{f/}}4.0
* Sigma ''f''&nbsp;=&nbsp;8&nbsp;mm {{f/}}3.5
* Samyang ''f''&nbsp;=&nbsp;7.5&nbsp;mm {{f/}}3.5
|
* Canon EF ''f''&nbsp;=&nbsp;15&nbsp;mm {{f/}}2.8 (1988)
* Minolta ''f''&nbsp;=&nbsp;16&nbsp;mm {{f/}}2.8 (1971)
* Nikkor ''f''&nbsp;=&nbsp;10.5&nbsp;mm {{f/}}2.8{{efn|For this lens, <math>k_1 = 1.47</math> and <math>1/k_2 = 0.713</math>, determined empirically.<ref name=Thoby>{{cite web |url=http://michel.thoby.free.fr/Blur_Panorama/Nikkor10-5mm_or_Sigma8mm/Sigma_or_Nikkor/Deux_Fiheyes_Compares.html |title=Comparaison de deux objectifs Fisheye: Sigma 8&nbsp;mm ''f''/4 et Nikkor 10,5&nbsp;mm ''f''/2,8 |trans-title=Comparison of two Fisheye lenses: Sigma 8&nbsp;mm ''f''/4 and Nikkor 10.5&nbsp;mm ''f''/2.8 |author=Thoby, Michel |date=20 December 2006 |access-date=14 November 2018}}</ref>}}
* Nikkor ''f''&nbsp;=&nbsp;16&nbsp;mm {{f/}}2.8 (1995)
* Sigma ''f''&nbsp;=&nbsp;4.5&nbsp;mm {{f/}}2.8
* Sigma ''f''&nbsp;=&nbsp;8&nbsp;mm {{f/}}4.0{{efn|In this case, <math>k_1 = 1.88</math> and <math>1/k_2 = 0.54</math>.<ref name=Thoby />}}
* Sigma ''f''&nbsp;=&nbsp;15&nbsp;mm {{f/}}2.8 (1990)
* Zuiko ''f''&nbsp;=&nbsp;8&nbsp;mm {{f/}}2.8
|
* Nikkor ''f''&nbsp;=&nbsp;10&nbsp;mm {{f/}}5.6 OP
* Yasuhara Madoka180 ''f''&nbsp;=&nbsp;7.3&nbsp;mm {{f/}}4
|}
{{notelist|25em}}

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

With appropriate software, the [[Curvilinear perspective|curvilinear]] images produced by a fisheye lens can be remapped to a conventional [[Engineering:Rectilinear lens|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==
{{Views}}
* [[Azimuthal equidistant projection]]
* [[Engineering:Dashcam|Dashcam]]
* [[Miniature faking]]
* [[Stereographic projection]]
* :de:Fischaugenobjektiv Fisheye lens with more information in German

==Notes==
{{Reflist|group=n}}
{{clear}}

==References==
{{Reflist}}

==External links==
* [http://wiki.panotools.org/Fisheye_Projection Fisheye projection theory]
* [https://docs.google.com/spreadsheets/d/1hqXPmpR_zBitA3g2aJknDAawy2pvvHnGHymTcJaGU9o/edit?usp=sharing List of all Current and all Affordable Photography Fisheye Lenses]
* T[https://www.thefisheyelist.com/ he New Fish List, Luca Vascon, 2021]
* [http://michel.thoby.free.fr/Fisheye_history_short/Projections/Various_lens_projection.html Various fisheye projections]
* {{cite conference |doi=10.1117/12.405226 |title=Fish-eye lens designs and their relative performance |author1=Kumler, James "Jay" |author2=Bauer, Martin |date=2000 |conference=International Symposium of Optical Science and Technology |location=San Diego, California |publisher=SPIE}} [https://web.archive.org/web/20050128213649/http://www.coastalopt.com/pdfs/FisheyeComparison_SPIE.pdf Alternative archived URL]

{{Photography}}

[[Category:Photographic lenses]]

{{Sourceattribution|Fisheye lens}}