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| [[File:Peirce quincuncial projection SW 20W.JPG|250px|thumb|Peirce quincuncial projection of the world. The red equator is a square whose corners are the only four points on the map which fail to be conformal.]]
| | The author is known by the name of Figures Wunder. His spouse doesn't like it the way he does but what he really likes doing is to do aerobics and he's been performing it for fairly a while. Minnesota is exactly where he's been living for many years. Bookkeeping is my occupation.<br><br>My blog ... [http://xrambo.com/blog/192034 xrambo.com] |
| [[File:Peirce quincuncial projection SW 20W tiles.JPG|thumb|250px|Tessellated version of the Peirce quincuncial map.]]
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| The '''Peirce quincuncial projection'''<ref>A Quincuncial Projection of the Sphere. By [[Charles Sanders Peirce]]. 1890.</ref><ref>[[I. Frischauf]]. Bemerkungen zu C. S. Peirce Quincuncial Projection. (Tr., Comments on C. S. Peirce Quincuncial Projection.)</ref><ref>A Treatise on Projections. By [[Thomas Craig (Author)|Thomas Craig]]. U.S. Government Printing Office, 1882. [http://books.google.com/books?id=9UU7AAAAMAAJ&pg=PA132 p132]</ref><ref>Science, Volume 11. Moses King, 1900. [http://books.google.com/books?id=n5oSAAAAYAAJ&pg=PA186 p186]</ref> is a [[conformal projection|conformal]] [[map projection]] (except for four points where its conformality fails) that presents the sphere as a square. It was developed by [[Charles Sanders Peirce]] in 1879.
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| ==History==
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| The maturation of [[complex analysis]] led to general techniques for [[conformal mapping]], where points of a flat surface are handled as numbers on the [[complex plane]].
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| While working at the [[U.S. Coast and Geodetic Survey]], the American philosopher [[Charles Sanders Peirce]] published his projection in 1879 (Peirce 1879),<ref>(Lee, 1976) gives 1877 as the year in which the projection was conceived, citing "US Coast Survey Report for the Year Ending with June 1877", 191–192.</ref> having been inspired by [[Hermann Amandus Schwarz|H.A. Schwarz's]] 1869 [[Schwarz–Christoffel mapping|conformal transformation of a circle onto a polygon of ''n'' sides]] (known as the Schwarz–Christoffel mapping). In the normal aspect, Peirce's projection presents the [[northern hemisphere]] in a square; the [[southern hemisphere]] is split into four isosceles triangles symmetrically surrounding the first one, akin to star-like projections. In effect, the whole map is a square, inspiring Peirce to call his projection ''[[quincuncial]]'', after the arrangement of five items in a [[quincunx]].
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| After Peirce presented his projection, two other cartographers developed similar projections of the hemisphere (or the whole sphere, after a suitable rearrangement) on a square: Guyou in 1887 and Adams in 1925 (Lee, 1976). The three projections are [[Map_projection#Aspects_of_the_projection|transversal]] versions of each other (see related projections below).
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| ==Formal description==
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| The Peirce quincuncial projection is "formed by transforming the [[Stereographic projection#Cartography|stereographic projection]] with a pole at infinity, by means of an elliptic function" (Peirce, 1879). The Peirce quincuncial is really a projection of the hemisphere, but its tessellation properties (see below) permit its use for the entire sphere. Peirce's projection maps the interior of a circle (corresponding to each hemisphere, which were created by projecting them using the stereographic projection) onto the interior of a square (using the [[Schwarz–Christoffel mapping]])<!--(the inner diamond on the map as shown in this article)--> (Lee, 1976).
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| A point ''P'' on the Earth's surface, a distance ''p'' from the [[north pole]] with [[longitude]] ''θ'' and [[latitude]] ''λ'' is first mapped to a point (''p'', ''θ'') of the plane through the equator, viewed as the complex plane with coordinate ''w''; this ''w'' coordinate is then mapped to another point (''x'', ''y'') of the complex plane (given the coordinate ''z'') by an elliptic function of the first kind. Using Gudermann's notation for [[Jacobi's elliptic functions]], the relationships are
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| :<math>\tan \left( \frac{p}{2} \right) e^{i \theta} = \mathrm{cn} \left( z, \frac{1}{2} \right), \text{ where } w = p e^{i \theta} \text{ and } z = x + i y.</math>
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| ==Properties==
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| According to Peirce, his projection has the following properties (Peirce, 1879):
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| * The sphere is presented in a square.
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| * The part where the exaggeration of scale amounts to double that at the centre is only 9% of the area of the sphere, against 13% for the [[Mercator projection]] and 50% for the stereographic projection.
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| * The curvature of lines representing great circles is, in every case, very slight, over the greater part of their length.
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| * It is conformal everywhere except at the four corners of the inner hemisphere (thus the midpoints of edges of the projection), where the equator and four meridians change direction abruptly (the equator is represented by a square). These are [[Mathematical singularity|singularities]] where [[differentiability]] fails.
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| * It can be tessellated in all directions.
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| ==Tiled Peirce quincuncial maps==
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| The projection [[tessellate]]s the plane; i.e., repeated copies can completely cover (tile) an arbitrary area, each copy's features exactly matching those of its neighbors. See [[:File:Peirce quincuncial projection SW 20W tiles.JPG|this image]] for an example.
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| Furthermore, the four triangles of the second hemisphere of Peirce quincuncial projection can be rearranged as another square that is placed next to the square that corresponds to the first hemisphere, resulting in a rectangle with aspect ratio of 2:1; this arrangement is equivalent to the transverse aspect of the [[Guyou hemisphere-in-a-square projection]] (Snyder, 1993).
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| ==Known uses==
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| [[Image:PeircePanorama2007.jpg|right|thumb|250px| Using the Peirce quincuncial projection to present a spherical panorama.]]
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| Like many other projections based upon complex numbers, the Peirce quincuncial has been rarely used for geographic purposes. One of the few recorded cases is in 1946, when it was used by the U.S. Coast and Geodetic Survey world map of air routes (Snyder, 1993).
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| It has been used recently to present spherical panoramas for practical as well as aesthetic purposes, where it can present the entire sphere with most areas being recognizable (German et al. 2007).
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| ==Related projections==
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| In transverse aspect, one hemisphere becomes the [[Adams hemisphere-in-a-square projection]] (the pole is placed at the corner of the square). Its four singularities are at the north pole, the south pole, on the equator at [[25th meridian west|25°W]], and on the equator at 155°E, in the Arctic, Atlantic, and Pacific oceans, and in Antarctica.<ref name="Furuti">
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| Carlos A. Furuti.
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| [http://www.progonos.com/furuti/MapProj/Normal/ProjConf/projConf.html Map Projections:Conformal Projections].
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| </ref> That great circle divides the traditional western and [[eastern hemisphere]]. | |
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| In [[Map_projection#Aspects_of_the_projection|oblique aspect]] (45 degrees) of one hemisphere becomes the [[Guyou hemisphere-in-a-square projection]] (the pole is placed in the middle of the edge of the square). Its four singularities are at 45 degrees north and south latitude on the great circle composed of the [[20th meridian west|20°W]] meridian and the 160°E meridians, in the Atlantic and Pacific oceans.<ref name="Furuti"/> That great circle divides the traditional western and eastern hemispheres.
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| ==See also==
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| {{Portal|Atlas}}
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| * [[List of map projections]]
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| ==References==
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| {{reflist}}
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| ;General
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| {{refbegin}}
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| *{{cite conference
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| | first = Daniel
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| | last = German
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| | authorlink =
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| | coauthors = d'Angelo, Pablo ; Gross, Michael and Postle, Bruno
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| | title = New Methods to Project Panoramas for Practical and Aesthetic Purposes
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| | booktitle = "Proceedings of Computational Aesthetics 2007"
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| | pages = 15–22
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| | publisher = Eurographics
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| | date = June 2007
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| | location = Banff}}
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| *{{cite book | author=Grattan-Guinness, I. | title=The Fontana History of the Mathematical Sciences | year=1997 | location=London | publisher=Fontana Press (Harper Collins) | isbn=0-00-686179-2 | oclc=222220485 }}
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| *{{cite journal | author=L.P. Lee | title = Conformal Projections based on Elliptic Functions | year=1976 | journal = Cartographica | volume=13 | issue = Monograph 16, supplement No. 1 to Canadian Cartographer}}
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| * Peirce, C. S. (1877/1879), "Appendix No. 15. A Quincuncial Projection of the Sphere", ''Report of the Superintendent of the United States Coast Survey Showing the Progress of the Survey for Fiscal Year Ending with June 1877'', pp. 191–194 followed by 25 progress sketches including (25th) the illustration (the map itself). Full ''Report'' submitted to the Senate December 26, 1877 and published 1880 (see further below).
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| ** Article first published December 1879, ''American Journal of Mathematics'' '''2''' (4): 394–397 (without the sketches except final map), Google Books [http://books.google.com/books?id=7a0EAAAAYAAJ&pg=PA394 Eprint] (Google version of map is partly botched), [http://jstor.org/stable/2369491 JSTOR Eprint], [[Digital Object Identifier|doi]]:[http://dx.doi.org/10.2307%2F2369491 10.2307/2369491]. ''AJM'' version reprinted in ''Writings of Charles S. Peirce'' '''4''':68–71.
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| ** Article reprinted 1880 including publication of all sketches, in the full ''Report'', by the U.S. Government Printing Office, Washington, D.C. [http://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0026.PDF#page=215 NOAA PDF Eprint], link goes to Peirce's article on ''Report'''s p. 191, PDF's p. 215. NOAA's PDF lacks the sketches and map and includes [http://historicals.ncd.noaa.gov/historicals/histmasp.asp broken link] to their planned online location, NOAA's [http://historicalcharts.noaa.gov/historicals/historical_zoom.asp Historical Map and Chart Collection], where they do not seem to be as of 7/19/2010. Google Books [http://books.google.com/books?id=nn3pAAAAMAAJ&pg=PA191 Eprint] (Google botched the sketches and partly botched the [http://books.google.com/books?id=nn3pAAAAMAAJ&pg=PA267 illustration (the map itself)].) Note: Other Google [http://books.google.com/books?id=TYpNAAAAYAAJ&pg=PA191 edition of 1877 Coast Survey Report] completely omits the pages of sketches including the illustration (the map).
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| * {{cite book
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| | last = [[John P. Snyder|Snyder]]
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| | first = John P.
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| | title = Flattening the Earth
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| | publisher = University of Chicago
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| | year = 1993
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| | isbn = 0-226-76746-9
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| | oclc = 26764604}}
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| * {{cite book
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| | last = Snyder
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| | first = John P.
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| | title = An Album of Map Projections, Professional Paper 1453
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| | publisher = US Geological Survey
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| | year = 1989}}
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| {{refend}}
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| ;Citations
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| {{reflist}}
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| == External links ==
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| *[http://www.uff.br/mapprojections/Peirce_en.html An interactive Java Applet to study the metric deformations of the Peirce Projection].
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| *[http://www.flickr.com/groups/quincuncial/ More examples of Peirce quincuncial panoramas]
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| {{Map Projections}}
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| [[Category:Cartographic projections]]
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| [[Category:Conformal mapping]]
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| [[Category:Charles Sanders Peirce]]
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The author is known by the name of Figures Wunder. His spouse doesn't like it the way he does but what he really likes doing is to do aerobics and he's been performing it for fairly a while. Minnesota is exactly where he's been living for many years. Bookkeeping is my occupation.
My blog ... xrambo.com