Peirce quincuncial projection 1

Peirce quincuncial projection

Peirce quincuncial projection of the world. The red equator is a square whose corners arethe only four points on the map which fail to be conformal.

The Peirce quincuncial projection isa conformal map projection (except forfour points where its conformalityfails) that presents the sphere as asquare. It was developed by CharlesSanders Peirce in 1879.

History

The maturation of complex analysisled to general techniques for conformalmapping, where points of a flat surfaceare handled as numbers on thecomplex plane. While working at theU.S. Coast and Geodetic Survey, theAmerican philosopher Charles SandersPeirce published his projection in 1879(Peirce 1879)[1], having been inspiredby H.A. Schwarz's 1869 conformaltransformation of a circle onto apolygon of n sides (known as theSchwarz–Christoffel mapping). In thenormal aspect, Peirce's projectionpresents the northern hemisphere in a square; the southern hemisphere is split into four isosceles trianglessymmetrically surrounding the first one, akin to star-like projections. In effect, the whole map is a square, inspiringPeirce to call his projection quincuncial, after the arrangement of five items in a quincunx.

After Peirce presented his projection, two other cartographers developed similar projections of the hemisphere (orthe whole sphere, after a suitable rearrangement) on a square: Guyou in 1887 and Adams in 1925 (Lee, 1976). Thethree projections are transversal versions of each other (see related projections below).

Peirce quincuncial projection 2

Tessellated version of the Peirce quincuncial map.

Formal description

The Peirce quincuncial projection is"formed by transforming thestereographic projection with a pole atinfinity, by means of an ellipticfunction" (Peirce, 1879). The Peircequincuncial is really a projection of thehemisphere, but its tessellationproperties (see below) permit its usefor the entire sphere. Peirce'sprojection maps the interior of a circle(corresponding to each hemisphere,which were created by projecting themusing the stereographic projection)onto the interior of a square (using theSchwarz–Christoffel mapping) (Lee,1976).

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 apoint (p, θ) of the plane through the equator, viewed as the complex plane with coordinate w; this w coordinate isthen mapped to another point (x, y) of the complex plane (given the coordinate z) by an elliptic function of the firstkind. Using Gudermann's notation for Jacobi's elliptic functions, the relationships are

PropertiesAccording to Peirce, his projection has the following properties (Peirce, 1879):•• The sphere is presented in a square.• 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.•• The curvature of lines representing great circles is, in every case, very slight, over the greater part of their length.• 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 asquare). These are singularities where differentiability fails.

•• It can be tessellated in all directions.

Peirce quincuncial projection 3

Tiled Peirce quincuncial mapsThe projection tessellates the plane; i.e., repeated copies can completely cover (tile) an arbitrary area, each copy'sfeatures exactly matching those of its neighbors. See this image [2] for an example. Furthermore, the four triangles ofthe second hemisphere of Peirce quincuncial projection can be rearranged as another square that is placed next to thesquare that corresponds to the first hemisphere, resulting in a rectangle with aspect ratio of 2:1; this arrangement isequivalent to the transverse aspect of the Guyou hemisphere-in-a-square projection (Snyder, 1993).

Known uses

Using the Peirce quincuncial projection to present a sphericalpanorama.

Like many other projections based upon complexnumbers, the Peirce quincuncial has been rarely usedfor geographic purposes. One of the few recorded casesis in 1946, when it was used by the U.S. Coast andGeodetic Survey world map of air routes (Snyder,1993). It has been used recently to present sphericalpanoramas for practical as well as aesthetic purposes,where it can present the entire sphere with most areasbeing recognizable (German et al. 2007).

Related projections

In transverse aspect, one hemisphere becomes theAdams hemisphere-in-a-square projection (the pole isplaced at the corner of the square). Its four singularitiesare at the north pole, the south pole, on the equator at25°W, and on the equator at 155°E, in the Arctic,Atlantic, and Pacific oceans, and in Antarctica.[3] Thatgreat circle divides the traditional western and easternhemisphere.

In oblique aspect (45 degrees) of one hemisphere becomes the Guyou hemisphere-in-a-square projection (the pole isplaced in the middle of the edge of the square). Its four singularities are at 45 degrees north and south latitude on thegreat circle composed of the 20°W meridian and the 160°E meridians, in the Atlantic and Pacific oceans.[3] Thatgreat circle divides the traditional western and eastern hemispheres.

Notes[1] (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.[2] http:/ / www. progonos. com/ furuti/ MapProj/ Normal/ ProjConf/ Img/ pqTiled. jpg[3] Carlos A. Furuti. Map Projections:Conformal Projections (http:/ / www. progonos. com/ furuti/ MapProj/ Normal/ ProjConf/ projConf. html).

References• German, Daniel; d'Angelo, Pablo ; Gross, Michael and Postle, Bruno (June 2007). "New Methods to Project

Panoramas for Practical and Aesthetic Purposes". "Proceedings of Computational Aesthetics 2007". Banff:Eurographics. pp. 15–22.

• Grattan-Guinness, I. (1997). The Fontana History of the Mathematical Sciences. London: Fontana Press (HarperCollins). ISBN 0-00-686179-2. OCLC 222220485.

Peirce quincuncial projection 4

• L.P. Lee (1976). "Conformal Projections based on Elliptic Functions". Cartographica 13 (Monograph 16,supplement No. 1 to Canadian Cartographer).

• Peirce, C. S. (1877/1879), "Appendix No. 15. A Quincuncial Projection of the Sphere", Report of theSuperintendent of the United States Coast Survey Showing the Progress of the Survey for Fiscal Year Ending withJune 1877, pp. 191–194 followed by 25 progress sketches including (25th) the illustration (the map itself). FullReport submitted to the Senate December 26, 1877 and published 1880 (see further below).• Article first published December 1879, American Journal of Mathematics 2 (4): 394–397 (without the sketches

except final map), Google Books Eprint (http:/ / books. google. com/ books?id=7a0EAAAAYAAJ&pg=PA394) (Google version of map is partly botched), JSTOR Eprint (http:/ / jstor. org/ stable/ 2369491), doi:10.2307/2369491 (http:/ / dx. doi. org/ 10. 2307/ 2369491). AJM version reprinted in Writings of Charles S.Peirce 4:68–71.

• Article reprinted 1880 including publication of all sketches, in the full Report, by the U.S. GovernmentPrinting Office, Washington, D.C. NOAA PDF Eprint (http:/ / docs. lib. noaa. gov/ rescue/ cgs/ 001_pdf/CSC-0026. PDF#page=215), link goes to Peirce's article on Report's p. 191, PDF's p. 215. NOAA's PDF lacksthe sketches and map and includes broken link (http:/ / historicals. ncd. noaa. gov/ historicals/ histmasp. asp) totheir planned online location, NOAA's Historical Map and Chart Collection (http:/ / historicalcharts. noaa. gov/historicals/ historical_zoom. asp), where they do not seem to be as of 7/19/2010. Google Books Eprint (http:/ /books. google. com/ books?id=nn3pAAAAMAAJ& pg=PA191) (Google botched the sketches and partlybotched the illustration (the map itself) (http:/ / books. google. com/ books?id=nn3pAAAAMAAJ&pg=PA267).) Note: Other Google edition of 1877 Coast Survey Report (http:/ / books. google. com/books?id=TYpNAAAAYAAJ& pg=PA191) completely omits the pages of sketches including the illustration(the map).

• Snyder, John P. (1993). Flattening the Earth. University of Chicago. ISBN 0-226-76746-9. OCLC 26764604.• Snyder, John P. (1989). An Album of Map Projections, Professional Paper 1453. US Geological Survey.

External links• An interactive Java Applet to study the metric deformations of the Peirce Projection (http:/ / www. uff. br/

mapprojections/ Peirce_en. html).• More examples of Peirce quincuncial panoramas (http:/ / www. flickr. com/ groups/ quincuncial/ )

Article Sources and Contributors 5

Article Sources and ContributorsPeirce quincuncial projection  Source: http://en.wikipedia.org/w/index.php?oldid=508239168  Contributors: Akulo, AnonMoos, Benizi, Blaxthos, Dmgerman, EdwardLane, Foobaz, Jbening,Klparrot, Mdf, Michael Hardy, Oleg Alexandrov, Powerslide, Rich Farmbrough, Rjwilmsi, Strebe, The Tetrast, Xnn, 7 anonymous edits

Image Sources, Licenses and ContributorsFile:Peirce quincuncial projection SW 20W.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Peirce_quincuncial_projection_SW_20W.JPG  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: StrebeFile:Peirce quincuncial projection SW 20W tiles.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Peirce_quincuncial_projection_SW_20W_tiles.JPG  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: StrebeImage:PeircePanorama2007.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:PeircePanorama2007.jpg  License: Creative Commons Attribution 2.5  Contributors: Dmgerman

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