A Peirce quincuncial projection of the Earth. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001

The Peirce quincuncial projection is a conformal map projection (except for four points where its conformality fails) that presents the sphere as a square. In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane A map projection is any method of representing the Surface of a sphere or other shape on a plane. It was developed by Charles Sanders Peirce in 1879. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher Year 1879 ( MDCCCLXXIX) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common

History

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. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis While working at the U.S. Coast and Geodetic Survey, the American philosopher Charles Sanders Peirce published his projection in 1879 (Peirce 1879)^[1], having been inspired by H.A. Schwarz's 1869 conformal transformation of a circle onto a polygon of n sides (known as the Schwarz-Christoffel mapping). The National Geodetic Survey and the Office of Coast Survey are the two successor agencies in the United States to the U Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher Year 1879 ( MDCCCLXXIX) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common See also Hermann Schwarz (philosopher (1864&ndash1951 A different Hermann Schwarz was a founder of Rohde & Schwarz, a German manufacturer In Complex analysis, a discipline within Mathematics, a Schwarz-Christoffel mapping is a transformation of the Complex plane that maps the Upper half-plane In the normal aspect, Peirce's projection presents the northern hemisphere in a square; the other hemisphere is split into four isosceles triangles symmetrically surrounding the first one, akin to star-like projections. Northern Hemisphere is the half of a Planet that is North of the Equator —the word hemisphere literally means 'half ball' In effect, the whole map is a square, inspiring Peirce to call his projection quincuncial, after the arrangement of five items in a quincunx. For Sir Francis Galton 's machine for demonstrating the Normal distribution named "quincunx" see Bean machine. For Sir Francis Galton 's machine for demonstrating the Normal distribution named "quincunx" see Bean machine.

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 transversal versions of each other (see related projections below).

Formal description

The Peirce quincuncial projection is "formed by transforming the stereographic projection with a pole at infinity, by means of an elliptic function" (Peirce, 1879). In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane 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) (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 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. The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is subject to the caveats explained below defined as the point in the northern Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the Using Gudermann's notation for Jacobi's elliptic functions, the relationships are

Properties

According to Peirce, his projection has the following properties (Peirce, 1879):

• The sphere is presented in a square. In Mathematics, the Jacobi elliptic functions are a set of basic Elliptic functions and auxiliary Theta functions that have historical importance with also
• The part where the exaggeration of scale amounts to double that at the centre is only 9% of the are of the sphere, against 13% for the Mercator and 50% for the stereographic
• 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 corners of the inner hemisphere (thus the midpoints of edges of the projection), where the Equator and 4 meridians change direction abruptly (the Equator is represented by a square).
• It can be tessellated in all directions.

Tiled Peirce quincuncial maps

Tessellated version of the Peirce quincuncial map above.

The projection tessellates the plane; i. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps e. , repeated copies can completely cover (tile) an arbitrary area, each copy's features exactly matching those of its neighbors. See this image for an example. 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 the transverse aspect of the Guyou hemisphere-in-a-square projection (Snyder, 1993). The Guyou hemisphere-in-a-square projection is a conformal Map projection for the hemisphere (except for four points where the conformality fails

Known uses

Using the Peirce quincuncial projection to present a spherical panorama.

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). 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).

Related projections

• It is based upon the stereographic projection
• Its transverse aspect of one hemisphere becomes the Adams hemisphere-in-a-square projection (the pole is placed at the corner of the square)
• Its 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). In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane The Adams-hemisphere-in-a-square is a Conformal map Map projection for a hemisphere (except for four points where the conformality fails The Guyou hemisphere-in-a-square projection is a conformal Map projection for the hemisphere (except for four points where the conformality fails

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.

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": 15--22, Banff: Eurographics.  
• Grattan-Guinness, I. (1997). The Fontana History of the Mathematical Sciences. London: Fontana Press (Harper Collins). ISBN 0-00-686179-2.  
• L. P. Lee (1976). "Conformal Projections based on Elliptic Functions". Cartographica 13 (Monograph 16, supplement No. 1 to Canadian Cartographer).  
• C. S. Peirce (Dec 1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics 2 (4): 394--396. doi:10.2307/2369491. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  

• Snyder, John P. John Parr Snyder ( 12 April 1926 – 28 April 1997) was an American cartographer most known for his work on Map projections (1993). Flattening the Earth. University of Chicago. ISBN 0-226-76746-9.  
• Snyder, John P. (1989). An Album of Map Projections, Professional Paper 1453. US Geological Survey.  

External links

• Map Projections:Conformal Projections
• An interactive Java Applet to study the metric deformations of the Peirce Projection.