Peirce quincuncial projection
The projection has the distinctive property that it forms a seamless square tiling of the plane, conformal except at four singular points along the equator.
Typically the projection is square and oriented such that the north pole lies at the center, but an oblique aspect in a rectangle was proposed by Émile Guyou in 1887, and a transverse aspect was proposed by Oscar S. Adams in 1925.
While working at the United States Coast and Geodetic Survey, the American philosopher Charles Sanders Peirce published his projection in 1879,[2] 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).
In effect, the whole map is a square, inspiring Peirce to call his projection quincuncial, after the arrangement of five items in a quincunx.
[4] The Peirce quincuncial is really a projection of the hemisphere, but its tessellation properties (see below) permit its use for the entire sphere.
where An elliptic integral of the first kind can be used to solve for w. The comma notation used for sd(u, k) means that
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.
[5] 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.
[6] In transverse aspect, one hemisphere becomes the Adams hemisphere-in-a-square projection (the pole is placed at the corner of the square).
In 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).
