Quincunx map projection

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Quincunx map projection of the earth's surface
Tile-like representation

The quincunx map projection is a conformal map projection (also called stereographic projection ) published by Charles S. Peirce in 1879 , which conforms in all areas with the exception of the corners of the inner hemisphere, i.e. is conformal .

Naming

Quincunx

The name Quincunx refers to the arrangement of five points in a square with one point in the center and four points in the corners. Peirce chose this name because this structure corresponds to the distribution of the poles in his map projection.

Structure of the quincunx projection

In Peirce's projection, the entire surface of the earth is captured in a square. The northern hemisphere is again projected into a square in the usual representation, the corners of which are in the middle of the edges of the outer square. The equator corresponds to the edges of this inner, pointed square. The southern hemisphere is divided into four isosceles triangles that lie in the outer corners of the overall square. While the north pole forms the center of the square, the south pole is indicated by the four corners of the outer square.

The surfaces of the southern hemisphere are arranged symmetrically in such a way that one can form a rectangle from two parts of the projection, which can be moved as desired. By quadrupling the projection you get a tile-like representation in which the south pole is also in the center of a map section.

In the case of conformal maps, the angles between the lines remain unchanged, i.e. the scale is constant in all directions from one point. The meridian and circles of latitude intersect perpendicularly. The shape of the surfaces is also retained locally. True-to-angle maps are mainly used in navigation and surveying . Peirce's projection is not true to length : it is strongly compressed at the poles and strongly stretched at the equator. The equator and four meridians are straight lines, all other longitudes and latitudes are complicated curves.

Notes on calculating the projection

The calculations of the quincunx cards are based on complex analysis ( function theory ), in which the points on a flat surface are treated like numbers on a complex plane . A conforming map is a function in which the angles are preserved. Formally, a function w  =  f ( z ) conforms if in z 0 the angles and the curves running through z 0 get their orientation. The computation of the projection requires elliptic integrals of the first kind.

Comparable projections

The French Emile Guyou (1886/87) and Oscar C. Adams , who, like Peirce, worked for the US Coast and Geodetic Survey, have published comparable concepts of conformal projections that are embedded in a square . The Peirce projection was proposed in a modified form by the US Coast and Geodetic Survey for use in tracking flight movements during World War II.

literature

  • Elisabeth Walter: Charles Sanders Peirce. Life and work. Agis, Baden-Baden 1989, ISBN 3-87007-035-8

Web links

Commons : Quincunx Map Projection  - collection of images, videos, and audio files