Peirce quincuncial projection

The quincunx map projection is a published by Charles S. Peirce in 1879 compliant map projection (also Stereographic projection), the compliant in all areas with the exception of the corners of the inner hemisphere, ie is conformal.

Naming

The name refers to the quincunx arrangement of five points in a square with a dot in the center and four points at the corners. Peirce chose this name because this structure, the distribution of the poles corresponds in its map projection.

Construction of projection quincunx

In the projection of Peirce of the entire planet is detected in a square. Northern hemisphere is again projected in the normal display in a square, the corners of which are each in the middle of the edges of the outer square. The equator corresponds to the edge of this inner, set at an angle square. The southern hemisphere is divided into four isosceles triangles that lie in the outer corners of the entire square. While the north pole is at the center of the square, the south pole is indicated by the four corners of the outer square.

The areas of the southern hemisphere are symmetrically arranged so that they can form two parts of a square projection which is to be moved freely. By quadrupling the projection gives a tile-like representation in which even the South Pole is located in the center of a map section.

In conformal maps the angle between lines remain the same, that is, the scale of a point in all directions constant. Meridian and parallels intersect at right angles. The shape of the surfaces is maintained locally. Conformal maps are mainly used in navigation and surveying. Scattering length of the projection Peirce is not: It is strongly crowded together at the poles and severely stretched at the equator. The equator and four meridians are straight lines, all other longitudes and latitudes are complex curves.

Notes on the calculation of the projection

The calculations of the quincunx - card based on the complex analysis (functional theory ), in which the points of a flat surface to be treated as a complex number plane. A compliant card is a function of the angle are kept. Formally, a function w = f ( z) -compliant if received at z0 the angles and curves passing through z0 their orientation. The calculation of the projection requires elliptic integrals of the first kind

Comparable projections

Publications of similar concepts conformal projections, which are integrated into a square, as submitted by the Frenchman Emile Guyou (1886 /87) and Oscar C. Adams, who, like Peirce worked with the U.S. Coast and Geodetic Survey. The Peirce projection has been proposed in a modified form by the U.S. Coast and Geodetic Survey in World War II for use in the prosecution of aircraft movements.