The Collignon projection is an equal-area

pseudocylindrical In cartography, map projection is the term used to describe a broad set of Transformation (function) , transformations employed to represent the two-dimensional curved Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In ...

map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...

first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. For the smallest choices of the parameters chosen for this projection, the sphere may be mapped either to a single diamond, a pair of squares, or a triangle. The projection is used in the polar areas as part of the

HEALPix HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-sphere, is an algorithm for pixelisation of the 2-sphere and the associated class of map projections. The pixelisation algorithm was de ...

spherical projection, which is widely used in

physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...

in making maps of the

cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...

, in particular by the

WMAP The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic mic ...

and

Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...

space missions.

Formulae

Let ''R'' be the radius of the sphere, ''φ'' the

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

, ''λ'' the

longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

, and ''λ₀'' the longitude of the central meridian (chosen as desired). Also, define

s = \sqrt = \sqrt \sin\left(\frac - \frac\right)

, where the two forms are equivalent for ''φ'' in the range of possible latitudes. Then the Collignon projection is given by: :

\begin
  x &= \fracR \left( \lambda - \lambda_0 \right) s, \\
  y &= \sqrt R \left( 1 - s \right).
\end

This formula gives the projection as pictured above, coming to a point at the

North Pole The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Mag ...

. For a projection coming to a point at the

South Pole The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...

, as in the bottom portion of the HEALPix projection, replace ''φ'' and ''y'' with ''-φ'' and ''-y''. The standard parallel is 15°51′N.

External links

* Map projections Equal-area projections {{cartography-stub

Formulae

See also

External links