The Collignon projection is an equal-area pseudocylindrical

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

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 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 fu ...

in making maps of the

cosmic microwave background The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...

, in particular by the WMAP and Planck space missions.

Formulae

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

latitude In geography, latitude is a geographic coordinate system, geographic 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 t ...

, ''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 lett ...

, 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 rotation, Earth's axis of rotation meets its surface. It is called the True North Pole to distingu ...

. For a projection coming to a point at the

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

, 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

* Equal-area projections {{cartography-stub

Formulae

See also

External links