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Equirectangular projection

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From Wikipedia, the free encyclopedia

Cylindrical equidistant map projection

Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).

Equirectangular projection withTissot's indicatrix of deformation and with the standard parallels lying on the equator

True-colour satellite image of Earth in equirectangular projection

Height map of planet Earth at 2km per pixel, including oceanicbathymetry information, normalized as 8-bit grayscale. Because of its easy conversion between x, y pixel information and lat-lon, maps like these are very useful for software map renderings.

Theequirectangular projection (also called theequidistant cylindrical projection orla carte parallélogrammatique projection), and which includes the special case of theplate carrée projection (also called thegeographic projection,lat/lon projection, orplane chart), is a simplemap projection attributed toMarinus of Tyre who,Ptolemy claims, invented the projection about AD 100.^[1]

The projection mapsmeridians to vertical straight lines of constant spacing (for meridional intervals of constant spacing), andcircles of latitude to horizontal straight lines of constant spacing (for constant intervals ofparallels). The projection is neitherequal area norconformal. Because of the distortions introduced by this projection, it has little use innavigation orcadastral mapping and finds its main use inthematic mapping. In particular, the plate carrée has become a standard for globalraster datasets, such asCelestia,NASA World Wind, theUSGS Astrogeology Research Program, andNatural Earth, because of the particularly simple relationship between the position of animage pixel on the map and its corresponding geographic location on Earth or other spherical solar system bodies. In addition it is frequently used in panoramic photography to represent a spherical panoramic image.^[2]

Definition

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The forward projection transforms spherical coordinates into planar coordinates. The reverse projection transforms from the plane back onto the sphere. The formulae presume aspherical model and use these definitions:

$\lambda$ is thelongitude of the location to project;
$\varphi$ is thelatitude of the location to project;
$\varphi _{1}$ are the standard parallels (north and south of the equator) where the scale of the projection is true;
$\varphi _{0}$ is the central parallel of the map;
$\lambda _{0}$ is the central meridian of the map;
$x {\displaystyle x}$ is the horizontal coordinate of the projected location on the map;
$y {\displaystyle y}$ is the vertical coordinate of the projected location on the map;
$R {\displaystyle R}$ is the radius of the globe.

Longitude and latitude variables are defined here in terms of radians.

Forward

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{\begin{aligned}x&=R(\lambda -\lambda _{0})\cos \varphi _{1}\\y&=R(\varphi -\varphi _{0})\end{aligned}}

Theplate carrée (French, forflat square),^[3] is the special case where $\varphi _{1}$ is zero. This projection mapsx to be the value of the longitude andy to be the value of the latitude,^[4] and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. Despite sometimes being called "unprojected",^{[by whom?]} it is actually projected.^{[citation needed]}

When the $\varphi _{1}$ is not zero, such asMarinus's $\varphi _{1}=36^{\circ }$ ,^[5] theGall isographic projection's $\varphi _{1}=45^{\circ }$ , or Ronald Miller's $\varphi _{1}=(37.5^{\circ },43.5^{\circ },50.5^{\circ })$ ,^[6] the projection can portray particular latitudes of interest at true scale.

While a projection with equally spaced parallels is possible for an ellipsoidal model, it would no longer be equidistant because the distance between parallels on an ellipsoid is not constant. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing.

Reverse

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{\begin{aligned}\lambda &={\frac {x}{R\cos \varphi _{1}}}+\lambda _{0}\\\varphi &={\frac {y}{R}}+\varphi _{0}\end{aligned}}

Alternative names

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In spherical panorama viewers, usually:

$\lambda$ is called "yaw";^[7]
$\varphi$ is called "pitch";^[8]

where both are defined in degrees.

References

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^Flattening the Earth: Two Thousand Years of Map Projections,John P. Snyder, 1993, pp. 5–8,ISBN 0-226-76747-7.
^"Equirectangular Projection - PanoTools.org Wiki".wiki.panotools.org. Retrieved2021-05-04.
^Farkas, Gábor."Plate Carrée - a simple example".O’Reilly Online Learning. Retrieved31 December 2022.
^Paul A. Longley; Michael F. Goodchild; David J. Maguire; David W. Rhind (2005).Geographic Information Systems and Science. John Wiley & Sons. p. 119.ISBN 9780470870013.
^Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 7,ISBN 0-226-76747-7.
^"Equidistant Cylindrical (Plate Carrée)".PROJ coordinate transformation software library. Retrieved25 August 2020.
^"Yaw - PanoTools.org Wiki".wiki.panotools.org. Retrieved2021-05-04.
^"Pitch - PanoTools.org Wiki".wiki.panotools.org. Retrieved2021-05-04.

External links

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Global MODIS based satellite map The blue marble: land surface, ocean color, and sea ice.
Table of examples and properties of all common projections, from radicalcartography.net.
Panoramic Equirectangular Projection, PanoTools wiki.
Equidistant Cylindrical (Plate Carrée) in proj4

Map projection

By surface

Cylindrical

Mercator-conformal	Gauss–Krüger Transverse Mercator Oblique Mercator
Equal-area	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert Smyth equal-surface Trystan Edwards
Cassini Central Equirectangular Gall stereographic Gall isographic Miller Space-oblique Mercator Web Mercator

Pseudocylindrical

Equal-area	Collignon Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Mollweide Sinusoidal Tobler hyperelliptical
Kavrayskiy VII Wagner VI Winkel I and II

Conical

Pseudoconical

Azimuthal
(planar)

General perspective	Gnomonic Orthographic Stereographic
Equidistant Lambert equal-area

Bonne	Sinusoidal Werner
Bottomley	Sinusoidal Werner
Cylindrical	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert cylindrical equal-area Smyth equal-surface Trystan Edwards
Tobler hyperelliptical	Collignon Mollweide
Albers Briesemeister Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Hammer Lambert azimuthal equal-area Quadrilateralized spherical cube Strebe 1995