Equirectangular projection: Difference between revisions

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{{Short description|Cylindrical equidistant map projection}}

[[File:Equirectangular projection SW.jpg|thumb|upright=1.75|Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).]]

[[File:Plate Carrée with Tissot's Indicatrices of Distortion.svg|thumb|upright=1.75|Equirectangular projection with [[Tissot's indicatrix]] of deformation and with the standard parallels lying on the equator]]

[[File:Blue Marble 2002.png|thumb|upright=1.75|True-colour satellite image of Earth in equirectangular projection]]

[[File:World elevation map.png|thumb|upright=1.75|[[Height map]] of planet Earth at 2km per pixel, including oceanic [[bathymetry]] 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.]]

The '''equirectangular projection''' (also called the '''equidistant cylindrical projection''', '''geographic projection''', or '''la carte parallélogrammatique projection''', and which includes the special case of the '''plate carrée projection''' or '''geographic projection''') is a simple [[map projection]] attributed to [[Marinus of Tyre]], who [[Ptolemy]] claims invented the projection about AD 100.<ref>''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&nbsp;5–8, {{ISBN|0-226-76747-7}}.</ref> The projection maps [[Meridian (geography)|meridians]] to vertical straight lines of constant spacing (for [[Meridian (geography)|meridional]] intervals of constant spacing), and [[circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[Circle of latitude|parallels]]). The projection is neither equal area nor [[Conformal map projection|conformal]]. Because of the distortions introduced by this projection, it has little use in [[navigation]] or [[cadastral]] mapping and finds its main use in [[thematic map]]ping. In particular, the plate carrée has become a standard for global [[Geographic information system#Raster|raster datasets]], such as [[Celestia]] and [[NASA World Wind]], because of the particularly simple relationship between the position of an [[pixel|image pixel]] on the map and its corresponding geographic ___location on Earth.

The '''equirectangular projection''' (also called the '''equidistant cylindrical projection''' or '''la carte parallélogrammatique projection'''), and which includes the special case of the '''plate carrée projection''' (also called the '''geographic projection''', '''lat/lon projection''', or '''plane chart'''), is a simple [[map projection]] attributed to [[Marinus of Tyre]] who, [[Ptolemy]] claims, invented the projection about AD 100.<ref>''Flattening the Earth: Two Thousand Years of Map Projections'', [[John P. Snyder]], 1993, pp.&nbsp;5–8, {{ISBN|0-226-76747-7}}.</ref>

The '''equirectangular projection''' (also called the '''equidistant cylindrical projection''', '''geographic projection''', or '''la carte parallélogrammatique projection''', and which includes the special case of the '''plate carrée projection''' or '''geographic projection''') is a simple [[map projection]] attributed to [[Marinus of Tyre]], who [[Ptolemy]] claims invented the projection about AD 100.<ref>''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&nbsp;5–8, {{ISBN|0-226-76747-7}}.</ref> The projection maps [[Meridianmeridian (geography)|meridians]] to vertical straight lines of constant spacing (for [[Meridian (geography)|meridional]] intervals of constant spacing), and [[circle of latitude|circles of latitude]] to horizontal straight lines of constant spacing (for constant intervals of [[Circlecircle of latitude|parallels]]). The projection is neither [[equal-area projection|equal area]] nor [[Conformalconformal map projection|conformal]]. Because of the distortions introduced by this projection, it has little use in [[navigation]] or [[cadastral]] mapping and finds its main use in [[thematic map]]ping. In particular, the plate carrée has become a standard for global [[Geographicgeographic information system#Raster|raster datasets]], such as [[Celestia]] and, [[NASA World Wind]], the [[USGS]] [[Astrogeology Research Program]], and [[Natural Earth]], because of the particularly simple relationship between the position of an [[pixel|image 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.<ref>{{cite web |title=Equirectangular Projection - PanoTools.org Wiki |url=https://wiki.panotools.org/Equirectangular_Projection |access-date=2021-05-04 |website=wiki.panotools.org}}</ref>

==Definition==

The forward projection transforms spherical coordinates into planar coordinates. The reverse projection transforms from the plane back onto the sphere. The formulae presume a [[Figurefigure of the Earth|spherical model]] and use these definitions:

:*<math>\lambda</math> is the [[longitude]] of the ___location to project;

:*<math>\varphi</math> is the [[latitude]] of the ___location to project;

:*<math>\varphi_1</math> are the standard parallels (north and south of the equator) where the scale of the projection is true;

:*<math>\varphi_0</math> is the central parallel of the map;

:*<math>\lambda_0</math> is the central meridian of the map;

:*<math>x</math> is the horizontal coordinate of the projected ___location on the map;

:*<math>y</math> is the vertical coordinate of the projected ___location on the map;

:*<math>R</math> is the radius of the globe.

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

===Forward===

:<math>\begin{align}

x &= R (\lambda - \lambda_0) \cos \varphi_1\\

y &= R (\varphi - \varphi_0)

\end{align}</math>

The {{lang|fr|plate carrée}} ([[French language|French]], for ''flat square''),<ref>{{Cite web |title=Plate Carrée - a simple example |last=Farkas |first=Gábor |work=O’Reilly Online Learning |date= |access-date=31 December 2022 |url= https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml}}</ref> is the special case where <math>\varphi_1</math> is zero. This projection maps ''x'' to be the value of the longitude and ''y'' to be the value of the latitude,<ref>{{cite book |url=https://books.google.com/books?id=-FbVI-2tSuYC&pg=PA119 |page=119 |title=Geographic Information Systems and Science |author1=Paul A. Longley |author2=Michael F. Goodchild |author3=David J. Maguire |author4=David W. Rhind |publisher=John Wiley & Sons |year=2005|isbn=9780470870013 }}</ref> and therefore is sometimes called the latitude/longitude or lat/lon(g) projection or is said to be “unprojected”. Despite sometimes being called “unprojected”"unprojected",{{by whom|date=December 2022}} it is actually projected.{{cn|date=December 2022}}

When the <math>\varphi_1</math> is not zero, such as [[Marinus_of_TyreMarinus of Tyre|Marinus]]'s <math>\varphi_1=36</math> ,<ref>''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&nbsp;7, {{ISBN|0-226-76747-7}}.</ref>, orthe [[Royal_Scottish_Geographical_Society|RonaldGall Millerisographic projection]]'s <math>\varphi_1=45</math>, or Ronald Miller's <math>\varphi_1=(37.5, 43.5, 50.5)</math> ,<ref>{{cite web |last1=PROJ Contributors |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |accessdateaccess-date=25 August 2020}}</ref>, 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.

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===Reverse===

:<math>\begin{align}

\lambda &= \frac{x} {R \cos \varphi_1} + \lambda_0\\

\varphi &= \frac{y} {R} + \varphi_0

\end{align}</math>

==See=Alternative alsonames===

In spherical panorama viewers, usually:

*<math>\lambda</math> is called "yaw";<ref>{{cite web |title=Yaw - PanoTools.org Wiki |url=https://wiki.panotools.org/Yaw |access-date=2021-05-04 |website=wiki.panotools.org}}</ref>

* [[List of map projections]]

*<math>\varphi</math> is called "pitch";<ref>{{cite web |title=Pitch - PanoTools.org Wiki |url=https://wiki.panotools.org/Pitch |access-date=2021-05-04 |website=wiki.panotools.org}}</ref>

where both are defined in degrees.

==See also==

*[[Cartography]]

*[[Cassini projection]]{{snd}}a transverse aspect of the equirectangular projection

*[[Gall–Peters projection]] with(mentions a resolution regardingrejecting the use of all rectangular world maps)

* [[List of map projections]]

*[[Mercator projection]]

*[[Spherical360 imagevideo projection]]

*[https://commons.wikimedia.org/wiki/Category:Maps%20of%20the%20world%20with%20equirectangular%20projection Wikimedia Gallery of Equirectangular World Maps]

==References==

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==External links==

* [https://visibleearth.nasa.gov/view.php?id=57730 Global MODIS based satellite map] The blue marble: land surface, ocean color, and sea ice.

* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net.

* [http://wiki.panotools.org/Equirectangular Panoramic Equirectangular Projection], PanoTools wiki.

* [https://proj4.org/operations/projections/eqc.html Equidistant Cylindrical (Plate Carrée) in proj4]

{{Map Projectionsprojections}}

[[Category:Map projections]]

[[Category:Equidistant projections]]

[[Category:MapCylindrical projections]]