Equirectangular projection: Difference between revisions

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[[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''' 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 projection maps [[meridian (geography)|meridians]] to vertical straight lines of constant spacing (for 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 map|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 datasets]], such as [[Celestia]], [[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>

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''' 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 projection maps [[meridian (geography)|meridians]] to vertical straight lines of constant spacing (for 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 mapprojection|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 datasets]], such as [[Celestia]], [[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==

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===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.co.ukcom/books?id=-FbVI-2tSuYC&pg=PA119 |ppage=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. Despite sometimes being called "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 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 [[RoyalGall Scottishisographic Geographicalprojection]]'s <math>\varphi_1=45</math>, or Society|Ronald Miller]]'s <math>\varphi_1=(37.5, 43.5, 50.5)</math>,<ref>{{cite web |title=Equidistant Cylindrical (Plate Carrée) |url=https://proj.org/operations/projections/eqc.html |website=PROJ coordinate transformation software library |access-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.

===Reverse===

:<math>\begin{align}

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

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

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==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]]

*[[360 video 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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{{Map projections}}

[[Category:Map projections]]

[[Category:Equidistant projections]]

[[Category:MapCylindrical projections]]