Equirectangular projection: Difference between revisions

Revision as of 19:08, 20 March 2022

Revision as of 23:34, 4 April 2022

Line 3:

[[File:Plate Carrée with Tissot's Indicatrices of Distortion.svg|thumb|upright=1.75|Equirectangular projection with [[Tissot's indicatrix]] of deformation]]

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

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 [[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 map|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 datasets]], such as [[Celestia]], [[NASA World Wind]], 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. In addition it is frequently used in panoramic photography to represent a spherical panoramic image.<ref>{{Citecite 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;

Line 15:

*<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''), 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, and therefore is sometimes called the latitude/longitude or lat/lon(g) projection or is said to be “unprojected”. Despite sometimes being called “unprojected”, it is actually projected.

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> or [[Royal_Scottish_Geographical_SocietyRoyal Scottish Geographical Society|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 |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.

Line 31:

===Reverse===

<math>\begin{align}

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

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

\end{align}</math>

=== Alternative names ===

In spherical panorama viewers, usually:

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

*<math>\varphi</math> is called "pitch";<ref>{{Citecite 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 ==

* [[List of map projections]]

*[[Cartography]]

*[[Cassini projection]]

*[[Gall–Peters projection]] with resolution regarding the use of rectangular world maps

* [[List of map projections]]

*[[Mercator projection]]

*[[Spherical360 imagevideo projection]]

==References==

Line 56 ⟶ 55:

==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]]