Equirectangular projection

<h2 id="definition">Definition</h2>
<p>The forward projection transforms spherical coordinates into planar coordinates. The reverse projection transforms from the plane back onto the sphere. The formulae presume a <a href="/facts/Figure_of_the_Earth/z1LvpVgd">spherical model</a> and use these definitions:
</p>
<ul><li>
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is the <a href="/facts/Longitude/HsC61ML9">longitude</a> of the location to project;</li>
<li>
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 is the <a href="/facts/Latitude/IBYqkjhU">latitude</a> of the location to project;</li>
<li>
  
    
      
        
          φ
          
            1
          
        
      
    
    {\displaystyle \varphi _{1}}
  
 are the standard parallels (north and south of the equator) where the scale of the projection is true;</li>
<li>
  
    
      
        
          φ
          
            0
          
        
      
    
    {\displaystyle \varphi _{0}}
  
 is the central parallel of the map;</li>
<li>
  
    
      
        
          λ
          
            0
          
        
      
    
    {\displaystyle \lambda _{0}}
  
 is the central meridian of the map;</li>
<li>
  
    
      
        x
      
    
    {\displaystyle x}
  
 is the horizontal coordinate of the projected location on the map;</li>
<li>
  
    
      
        y
      
    
    {\displaystyle y}
  
 is the vertical coordinate of the projected location on the map;</li>
<li>
  
    
      
        R
      
    
    {\displaystyle R}
  
 is the radius of the globe.</li></ul>
<p>Longitude and latitude variables are defined here in terms of radians.
</p>
<h3>Forward</h3>

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

<p>The <i>plate carrée</i> (<a href="/facts/French_language/qe1DnaQh">French</a>, for <i>flat square</i>),<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> is the special case where 
  
    
      
        
          φ
          
            1
          
        
      
    
    {\displaystyle \varphi _{1}}
  
 is zero. This projection maps <i>x</i> to be the value of the longitude and <i>y</i> to be the value of the latitude,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> and therefore is sometimes called the latitude/longitude or lat/lon(g) projection. Despite sometimes being called "unprojected",[<i>by whom?</i>] it is actually projected.
</p><p>When the 
  
    
      
        
          φ
          
            1
          
        
      
    
    {\displaystyle \varphi _{1}}
  
 is not zero, such as <a href="/facts/Marinus_of_Tyre/VFDbWhvv">Marinus</a>'s 
  
    
      
        
          φ
          
            1
          
        
        =
        36
      
    
    {\displaystyle \varphi _{1}=36}
  
,<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> the <a href="/facts/Gall_isographic_projection/VgB7V8Eg">Gall isographic projection</a>'s 
  
    
      
        
          φ
          
            1
          
        
        =
        45
      
    
    {\displaystyle \varphi _{1}=45}
  
, or Ronald Miller's 
  
    
      
        
          φ
          
            1
          
        
        =
        (
        37.5
        ,
        43.5
        ,
        50.5
        )
      
    
    {\displaystyle \varphi _{1}=(37.5,43.5,50.5)}
  
,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> the projection can portray particular latitudes of interest at true scale.
</p><p>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.
</p>
<h3>Reverse</h3>

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

<h3>Alternative names</h3>
<p>In spherical panorama viewers, usually:
</p>
<ul><li>
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is called "yaw";<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li>
<li>
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 is called "pitch";<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li></ul>
<p>where both are defined in degrees.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cartography/8YdDAAeX">Cartography</a></li>
<li><a href="/facts/Cassini_projection/Df8DFXsu">Cassini projection</a></li>
<li><a href="/facts/Gall%25E2%2580%2593Peters_projection/5LD6oSL3">Gall–Peters projection</a> (mentions a resolution rejecting the use of all rectangular world maps)</li>
<li><a href="/facts/List_of_map_projections/cGaBu5WV">List of map projections</a></li>
<li><a href="/facts/Mercator_projection/bYVL5bP9">Mercator projection</a></li>
<li><a href="/facts/360_video_projection/vhrtajcv">360 video projection</a></li>
<li><a href="https://commons.wikimedia.org/wiki/Category:Maps%20of%20the%20world%20with%20equirectangular%20projection">Wikimedia Gallery of Equirectangular World Maps</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://visibleearth.nasa.gov/view.php?id=57730">Global MODIS based satellite map</a> The blue marble: land surface, ocean color, and sea ice.</li>
<li><a href="http://www.radicalcartography.net/?projectionref">Table of examples and properties of all common projections</a>, from radicalcartography.net.</li>
<li><a href="http://wiki.panotools.org/Equirectangular">Panoramic Equirectangular Projection</a>, PanoTools wiki.</li>
<li><a href="https://proj4.org/operations/projections/eqc.html">Equidistant Cylindrical (Plate Carrée) in proj4</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 5–8, ISBN 0-226-76747-7. <a href="/wiki/John_P._Snyder" target="_blank">/wiki/John_P._Snyder</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Equirectangular Projection - PanoTools.org Wiki". wiki.panotools.org. Retrieved 2021-05-04. <a href="https://wiki.panotools.org/Equirectangular_Projection" target="_blank">https://wiki.panotools.org/Equirectangular_Projection</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Farkas, Gábor. "Plate Carrée - a simple example". O’Reilly Online Learning. Retrieved 31 December 2022. <a href="https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml" target="_blank">https://www.oreilly.com/library/view/practical-gis/9781787123328/Text/b21938a9-09f7-46fa-b905-58a0a4ed7d8f.xhtml</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="9780470870013" target="_blank">9780470870013</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 7, ISBN 0-226-76747-7. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>"Equidistant Cylindrical (Plate Carrée)". PROJ coordinate transformation software library. Retrieved 25 August 2020. <a href="https://proj.org/operations/projections/eqc.html" target="_blank">https://proj.org/operations/projections/eqc.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>"Yaw - PanoTools.org Wiki". wiki.panotools.org. Retrieved 2021-05-04. <a href="https://wiki.panotools.org/Yaw" target="_blank">https://wiki.panotools.org/Yaw</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>"Pitch - PanoTools.org Wiki". wiki.panotools.org. Retrieved 2021-05-04. <a href="https://wiki.panotools.org/Pitch" target="_blank">https://wiki.panotools.org/Pitch</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Equirectangular projection open-in-new

Equirectangular projection