View source for Web Mercator projection

{{Short description|Mercator variant map projection}}
[[File:Web_maps_Mercator_projection_SW.jpg|thumb|300px|The Web Mercator projection is almost indistinguishable at global scale from a Mercator projection cropped to around 85°N to 85°S]]
[[File:OpenStreetMap homepage 2018 en.png|thumb|400px|Homepage of OpenStreetMap in 2018. The standard style for OpenStreetMap, like most Web maps, uses the Web Mercator projection]]
'''Web Mercator''', '''Google Web Mercator''', '''Spherical Mercator''', '''WGS 84 Web Mercator'''<ref>{{Cite web|date=2014-05-16|title=WGS 84 and the Web Mercator Projection NGA Office of Geomatics|url=http://earth-info.nga.mil/GandG/wgs84/web_mercator/%28U%29%20NGA_SIG_0011_1.0.0_WEBMERC.pdf|url-status=dead|archive-url=https://web.archive.org/web/20141009142830/http://earth-info.nga.mil/GandG/wgs84/web_mercator/(U)%20NGA_SIG_0011_1.0.0_WEBMERC.pdf|archive-date=2014-10-09|access-date=2014-08-06|publisher=[[National Geospatial-Intelligence Agency]]}}</ref> or '''WGS 84/Pseudo-Mercator''' is a variant of the [[Mercator projection|Mercator]] [[map projection]] and is the [[de facto standard]] for [[World Wide Web|Web]] mapping applications. It rose to prominence when [[Google Maps]] adopted it in 2005.<ref>{{cite web|url=https://productforums.google.com/d/msg/maps/A2ygEJ5eG-o/KbZr_B0h2hkJ|title=Google Maps & Earth Help Forum|access-date=29 August 2017|archive-url=https://web.archive.org/web/20200101203940/https://support.google.com/maps/forum/AAAAQuUrST8A2ygEJ5eG-o/?hl=en&msgid=KbZr_B0h2hkJ&gpf=d/msg/maps/A2ygEJ5eG-o/KbZr_B0h2hkJ|archive-date=January 1, 2020}}</ref> It is used by virtually all major online map providers, including Google Maps, [[CartoDB|CARTO]], [[Mapbox]],<ref name="MapBox">{{cite web
 | url         = https://www.mapbox.com/help/how-mapbox-data-works/
 | title       = Our Map Data
 | publisher   = MapBox
 | access-date = June 20, 2018
 | quote       = Mapbox supports the popular Web Mercator projection, and currently does not support any other projections for display.
 }}</ref> [[Bing Maps]], [[OpenStreetMap]], [[Mapquest]], [[Esri]], and many others.<ref name="Battersby">{{cite journal
  | journal    = Cartographica
  | title      = Implications of Web Mercator and Its Use in Online Mapping
  | last1      = Battersby
  | first1     = Sarah E.  
  | last2      = Finn
  | first2     = Michael P.
  | last3      = Usery
  | first3     = E. Lynn
  | last4      = Yamamoto
  | first4     = Kristina H.
  | year       = 2014
  | volume     = 49
  | issue      = 2
  | page       = 92
  | doi        = 10.3138/carto.49.2.2313
  | s2cid = 6403891
 | url        = https://utpjournals.press/doi/10.3138/carto.49.2.2313
 | archive-url = https://web.archive.org/web/20150215113220/http://cegis.usgs.gov/projection/pdf/Battersby_Implications%20of%20Web%20Mecator%20and%20Its%20Use%20in%20Online%20Mapping.pdf
| archive-date = 15 February 2015
}}
</ref> Its official [[EPSG identifier]] is EPSG:3857, although others have been used historically.

== Properties ==
Web Mercator is a slight variant of the Mercator projection, one used primarily in Web-based mapping programs. It uses the same formulas as the standard Mercator as used for [[Scale (map)#Large scale, medium scale, small scale|small-scale maps]]. However, the Web Mercator uses the spherical formulas at all scales whereas [[Scale (map)#Large scale, medium scale, small scale|large-scale]] Mercator maps normally use the ellipsoidal form of the projection.{{citation needed|date=June 2023}} The discrepancy is imperceptible at the global scale but causes maps of local areas to deviate slightly from true ellipsoidal Mercator maps at the same scale. 

While the Web Mercator's formulas are for the spherical form of the Mercator, geographical coordinates are required to be in the [[World Geodetic System#WGS 84|WGS 84]] ellipsoidal datum. This discrepancy causes the projection to be slightly non-[[Conformal map projection|conformal]]. General lack of understanding that the Web Mercator differs from standard Mercator usage has caused considerable confusion and misuse.<ref name="Battersby"/>{{rp|87}} Mistaking Web Mercator for the standard Mercator during coordinate conversion can lead to deviations as much as 40&nbsp;km on the ground.<ref name="NGAnotice" /><ref name="al">{{cite web |date=23 January 2011 |title=The Google Maps / Bing Maps Spherical Mercator Projection |url=http://alastaira.wordpress.com/2011/01/23/the-google-maps-bing-maps-spherical-mercator-projection/ |access-date=4 October 2014 |work=Alastair Aitchison}}</ref> For all these reasons, the [[United States Department of Defense]] through the [[National Geospatial-Intelligence Agency]] has declared this map projection to be unacceptable for any official use.<ref name=NGAnotice>{{cite web|title=NGA: (U) NGA Advisory Notice on "Web Mercator" (UNCLASSIFIED)|url=http://earth-info.nga.mil/GandG/wgs84/web_mercator/index.html|url-status=dead|archive-url= https://web.archive.org/web/20170329065451/https://earth-info.nga.mil/GandG/wgs84/web_mercator/index.html |archive-date=2017-03-29|access-date=4 October 2014|publisher=[[National Geospatial-Intelligence Agency]] Office of Geomatics|publication-date=2014-02-18}}</ref>

Unlike most map projections for the sphere, the Web Mercator uses the equatorial radius of the WGS 84 spheroid, rather than some [[Earth radius#Global radii|compromise between the equatorial and polar radii]]. This results in a slightly larger map compared to the map's [[Scale (map)|stated (nominal) scale]] than for most maps.

=== Formulas ===
Formulas for the Web Mercator are fundamentally the same as for the standard spherical Mercator, but before applying zoom, the "world coordinates" are adjusted such that the upper left corner is {{nowrap|(0, 0)}} and the lower right corner is {{nowrap|( <math>2^{\text{zoom level}}-1</math>, <math>2^{\text{zoom level}}-1</math>)}}:<ref>{{cite web|url=https://wiki.openstreetmap.org/wiki/Slippy_map_tilenames |title=OpenStreetMap Slippy map tilenames |publisher=openstreetmap.org |access-date=2018-11-14}}</ref>
<math display="block">\begin{align}
  x &= \left\lfloor\frac{1}{2\pi} \cdot 2^{\text{zoom level}} \left(\pi + \lambda \right)\right\rfloor \text{ pixels}\\[5pt]
  y &= \left\lfloor\frac{1}{2\pi} \cdot 2^{\text{zoom level}} \left(\pi - \ln \left[\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right) \right]\right)\right\rfloor \text{ pixels}
\end{align}</math>
where <math>\lambda</math> is the [[longitude]] in radians and <math>\varphi</math>  is [[geodetic latitude]] in radians.<ref>See [[Gudermannian function#Inverse|Inverse Gudermannian_function]]: <math>\operatorname{gd}^{-1} x  = \ln \left| \tan\left(\frac{x}{2} + \frac{\pi}{4}\right) \right|</math>.</ref>

Because the Mercator projects the poles at infinity, a map using the Web Mercator projection cannot show the poles. Services such as Google Maps cut off coverage at 85.051129° north and south. This is not a limitation for street maps, which is the primary purpose for such services. The value 85.051129° is the latitude at which the full projected map becomes a square, and is computed as <math>\varphi</math> given {{nowrap|1= ''y'' = 0}}:
<math display="block">\begin{align}
\varphi_{\text{max}} = \left[2\arctan(e^{\pi}) - \frac{\pi}{2}\right]
\end{align}</math>

=== Spherical and ellipsoidal mix ===
The projection is neither strictly ellipsoidal nor strictly spherical. EPSG's definition says the projection "uses spherical development of ellipsoidal coordinates".<ref>{{cite web |url=http://www.epsg-registry.org/report.htm?type=selection&entity=urn:ogc:def:crs:EPSG::3857&reportDetail=short&style=urn:uuid:report-style:default-with-code&style_name=OGP%20Default%20With%20Code&title=EPSG:3857 |title=Custom report |website=www.epsg-registry.org |url-status=dead |archive-url=https://web.archive.org/web/20120622004633/http://www.epsg-registry.org/report.htm?type=selection&entity=urn:ogc:def:crs:EPSG::3857&reportDetail=short&style=urn:uuid:report-style:default-with-code&style_name=OGP%20Default%20With%20Code&title=EPSG:3857 |archive-date=2012-06-22}}</ref> The underlying geographic coordinates are defined using the [[WGS84]] ellipsoidal model of the Earth's surface, but are projected as if defined on a sphere.<ref name="al"/> This practice is uncontroversial for small-scale maps (such as of the entire world), but has little precedent in large-scale maps (such as of a city or province).<ref name ="Strebe2018">{{cite journal
| last = Strebe
| first = Daniel "daan"
| journal = Cartography and Geographic Information Science
| title = A bevy of area-preserving transforms for map projection designers
| doi = 10.1080/15230406.2018.1452632
| year = 2019
| volume = 46
| issue = 3 | pages = 260–276
| bibcode = 2019CGISc..46..260S
| s2cid = 134864785
}}</ref>

=== Advantages and disadvantages ===
Web Mercator is a spherical Mercator projection, and so it has the same properties as a spherical Mercator: north is up everywhere, meridians are equally spaced vertical lines, angles are locally correct (assuming spherical coordinates), and areas inflate with distance from the equator such that the polar regions are grossly exaggerated. The ellipsoidal Mercator has these same properties, but models the earth as an ellipsoid.

Unlike the ellipsoidal Mercator, however, the Web Mercator is not quite conformal. This means that angles between lines on the surface will not be drawn to the same angles in the map, although they will not deviate enough to be noticeable by eye. Lines deviate because Web Mercator specifies that coordinates be given as surveyed on the WGS 84 ellipsoidal model. By projecting coordinates surveyed against the ellipsoid as if they were surveyed on a sphere, angular relationships change slightly. This is standard practice on the standard spherical Mercator projection, but unlike Web Mercator, the spherical Mercator is not normally used for maps of local areas, such as street maps, and so the accuracy of positions needed for plotting is typically less than the angular deviation caused by using spherical formulas. The benefit the Web Mercator gains is that the spherical form is much simpler to calculate than the ellipsoidal form, and so requires only a fraction of the computing resources.<ref>{{Cite web|title=The Earth is Not Round! Utah, NAD83 and WebMercator Projections|url=https://gis.utah.gov/nad83-and-webmercator-projections/|access-date=2020-06-20|website=Utah GIS Portal|language=en}}</ref>

== Identifiers ==
Due to slow adoption by the [[EPSG registry]], the Web Mercator is represented by several different names and [[SRID|spatial reference system identifiers]] (SRIDs), including EPSG:900913, EPSG:3785 and EPSG:3857, the latter being the official EPSG identifier since 2009.<ref name="epsg.org">{{Cite web|url=https://epsg.org/crs_3857/WGS-84-Pseudo-Mercator.html|title=WGS 84 / Pseudo-Mercator|author=EPSG registry|website=epsg.org}}</ref>

=== EPSG:900913 ===
The projected coordinate reference system originally lacked an official spatial reference identifier ([[Spatial reference system#Identifier|SRID]]), and the Geodesy subcommittee of the OGP's Geomatics committee (also known as EPSG) refused to provide it with one, declaring "We have reviewed the coordinate reference system used by Microsoft, Google, etc. and believe that it is technically flawed. We will not devalue the EPSG dataset by including such inappropriate geodesy and cartography."<ref>{{Cite web|url=http://www.sharpgis.net/post/SphericalWeb-Mercator-EPSG-code-3785|title=dotMorten &#124; Spherical/Web Mercator: EPSG code 3785|website=www.sharpgis.net}}</ref> The unofficial code "EPSG:900913" (GOOGLE [[Leet|transliterated to numbers]]) came to be used. It was originally defined by Christopher Schmidt in his Technical Ramblings blog<ref name="Schmidt">{{Cite web|url=https://crschmidt.net/blog/archives/243/google-projection-900913/|title=Google Projection: 900913|date=August 7, 2007}}</ref> and became codified in OpenLayers 2,<ref>{{Cite web|url=http://docs.openlayers.org/library/spherical_mercator.html|title=Spherical Mercator — OpenLayers|website=docs.openlayers.org|access-date=2018-11-12|archive-date=2021-04-18|archive-url=https://web.archive.org/web/20210418033655/http://docs.openlayers.org/library/spherical_mercator.html|url-status=dead}}</ref> which, technically, would make OpenLayers the SRID authority.

=== EPSG:3785 ===
In 2008, EPSG provided the official identifier EPSG:3785 with the official name "Popular Visualisation [[Spatial reference system|CRS]] / Mercator", but noted "It is not an official [[geodesy|geodetic]] system".<ref name="al"/> This definition used a spherical (rather than ellipsoidal) model of the Earth.

=== EPSG:3857 ===
Later that year, EPSG provided an updated identifier, EPSG:3857 with the official name "WGS 84 / Pseudo-Mercator".<ref name="al" /> The definition switched to using the [[WGS84]] ellipsoid (EPSG:4326), rather than the sphere.
 
Although the projection is closely associated with Google, Microsoft is listed as the "information source" in EPSG's standards.<ref name="epsg.org"/>

=== Other identifiers ===
Other identifiers that have been used include ESRI:102113, ESRI:102100, and OSGEO:41001.<ref name="epsg.io">{{Cite web|url=http://epsg.io/|title=WGS 84 / Pseudo-Mercator - Spherical Mercator, Google Maps, OpenStreetMap, Bing, ArcGIS, ESRI - EPSG:3857|author=Klokan Technologies GmbH |website=epsg.io}}</ref><ref name="Schmidt" />

ESRI:102113 corresponds to EPSG:3785 while ESRI:102100 corresponds to EPSG:3857.<ref>{{Cite web|url=https://community.esri.com/t5/arcgis-rest-api-questions/102100-vs-102113-is-mixed-up/m-p/426380#M2050|title=102100 vs. 102113 is mixed up?|date=October 3, 2012|website=GeoNet, The Esri Community}}</ref>

== WKT definition ==
The projection covers the Earth from −180° to 180° longitude, and 85.05° north and south. Using [[well-known text representation of coordinate reference systems]] (WKT), EPSG:3857 is defined as follows:<ref name="epsg.org"/>
<pre>
PROJCRS["WGS 84 / Pseudo-Mercator",
    BASEGEOGCRS["WGS 84",
        ENSEMBLE["World Geodetic System 1984 ensemble",
            MEMBER["World Geodetic System 1984 (Transit)", ID["EPSG",1166]],
            MEMBER["World Geodetic System 1984 (G730)",    ID["EPSG",1152]],
            MEMBER["World Geodetic System 1984 (G873)",    ID["EPSG",1153]],
            MEMBER["World Geodetic System 1984 (G1150)",   ID["EPSG",1154]],
            MEMBER["World Geodetic System 1984 (G1674)",   ID["EPSG",1155]],
            MEMBER["World Geodetic System 1984 (G1762)",   ID["EPSG",1156]],
            MEMBER["World Geodetic System 1984 (G2139)",   ID["EPSG",1309]],
            ELLIPSOID["WGS 84", 6378137, 298.257223563, LENGTHUNIT["metre", 1, ID["EPSG",9001]], ID["EPSG",7030]],
            ENSEMBLEACCURACY[2], ID["EPSG",6326]],
        ID["EPSG",4326]],
    CONVERSION["Popular Visualisation Pseudo-Mercator",
        METHOD["Popular Visualisation Pseudo Mercator", ID["EPSG",1024]],
        PARAMETER["Latitude of natural origin",  0, ANGLEUNIT["degree", 0.0174532925199433, ID["EPSG",9102]], ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin", 0, ANGLEUNIT["degree", 0.0174532925199433, ID["EPSG",9102]], ID["EPSG",8802]],
        PARAMETER["False easting",               0, LENGTHUNIT["metre", 1,                  ID["EPSG",9001]], ID["EPSG",8806]],
        PARAMETER["False northing",              0, LENGTHUNIT["metre", 1,                  ID["EPSG",9001]], ID["EPSG",8807]],
        ID["EPSG",3856]],
    CS[Cartesian, 2, ID["EPSG",4499]],
    AXIS["Easting (X)", east],
    AXIS["Northing (Y)", north],
    LENGTHUNIT["metre", 1, ID["EPSG",9001]],
    ID["EPSG",3857]]
</pre>

==References==
<references />

[[Category:Geographic coordinate systems]]
[[Category:Web mapping]]
[[Category:Google Maps]]
[[Category:Conformal projections]]