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<li class="toctree-l1 current"><a class="reference internal" href="../index.html">Coordinate operations</a><ul class="current">
<li class="toctree-l2 current"><a class="reference internal" href="index.html">Projections</a><ul class="current">
<li class="toctree-l3"><a class="reference internal" href="adams_hemi.html">Adams Hemisphere in a Square</a></li>
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<li class="toctree-l3"><a class="reference internal" href="aea.html">Albers Equal Area</a></li>
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<li class="toctree-l3"><a class="reference internal" href="alsk.html">Modified Stereographic of Alaska</a></li>
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<li class="toctree-l3"><a class="reference internal" href="bipc.html">Bipolar conic of western hemisphere</a></li>
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<li class="toctree-l3"><a class="reference internal" href="cc.html">Central Cylindrical</a></li>
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<li class="toctree-l3"><a class="reference internal" href="cea.html">Equal Area Cylindrical</a></li>
<li class="toctree-l3"><a class="reference internal" href="chamb.html">Chamberlin Trimetric</a></li>
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<li class="toctree-l3"><a class="reference internal" href="crast.html">Craster Parabolic (Putnins P4)</a></li>
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<li class="toctree-l3"><a class="reference internal" href="eck1.html">Eckert I</a></li>
<li class="toctree-l3"><a class="reference internal" href="eck2.html">Eckert II</a></li>
<li class="toctree-l3"><a class="reference internal" href="eck3.html">Eckert III</a></li>
<li class="toctree-l3"><a class="reference internal" href="eck4.html">Eckert IV</a></li>
<li class="toctree-l3"><a class="reference internal" href="eck5.html">Eckert V</a></li>
<li class="toctree-l3"><a class="reference internal" href="eck6.html">Eckert VI</a></li>
<li class="toctree-l3"><a class="reference internal" href="eqc.html">Equidistant Cylindrical (Plate Carrée)</a></li>
<li class="toctree-l3"><a class="reference internal" href="eqdc.html">Equidistant Conic</a></li>
<li class="toctree-l3"><a class="reference internal" href="eqearth.html">Equal Earth</a></li>
<li class="toctree-l3"><a class="reference internal" href="euler.html">Euler</a></li>
<li class="toctree-l3"><a class="reference internal" href="fahey.html">Fahey</a></li>
<li class="toctree-l3"><a class="reference internal" href="fouc.html">Foucaut</a></li>
<li class="toctree-l3"><a class="reference internal" href="fouc_s.html">Foucaut Sinusoidal</a></li>
<li class="toctree-l3"><a class="reference internal" href="gall.html">Gall (Gall Stereographic)</a></li>
<li class="toctree-l3"><a class="reference internal" href="geos.html">Geostationary Satellite View</a></li>
<li class="toctree-l3"><a class="reference internal" href="gins8.html">Ginsburg VIII (TsNIIGAiK)</a></li>
<li class="toctree-l3"><a class="reference internal" href="gn_sinu.html">General Sinusoidal Series</a></li>
<li class="toctree-l3"><a class="reference internal" href="gnom.html">Gnomonic</a></li>
<li class="toctree-l3"><a class="reference internal" href="goode.html">Goode Homolosine</a></li>
<li class="toctree-l3"><a class="reference internal" href="gs48.html">Modified Stereographic of 48 U.S.</a></li>
<li class="toctree-l3"><a class="reference internal" href="gs50.html">Modified Stereographic of 50 U.S.</a></li>
<li class="toctree-l3"><a class="reference internal" href="guyou.html">Guyou</a></li>
<li class="toctree-l3"><a class="reference internal" href="hammer.html">Hammer & Eckert-Greifendorff</a></li>
<li class="toctree-l3"><a class="reference internal" href="hatano.html">Hatano Asymmetrical Equal Area</a></li>
<li class="toctree-l3"><a class="reference internal" href="healpix.html">HEALPix</a></li>
<li class="toctree-l3"><a class="reference internal" href="rhealpix.html">rHEALPix</a></li>
<li class="toctree-l3"><a class="reference internal" href="igh.html">Interrupted Goode Homolosine</a></li>
<li class="toctree-l3"><a class="reference internal" href="igh_o.html">Interrupted Goode Homolosine (Oceanic View)</a></li>
<li class="toctree-l3"><a class="reference internal" href="imw_p.html">International Map of the World Polyconic</a></li>
<li class="toctree-l3"><a class="reference internal" href="isea.html">Icosahedral Snyder Equal Area</a></li>
<li class="toctree-l3"><a class="reference internal" href="kav5.html">Kavrayskiy V</a></li>
<li class="toctree-l3"><a class="reference internal" href="kav7.html">Kavrayskiy VII</a></li>
<li class="toctree-l3"><a class="reference internal" href="krovak.html">Krovak</a></li>
<li class="toctree-l3"><a class="reference internal" href="labrd.html">Laborde</a></li>
<li class="toctree-l3"><a class="reference internal" href="laea.html">Lambert Azimuthal Equal Area</a></li>
<li class="toctree-l3"><a class="reference internal" href="lagrng.html">Lagrange</a></li>
<li class="toctree-l3"><a class="reference internal" href="larr.html">Larrivee</a></li>
<li class="toctree-l3"><a class="reference internal" href="lask.html">Laskowski</a></li>
<li class="toctree-l3"><a class="reference internal" href="lcc.html">Lambert Conformal Conic</a></li>
<li class="toctree-l3"><a class="reference internal" href="lcca.html">Lambert Conformal Conic Alternative</a></li>
<li class="toctree-l3"><a class="reference internal" href="leac.html">Lambert Equal Area Conic</a></li>
<li class="toctree-l3"><a class="reference internal" href="lee_os.html">Lee Oblated Stereographic</a></li>
<li class="toctree-l3"><a class="reference internal" href="loxim.html">Loximuthal</a></li>
<li class="toctree-l3"><a class="reference internal" href="lsat.html">Space oblique for LANDSAT</a></li>
<li class="toctree-l3"><a class="reference internal" href="mbt_s.html">McBryde-Thomas Flat-Polar Sine (No. 1)</a></li>
<li class="toctree-l3"><a class="reference internal" href="mbt_fps.html">McBryde-Thomas Flat-Pole Sine (No. 2)</a></li>
<li class="toctree-l3"><a class="reference internal" href="mbtfpp.html">McBride-Thomas Flat-Polar Parabolic</a></li>
<li class="toctree-l3"><a class="reference internal" href="mbtfpq.html">McBryde-Thomas Flat-Polar Quartic</a></li>
<li class="toctree-l3"><a class="reference internal" href="mbtfps.html">McBryde-Thomas Flat-Polar Sinusoidal</a></li>
<li class="toctree-l3"><a class="reference internal" href="merc.html">Mercator</a></li>
<li class="toctree-l3"><a class="reference internal" href="mil_os.html">Miller Oblated Stereographic</a></li>
<li class="toctree-l3"><a class="reference internal" href="mill.html">Miller Cylindrical</a></li>
<li class="toctree-l3"><a class="reference internal" href="misrsom.html">Space oblique for MISR</a></li>
<li class="toctree-l3"><a class="reference internal" href="moll.html">Mollweide</a></li>
<li class="toctree-l3"><a class="reference internal" href="murd1.html">Murdoch I</a></li>
<li class="toctree-l3"><a class="reference internal" href="murd2.html">Murdoch II</a></li>
<li class="toctree-l3"><a class="reference internal" href="murd3.html">Murdoch III</a></li>
<li class="toctree-l3"><a class="reference internal" href="natearth.html">Natural Earth</a></li>
<li class="toctree-l3"><a class="reference internal" href="natearth2.html">Natural Earth II</a></li>
<li class="toctree-l3"><a class="reference internal" href="nell.html">Nell</a></li>
<li class="toctree-l3"><a class="reference internal" href="nell_h.html">Nell-Hammer</a></li>
<li class="toctree-l3"><a class="reference internal" href="nicol.html">Nicolosi Globular</a></li>
<li class="toctree-l3"><a class="reference internal" href="nsper.html">Near-sided perspective</a></li>
<li class="toctree-l3"><a class="reference internal" href="nzmg.html">New Zealand Map Grid</a></li>
<li class="toctree-l3"><a class="reference internal" href="ob_tran.html">General Oblique Transformation</a></li>
<li class="toctree-l3"><a class="reference internal" href="ocea.html">Oblique Cylindrical Equal Area</a></li>
<li class="toctree-l3"><a class="reference internal" href="oea.html">Oblated Equal Area</a></li>
<li class="toctree-l3"><a class="reference internal" href="omerc.html">Oblique Mercator</a></li>
<li class="toctree-l3"><a class="reference internal" href="ortel.html">Ortelius Oval</a></li>
<li class="toctree-l3"><a class="reference internal" href="ortho.html">Orthographic</a></li>
<li class="toctree-l3"><a class="reference internal" href="patterson.html">Patterson</a></li>
<li class="toctree-l3"><a class="reference internal" href="pconic.html">Perspective Conic</a></li>
<li class="toctree-l3"><a class="reference internal" href="peirce_q.html">Peirce Quincuncial</a></li>
<li class="toctree-l3"><a class="reference internal" href="poly.html">Polyconic (American)</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp1.html">Putnins P1</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp2.html">Putnins P2</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp3.html">Putnins P3</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp3p.html">Putnins P3’</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp4p.html">Putnins P4’</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp5.html">Putnins P5</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp5p.html">Putnins P5’</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp6.html">Putnins P6</a></li>
<li class="toctree-l3"><a class="reference internal" href="putp6p.html">Putnins P6’</a></li>
<li class="toctree-l3"><a class="reference internal" href="qua_aut.html">Quartic Authalic</a></li>
<li class="toctree-l3"><a class="reference internal" href="qsc.html">Quadrilateralized Spherical Cube</a></li>
<li class="toctree-l3"><a class="reference internal" href="robin.html">Robinson</a></li>
<li class="toctree-l3"><a class="reference internal" href="rouss.html">Roussilhe Stereographic</a></li>
<li class="toctree-l3"><a class="reference internal" href="rpoly.html">Rectangular Polyconic</a></li>
<li class="toctree-l3"><a class="reference internal" href="s2.html">S2</a></li>
<li class="toctree-l3"><a class="reference internal" href="sch.html">Spherical Cross-track Height</a></li>
<li class="toctree-l3"><a class="reference internal" href="sinu.html">Sinusoidal (Sanson-Flamsteed)</a></li>
<li class="toctree-l3"><a class="reference internal" href="somerc.html">Swiss Oblique Mercator</a></li>
<li class="toctree-l3"><a class="reference internal" href="stere.html">Stereographic</a></li>
<li class="toctree-l3"><a class="reference internal" href="sterea.html">Oblique Stereographic Alternative</a></li>
<li class="toctree-l3"><a class="reference internal" href="gstmerc.html">Gauss-Schreiber Transverse Mercator (aka Gauss-Laborde Reunion)</a></li>
<li class="toctree-l3"><a class="reference internal" href="tcc.html">Transverse Central Cylindrical</a></li>
<li class="toctree-l3"><a class="reference internal" href="tcea.html">Transverse Cylindrical Equal Area</a></li>
<li class="toctree-l3"><a class="reference internal" href="times.html">Times</a></li>
<li class="toctree-l3"><a class="reference internal" href="tissot.html">Tissot</a></li>
<li class="toctree-l3"><a class="reference internal" href="tmerc.html">Transverse Mercator</a></li>
<li class="toctree-l3"><a class="reference internal" href="tobmerc.html">Tobler-Mercator</a></li>
<li class="toctree-l3"><a class="reference internal" href="tpeqd.html">Two Point Equidistant</a></li>
<li class="toctree-l3"><a class="reference internal" href="tpers.html">Tilted perspective</a></li>
<li class="toctree-l3"><a class="reference internal" href="ups.html">Universal Polar Stereographic</a></li>
<li class="toctree-l3"><a class="reference internal" href="urm5.html">Urmaev V</a></li>
<li class="toctree-l3"><a class="reference internal" href="urmfps.html">Urmaev Flat-Polar Sinusoidal</a></li>
<li class="toctree-l3"><a class="reference internal" href="utm.html">Universal Transverse Mercator (UTM)</a></li>
<li class="toctree-l3"><a class="reference internal" href="vandg.html">van der Grinten (I)</a></li>
<li class="toctree-l3"><a class="reference internal" href="vandg2.html">van der Grinten II</a></li>
<li class="toctree-l3"><a class="reference internal" href="vandg3.html">van der Grinten III</a></li>
<li class="toctree-l3"><a class="reference internal" href="vandg4.html">van der Grinten IV</a></li>
<li class="toctree-l3"><a class="reference internal" href="vitk1.html">Vitkovsky I</a></li>
<li class="toctree-l3"><a class="reference internal" href="wag1.html">Wagner I (Kavrayskiy VI)</a></li>
<li class="toctree-l3"><a class="reference internal" href="wag2.html">Wagner II</a></li>
<li class="toctree-l3"><a class="reference internal" href="wag3.html">Wagner III</a></li>
<li class="toctree-l3"><a class="reference internal" href="wag4.html">Wagner IV</a></li>
<li class="toctree-l3"><a class="reference internal" href="wag5.html">Wagner V</a></li>
<li class="toctree-l3"><a class="reference internal" href="wag6.html">Wagner VI</a></li>
<li class="toctree-l3"><a class="reference internal" href="wag7.html">Wagner VII</a></li>
<li class="toctree-l3 current"><a class="current reference internal" href="#">Web Mercator / Pseudo Mercator</a><ul>
<li class="toctree-l4"><a class="reference internal" href="#usage">Usage</a></li>
<li class="toctree-l4"><a class="reference internal" href="#parameters">Parameters</a></li>
<li class="toctree-l4"><a class="reference internal" href="#mathematical-definition">Mathematical definition</a></li>
<li class="toctree-l4"><a class="reference internal" href="#further-reading">Further reading</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="weren.html">Werenskiold I</a></li>
<li class="toctree-l3"><a class="reference internal" href="wink1.html">Winkel I</a></li>
<li class="toctree-l3"><a class="reference internal" href="wink2.html">Winkel II</a></li>
<li class="toctree-l3"><a class="reference internal" href="wintri.html">Winkel Tripel</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../conversions/index.html">Conversions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../transformations/index.html">Transformations</a></li>
<li class="toctree-l2"><a class="reference internal" href="../pipeline.html">The pipeline operator</a></li>
<li class="toctree-l2"><a class="reference internal" href="../operations_computation.html">Computation of coordinate operations between two CRS</a></li>
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<section id="web-mercator-pseudo-mercator">
<span id="webmerc"></span><h1>Web Mercator / Pseudo Mercator<a class="headerlink" href="#web-mercator-pseudo-mercator" title="Permalink to this headline">¶</a></h1>
<div class="versionadded">
<p><span class="versionmodified added">New in version 5.1.0.</span></p>
</div>
<p>The Web Mercator / Pseudo Mercator projection is a cylindrical map projection.
This is a variant of the regular <a class="reference internal" href="merc.html#merc"><span class="std std-ref">Mercator</span></a> projection, except that the computation
is done on a sphere, using the semi-major axis of the ellipsoid.</p>
<p>From <a class="reference external" href="https://en.wikipedia.org/wiki/Web_Mercator">Wikipedia</a>:</p>
<blockquote>
<div><p>This projection is widely used by the Web Mercator, Google Web Mercator,
Spherical Mercator, WGS 84 Web Mercator[1] or WGS 84/Pseudo-Mercator is a
variant of the Mercator projection and is the de facto standard for Web
mapping applications. […]
It is used by virtually all major online map providers […]
Its official EPSG identifier is EPSG:3857, although others have been used
historically.</p>
</div></blockquote>
<table class="docutils align-default">
<colgroup>
<col style="width: 27%" />
<col style="width: 73%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><strong>Classification</strong></p></td>
<td><p>Cylindrical (non conformant if used with ellipsoid)</p></td>
</tr>
<tr class="row-even"><td><p><strong>Available forms</strong></p></td>
<td><p>Forward and inverse</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Defined area</strong></p></td>
<td><p>Global</p></td>
</tr>
<tr class="row-even"><td><p><strong>Alias</strong></p></td>
<td><p>webmerc</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Domain</strong></p></td>
<td><p>2D</p></td>
</tr>
<tr class="row-even"><td><p><strong>Input type</strong></p></td>
<td><p>Geodetic coordinates</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Output type</strong></p></td>
<td><p>Projected coordinates</p></td>
</tr>
</tbody>
</table>
<section id="usage">
<h2>Usage<a class="headerlink" href="#usage" title="Permalink to this headline">¶</a></h2>
<p>Example:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>$ echo 2 49 | proj +proj=webmerc +datum=WGS84
222638.98 6274861.39
</pre></div>
</div>
</section>
<section id="parameters">
<h2>Parameters<a class="headerlink" href="#parameters" title="Permalink to this headline">¶</a></h2>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>All parameters for the projection are optional, except the ellipsoid
definition, which is WGS84 for the typical use case of EPSG:3857.
In which case, the other parameters are set to their default 0 value.</p>
</div>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-ellps">
<span class="sig-name descname"><span class="pre">+ellps</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-ellps" title="Permalink to this definition">¶</a></dt>
<dd><p>The name of a built-in ellipsoid definition.</p>
<p>See <a class="reference internal" href="../../usage/ellipsoids.html#ellipsoids"><span class="std std-ref">Ellipsoids</span></a> for more information, or execute
<a class="reference internal" href="../../apps/proj.html#cmdoption-proj-le"><code class="xref std std-option docutils literal notranslate"><span class="pre">proj</span> <span class="pre">-le</span></code></a> for a list of built-in ellipsoid names.</p>
<p><em>Defaults to “GRS80”.</em></p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-lon_0">
<span id="cmdoption-arg-lon-0"></span><span class="sig-name descname"><span class="pre">+lon_0</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-lon_0" title="Permalink to this definition">¶</a></dt>
<dd><p>Longitude of projection center.</p>
<p><em>Defaults to 0.0.</em></p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-x_0">
<span id="cmdoption-arg-x-0"></span><span class="sig-name descname"><span class="pre">+x_0</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-x_0" title="Permalink to this definition">¶</a></dt>
<dd><p>False easting.</p>
<p><em>Defaults to 0.0.</em></p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-y_0">
<span id="cmdoption-arg-y-0"></span><span class="sig-name descname"><span class="pre">+y_0</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-y_0" title="Permalink to this definition">¶</a></dt>
<dd><p>False northing.</p>
<p><em>Defaults to 0.0.</em></p>
</dd></dl>
</section>
<section id="mathematical-definition">
<h2>Mathematical definition<a class="headerlink" href="#mathematical-definition" title="Permalink to this headline">¶</a></h2>
<p>The formulas describing the Mercator projection are all taken from G. Evenden’s libproj manuals <span id="id1">[<a class="reference internal" href="../../zreferences.html#id11" title="Evenden, G. I. libproj4: A Comprehensive Library of Cartographic Projection Functions (Preliminary Draft). 2005. URL: https://github.com/OSGeo/PROJ/blob/master/docs/old/libproj.pdf.">Evenden2005</a>]</span>.</p>
<section id="forward-projection">
<h3>Forward projection<a class="headerlink" href="#forward-projection" title="Permalink to this headline">¶</a></h3>
<div class="math notranslate nohighlight">
\[x = \lambda\]</div>
<div class="math notranslate nohighlight">
\[y = \ln \left[ \tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right]\]</div>
</section>
<section id="inverse-projection">
<h3>Inverse projection<a class="headerlink" href="#inverse-projection" title="Permalink to this headline">¶</a></h3>
<div class="math notranslate nohighlight">
\[\lambda = {x}\]</div>
<div class="math notranslate nohighlight">
\[\phi = \frac{\pi}{2} - 2 \arctan \left[ e^{-y} \right]\]</div>
</section>
</section>
<section id="further-reading">
<h2>Further reading<a class="headerlink" href="#further-reading" title="Permalink to this headline">¶</a></h2>
<ol class="arabic simple">
<li><p><a class="reference external" href="https://en.wikipedia.org/wiki/Web_Mercator">Wikipedia</a></p></li>
</ol>
</section>
</section>
</div>
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