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<!DOCTYPE html>
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<li class="toctree-l1"><a class="reference internal" href="../../about.html">About</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../../download.html">Download</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../install.html">Installation</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../usage/index.html">Using PROJ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../apps/index.html">Applications</a></li>
<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>
<li class="toctree-l3"><a class="reference internal" href="adams_ws1.html">Adams World in a Square I</a></li>
<li class="toctree-l3"><a class="reference internal" href="adams_ws2.html">Adams World in a Square II</a></li>
<li class="toctree-l3"><a class="reference internal" href="aea.html">Albers Equal Area</a></li>
<li class="toctree-l3"><a class="reference internal" href="aeqd.html">Azimuthal Equidistant</a></li>
<li class="toctree-l3"><a class="reference internal" href="airy.html">Airy</a></li>
<li class="toctree-l3"><a class="reference internal" href="aitoff.html">Aitoff</a></li>
<li class="toctree-l3"><a class="reference internal" href="alsk.html">Modified Stereographic of Alaska</a></li>
<li class="toctree-l3"><a class="reference internal" href="apian.html">Apian Globular I</a></li>
<li class="toctree-l3"><a class="reference internal" href="august.html">August Epicycloidal</a></li>
<li class="toctree-l3"><a class="reference internal" href="bacon.html">Bacon Globular</a></li>
<li class="toctree-l3"><a class="reference internal" href="bertin1953.html">Bertin 1953</a></li>
<li class="toctree-l3"><a class="reference internal" href="bipc.html">Bipolar conic of western hemisphere</a></li>
<li class="toctree-l3"><a class="reference internal" href="boggs.html">Boggs Eumorphic</a></li>
<li class="toctree-l3"><a class="reference internal" href="bonne.html">Bonne (Werner lat_1=90)</a></li>
<li class="toctree-l3"><a class="reference internal" href="calcofi.html">Cal Coop Ocean Fish Invest Lines/Stations</a></li>
<li class="toctree-l3"><a class="reference internal" href="cass.html">Cassini (Cassini-Soldner)</a></li>
<li class="toctree-l3"><a class="reference internal" href="cc.html">Central Cylindrical</a></li>
<li class="toctree-l3"><a class="reference internal" href="ccon.html">Central Conic</a></li>
<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>
<li class="toctree-l3"><a class="reference internal" href="collg.html">Collignon</a></li>
<li class="toctree-l3"><a class="reference internal" href="col_urban.html">Colombia Urban</a></li>
<li class="toctree-l3"><a class="reference internal" href="comill.html">Compact Miller</a></li>
<li class="toctree-l3"><a class="reference internal" href="crast.html">Craster Parabolic (Putnins P4)</a></li>
<li class="toctree-l3"><a class="reference internal" href="denoy.html">Denoyer Semi-Elliptical</a></li>
<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 current"><a class="current reference internal" href="#">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="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"><a class="reference internal" href="webmerc.html">Web Mercator / Pseudo Mercator</a></li>
<li class="toctree-l3"><a class="reference internal" href="weren.html">Werenskiold I</a></li>
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<section id="mercator">
<span id="merc"></span><h1>Mercator<a class="headerlink" href="#mercator" title="Permalink to this headline">¶</a></h1>
<p>The Mercator projection is a cylindrical map projection that origins
from the 16th century. It is widely recognized as the first regularly
used map projection. It is a conformal projection in which the equator
projects to a straight line at constant scale. The projection has the
property that a rhumb line, a course of constant heading, projects to a
straight line. This makes it suitable for navigational purposes.</p>
<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>Conformal cylindrical</p></td>
</tr>
<tr class="row-even"><td><p><strong>Available forms</strong></p></td>
<td><p>Forward and inverse, spherical and ellipsoidal</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Defined area</strong></p></td>
<td><p>Global, but best used near the equator</p></td>
</tr>
<tr class="row-even"><td><p><strong>Alias</strong></p></td>
<td><p>merc</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>
<figure class="align-center" id="id5">
<a class="reference internal image-reference" href="../../_images/merc.png"><img alt="Mercator" src="../../_images/merc.png" style="width: 500px;" /></a>
<figcaption>
<p><span class="caption-text">proj-string: <code class="docutils literal notranslate"><span class="pre">+proj=merc</span></code></span><a class="headerlink" href="#id5" title="Permalink to this image">¶</a></p>
</figcaption>
</figure>
<section id="usage">
<h2>Usage<a class="headerlink" href="#usage" title="Permalink to this headline">¶</a></h2>
<p>Applications should be limited to equatorial regions, but is frequently
used for navigational charts with latitude of true scale (<a class="reference internal" href="wink2.html#cmdoption-arg-lat_ts"><code class="xref std std-option docutils literal notranslate"><span class="pre">+lat_ts</span></code></a>) specified within
or near chart’s boundaries.
It is considered to be inappropriate for world maps because of the gross
distortions in area; for example the projected area of Greenland is
larger than that of South America, despite the fact that Greenland’s
area is <span class="math notranslate nohighlight">\(\frac18\)</span> that of South America <span id="id1">[<a class="reference internal" href="../../zreferences.html#id37" title="Snyder, J. P. Map projections — A working manual. Professional Paper 1395, U.S. Geological Survey, 1987. doi:10.3133/pp1395.">Snyder1987</a>]</span>.</p>
<p>Example using latitude of true scale:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>$ echo 56.35 12.32 | proj +proj=merc +lat_ts=56.5
3470306.37 759599.90
</pre></div>
</div>
<p>Example using scaling factor:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>echo 56.35 12.32 | proj +proj=merc +k_0=2
12545706.61 2746073.80
</pre></div>
</div>
<p>Note that <a class="reference internal" href="wink2.html#cmdoption-arg-lat_ts"><code class="xref std std-option docutils literal notranslate"><span class="pre">+lat_ts</span></code></a> and <a class="reference internal" href="tobmerc.html#cmdoption-arg-k_0"><code class="xref std std-option docutils literal notranslate"><span class="pre">+k_0</span></code></a> are mutually exclusive.
If used together, <a class="reference internal" href="wink2.html#cmdoption-arg-lat_ts"><code class="xref std std-option docutils literal notranslate"><span class="pre">+lat_ts</span></code></a> takes precedence over <a class="reference internal" href="tobmerc.html#cmdoption-arg-k_0"><code class="xref std std-option docutils literal notranslate"><span class="pre">+k_0</span></code></a>.</p>
</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.</p>
</div>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-lat_ts">
<span id="cmdoption-arg-lat-ts"></span><span class="sig-name descname"><span class="pre">+lat_ts</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-lat_ts" title="Permalink to this definition">¶</a></dt>
<dd><p>Latitude of true scale. Defines the latitude where scale is not distorted.
Takes precedence over <code class="docutils literal notranslate"><span class="pre">+k_0</span></code> if both options are used together.</p>
<p><em>Defaults to 0.0.</em></p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-k_0">
<span id="cmdoption-arg-k-0"></span><span class="sig-name descname"><span class="pre">+k_0</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-k_0" title="Permalink to this definition">¶</a></dt>
<dd><p>Scale factor. Determines scale factor used in the projection.</p>
<p><em>Defaults to 1.0.</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>
<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-R">
<span id="cmdoption-arg-r"></span><span class="sig-name descname"><span class="pre">+R</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-R" title="Permalink to this definition">¶</a></dt>
<dd><p>Radius of the sphere, given in meters. If used in conjunction with
<code class="docutils literal notranslate"><span class="pre">+ellps</span></code>, <a class="reference internal" href="../../usage/ellipsoids.html#cmdoption-arg-R"><code class="xref std std-option docutils literal notranslate"><span class="pre">+R</span></code></a> takes precedence.</p>
<p>See <a class="reference internal" href="../../usage/ellipsoids.html#ellipsoid-size-parameters"><span class="std std-ref">Ellipsoid size parameters</span></a> for more information.</p>
</dd></dl>
</section>
<section id="mathematical-definition">
<h2>Mathematical definition<a class="headerlink" href="#mathematical-definition" title="Permalink to this headline">¶</a></h2>
<section id="spherical-form">
<h3>Spherical form<a class="headerlink" href="#spherical-form" title="Permalink to this headline">¶</a></h3>
<p>For the spherical form of the projection we introduce the scaling factor:</p>
<div class="math notranslate nohighlight">
\[k_0 = \cos \phi_{ts}\]</div>
<section id="forward-projection">
<h4>Forward projection<a class="headerlink" href="#forward-projection" title="Permalink to this headline">¶</a></h4>
<div class="math notranslate nohighlight">
\[x = k_0R \lambda; \qquad y = k_0R \psi\]</div>
<div class="math notranslate nohighlight">
\[\begin{split}\psi &= \ln \tan \biggl(\frac{\pi}{4} + \frac{\phi}{2} \biggr)\\
&= \sinh^{-1}\tan\phi\end{split}\]</div>
<p>The quantity <span class="math notranslate nohighlight">\(\psi\)</span> is the isometric latitude.</p>
</section>
<section id="inverse-projection">
<h4>Inverse projection<a class="headerlink" href="#inverse-projection" title="Permalink to this headline">¶</a></h4>
<div class="math notranslate nohighlight">
\[\lambda = \frac{x}{k_0R}; \qquad \psi = \frac{y}{k_0R}\]</div>
<div class="math notranslate nohighlight">
\[\begin{split}\phi &= \frac{\pi}{2} - 2 \tan^{-1} \exp(-\psi)\\
&= \tan^{-1}\sinh\psi\end{split}\]</div>
</section>
</section>
<section id="ellipsoidal-form">
<h3>Ellipsoidal form<a class="headerlink" href="#ellipsoidal-form" title="Permalink to this headline">¶</a></h3>
<p>For the ellipsoidal form of the projection we introduce the scaling factor:</p>
<div class="math notranslate nohighlight">
\[k_0 = m( \phi_{ts} )\]</div>
<p>where</p>
<div class="math notranslate nohighlight">
\[m(\phi) = \frac{\cos\phi}{\sqrt{1 - e^2\sin^2\phi}}\]</div>
<p><span class="math notranslate nohighlight">\(a\,m(\phi)\)</span> is the radius of the circle of latitude <span class="math notranslate nohighlight">\(\phi\)</span>.</p>
<section id="id2">
<h4>Forward projection<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h4>
<div class="math notranslate nohighlight">
\[x = k_0 a \lambda; \qquad y = k_0 a \psi\]</div>
<div class="math notranslate nohighlight">
\[\begin{split}\psi &= \ln\tan\biggl(\frac\pi4 + \frac{\phi}2\biggr)
-\frac12 e
\ln \biggl(\frac{1 + e \sin\phi}{1 - e \sin\phi}\biggr)\\
&= \sinh^{-1}\tan\phi - e \tanh^{-1}(e \sin\phi)\end{split}\]</div>
</section>
<section id="id3">
<h4>Inverse projection<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h4>
<div class="math notranslate nohighlight">
\[\lambda = \frac{x}{k_0 a}; \quad \psi = \frac{y}{k_0 a}\]</div>
<p>The latitude <span class="math notranslate nohighlight">\(\phi\)</span> is found by inverting the equation for
<span class="math notranslate nohighlight">\(\psi\)</span>. This follows the method given by <span id="id4">[<a class="reference internal" href="../../zreferences.html#id21" title="Karney, C. F. F. Transverse Mercator with an accuracy of a few nanometers. J. Geod., 85(8):475-485, August 2011. arXiv:1002.1417, doi:10.1007/s00190-011-0445-3.">Karney2011tm</a>]</span>.
Start by introducing the conformal latitude</p>
<div class="math notranslate nohighlight">
\[\chi = \tan^{-1}\sinh\psi\]</div>
<p>The tangents of the latitudes <span class="math notranslate nohighlight">\(\tau = \tan\phi\)</span> and <span class="math notranslate nohighlight">\(\tau' =
\tan\chi = \sinh\psi\)</span> are related by</p>
<div class="math notranslate nohighlight">
\[\tau' = \tau \sqrt{1 + \sigma^2} - \sigma \sqrt{1 + \tau^2}\]</div>
<p>where</p>
<div class="math notranslate nohighlight">
\[\sigma = \sinh\bigl(e \tanh^{-1}(e \tau/\sqrt{1 + \tau^2}) \bigr)\]</div>
<p>This is obtained by taking the <span class="math notranslate nohighlight">\(\sinh\)</span> of the equation for
<span class="math notranslate nohighlight">\(\psi\)</span> and using the multiple argument formula. The equation for
<span class="math notranslate nohighlight">\(\tau'\)</span> can be solved to give <span class="math notranslate nohighlight">\(\tau\)</span> using Newton’s method
using <span class="math notranslate nohighlight">\(\tau = \tau'/(1 - e^2)\)</span> as an initial guess and with the
needed derivative given by</p>
<div class="math notranslate nohighlight">
\[\frac{d\tau'}{d\tau} = \frac{1 - e^2}{1 + (1 - e^2)\tau^2}
\sqrt{1 + \tau'^2} \sqrt{1 + \tau^2}\]</div>
<p>This converges after no more than 2 iterations. Finally set
<span class="math notranslate nohighlight">\(\phi=\tan^{-1}\tau\)</span>.</p>
</section>
</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/Mercator_projection">Wikipedia</a></p></li>
<li><p><a class="reference external" href="http://mathworld.wolfram.com/MercatorProjection.html">Wolfram Mathworld</a></p></li>
</ol>
</section>
</section>
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