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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>
<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>
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<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>
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<li class="toctree-l3 current"><a class="current reference internal" href="#">Cassini (Cassini-Soldner)</a><ul>
<li class="toctree-l4"><a class="reference internal" href="#usage">Usage</a></li>
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<li class="toctree-l4"><a class="reference internal" href="#further-reading">Further reading</a></li>
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<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"><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"><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="cassini-cassini-soldner">
<span id="cass"></span><h1>Cassini (Cassini-Soldner)<a class="headerlink" href="#cassini-cassini-soldner" title="Permalink to this headline">¶</a></h1>
<p>Although the Cassini projection has been largely replaced by the Transverse Mercator, it is still in limited use outside the United States and was one of the major topographic mapping projections until the early 20th century.</p>
<table class="docutils align-default">
<colgroup>
<col style="width: 22%" />
<col style="width: 78%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><strong>Classification</strong></p></td>
<td><p>Transverse and oblique 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 central meridian with long, narrow areas</p></td>
</tr>
<tr class="row-even"><td><p><strong>Alias</strong></p></td>
<td><p>cass</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="id6">
<a class="reference internal image-reference" href="../../_images/cass.png"><img alt="Cassini" src="../../_images/cass.png" style="width: 500px;" /></a>
<figcaption>
<p><span class="caption-text">proj-string: <code class="docutils literal notranslate"><span class="pre">+proj=cass</span></code></span><a class="headerlink" href="#id6" 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>There has been little usage of the spherical version of the Cassini, but the ellipsoidal Cassini-Soldner version was adopted by the Ordnance Survey for the official survey of Great Britain during the second half of the 19th century <span id="id1">[<a class="reference internal" href="../../zreferences.html#id38" title="Steers, J. A. An introduction to the study of map projections. University of London Press, 15th edition, 1970.">Steers1970</a>]</span>.
Many of these maps were prepared at a scale of 1:2,500.
The Cassini-Soldner was also used for the detailed mapping of many German states during the same period.</p>
<p>Example using EPSG 30200 (Trinidad 1903, units in clarke’s links):</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>$ echo 0.17453293 -1.08210414 | proj +proj=cass +lat_0=10.44166666666667 +lon_0=-61.33333333333334 +x_0=86501.46392051999 +y_0=65379.0134283 +a=6378293.645208759 +b=6356617.987679838 +to_meter=0.201166195164
66644.94 82536.22
</pre></div>
</div>
<p>Example using EPSG 3068 (Soldner Berlin):</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>$ echo 13.5 52.4 | proj +proj=cass +lat_0=52.41864827777778 +lon_0=13.62720366666667 +x_0=40000 +y_0=10000 +ellps=bessel +units=m
31343.05 7932.76
</pre></div>
</div>
</section>
<section id="options">
<h2>Options<a class="headerlink" href="#options" title="Permalink to this headline">¶</a></h2>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>All options are optional for the Cassini projection.</p>
</div>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-lat_0">
<span id="cmdoption-arg-lat-0"></span><span class="sig-name descname"><span class="pre">+lat_0</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-lat_0" title="Permalink to this definition">¶</a></dt>
<dd><p>Latitude 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-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>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-hyperbolic">
<span class="sig-name descname"><span class="pre">+hyperbolic</span></span><span class="sig-prename descclassname"></span><a class="headerlink" href="#cmdoption-arg-hyperbolic" title="Permalink to this definition">¶</a></dt>
<dd><p>Use modified form of the standard Cassini-Soldner projection known as the Hyperbolic Cassini-Soldner.
This is used in particular for the “Vanua Levu Grid” of the island of Vanua Levu, Fiji (EPSG:3139)</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 Cassini projection are taken from <span id="id2">[<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><span class="math notranslate nohighlight">\(\phi_0\)</span> is the latitude of origin that match the center of the map (default to 0). It can be set with <code class="docutils literal notranslate"><span class="pre">+lat_0</span></code>.</p>
<section id="spherical-form">
<h3>Spherical form<a class="headerlink" href="#spherical-form" title="Permalink to this headline">¶</a></h3>
<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">
\[x = \arcsin(\cos(\phi) \sin(\lambda))\]</div>
<div class="math notranslate nohighlight">
\[y = \arctan2(\tan(\phi), \cos(\lambda)) - \phi_0\]</div>
</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">
\[\phi = \arcsin(\sin(y+\phi_0) \cos(x))\]</div>
<div class="math notranslate nohighlight">
\[\lambda = \arctan2(\tan(x), \cos(y+\phi_0))\]</div>
</section>
</section>
<section id="ellipsoidal-form">
<h3>Ellipsoidal form<a class="headerlink" href="#ellipsoidal-form" title="Permalink to this headline">¶</a></h3>
<section id="id3">
<h4>Forward projection<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h4>
<div class="math notranslate nohighlight">
\[N = (1 - e^2 \sin^2(\phi))^{-1/2}\]</div>
<div class="math notranslate nohighlight">
\[T = \tan^2(\phi)\]</div>
<div class="math notranslate nohighlight">
\[A = \lambda \cos(\phi)\]</div>
<div class="math notranslate nohighlight">
\[C = \frac{e^2}{1-e^2} cos^2(\phi)\]</div>
<div class="math notranslate nohighlight">
\[x = N ( A - T \frac{A^3}{6} - (8-T+8C)T\frac{A^5}{120})\]</div>
<div class="math notranslate nohighlight">
\[y = M(\phi) - M(\phi_0) + N \tan(\phi)(\frac{A^2}{2} + (5-T+6C)\frac{A^4}{24})\]</div>
<p>and M() is the meridional distance function.</p>
</section>
<section id="id4">
<h4>Inverse projection<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h4>
<div class="math notranslate nohighlight">
\[\phi' = M^{-1}(M(\phi_0)+y)\]</div>
<p>if <span class="math notranslate nohighlight">\(\phi' = \frac{\pi}{2}\)</span> then <span class="math notranslate nohighlight">\(\phi=\phi'\)</span> and <span class="math notranslate nohighlight">\(\lambda=0\)</span></p>
<p>otherwise evaluate T and N above using <span class="math notranslate nohighlight">\(\phi'\)</span> and</p>
<div class="math notranslate nohighlight">
\[R = (1 - e^2)(1 - e^2 sin^2 \phi')^{-3/2}\]</div>
<div class="math notranslate nohighlight">
\[D = x/N\]</div>
<div class="math notranslate nohighlight">
\[\phi = \phi' - \tan \phi' \frac{N}{R}(\frac{D^2}{2}-(1+3T)\frac{D^4}{24})\]</div>
<div class="math notranslate nohighlight">
\[\lambda = \frac{(D - T\frac{D^3}{3} + (1+3T)T\frac{D^5}{15})}{\cos \phi'}\]</div>
</section>
</section>
</section>
<section id="further-reading">
<span id="id5"></span><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/Equirectangular_projection">Wikipedia</a></p></li>
<li><p><a class="reference external" href="http://www.ihsenergy.com/epsg/guid7.pdf">EPSG, POSC literature pertaining to Coordinate Conversions and Transformations including Formulas</a></p></li>
</ol>
</section>
</section>
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