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<section id="geocentric-latitude">
<span id="geoc"></span><h1>Geocentric Latitude<a class="headerlink" href="#geocentric-latitude" title="Permalink to this headline">¶</a></h1>
<div class="versionadded">
<p><span class="versionmodified added">New in version 5.0.0.</span></p>
</div>
<p>Convert from Geodetic Latitude to Geocentric Latitude (in the forward path).</p>
<table class="docutils align-default">
<colgroup>
<col style="width: 27%" />
<col style="width: 73%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><strong>Alias</strong></p></td>
<td><p>geoc</p></td>
</tr>
<tr class="row-even"><td><p><strong>Domain</strong></p></td>
<td><p>2D</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Input type</strong></p></td>
<td><p>Geodetic coordinates</p></td>
</tr>
<tr class="row-even"><td><p><strong>Output type</strong></p></td>
<td><p>Geocentric angular coordinates</p></td>
</tr>
</tbody>
</table>
<p>The geodetic (or geographic) latitude (also called planetographic latitude
in the context of non-Earth bodies) is the angle between the equatorial plane
and the normal (vertical) to the ellipsoid surface at the considered point.
The geodetic latitude is what is normally used everywhere in PROJ when angular
coordinates are expected or produced.</p>
<p>The geocentric latitude (also called planetocentric latitude in the context of
non-Earth bodies) is the angle between the equatorial plane and a line joining
the body centre to the considered point.</p>
<a class="reference internal image-reference" href="../../_images/geocentric_latitude.png"><img alt="Geocentric latitude" class="align-center" src="../../_images/geocentric_latitude.png" style="width: 437.0px; height: 257.0px;" /></a>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This conversion must be distinguished from the <a class="reference internal" href="cart.html#cart"><span class="std std-ref">Geodetic to cartesian conversion</span></a>
which converts geodetic coordinates to geocentric coordinates in the cartesian
domain.</p>
</div>
<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 conversion are taken from <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>
(equation 3-28)</p>
<p>Let <span class="math notranslate nohighlight">\(\phi'\)</span> to be the geocentric latitude and <span class="math notranslate nohighlight">\(\phi\)</span> the
geodetic latitude, then</p>
<div class="math notranslate nohighlight">
\[\phi' = \arctan \left[ (1 - e^2) \tan \left( \phi \right) \right]\]</div>
<p>The geocentric latitude is consequently lesser (in absolute value) than the geodetic
latitude, except at the equator and the poles where they are equal.</p>
<p>On a sphere, they are always equal.</p>
</section>
<section id="usage">
<h2>Usage<a class="headerlink" href="#usage" title="Permalink to this headline">¶</a></h2>
<p>Converting from geodetic latitude to geocentric latitude:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>+proj=geoc +ellps=GRS80
</pre></div>
</div>
<p>Converting from geocentric latitude to geodetic latitude:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>+proj=pipeline +step +proj=geoc +inv +ellps=GRS80
</pre></div>
</div>
</section>
<section id="parameters">
<h2>Parameters<a class="headerlink" href="#parameters" title="Permalink to this headline">¶</a></h2>
<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>
</section>
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