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<section id="ellipsoids">
<span id="id1"></span><h1>Ellipsoids<a class="headerlink" href="#ellipsoids" title="Permalink to this headline">¶</a></h1>
<p>An ellipsoid is a mathematically defined surface which approximates the <em>geoid</em>:
the surface of the Earth’s gravity field, which is approximately the same as
mean sea level.</p>
<figure class="align-center" id="id5">
<a class="reference internal image-reference" href="../_images/general_ellipsoid.png"><img alt="Global and local fitting of the ellipsoid" src="../_images/general_ellipsoid.png" style="width: 500px;" /></a>
<figcaption>
<p><span class="caption-text">Global and local fitting of the ellipsoid</span><a class="headerlink" href="#id5" title="Permalink to this image">¶</a></p>
</figcaption>
</figure>
<p>A complete ellipsoid definition comprises a size (primary) and a shape (secondary)
parameter.</p>
<section id="ellipsoid-size-parameters">
<span id="id2"></span><h2>Ellipsoid size parameters<a class="headerlink" href="#ellipsoid-size-parameters" title="Permalink to this headline">¶</a></h2>
<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, <span class="math notranslate nohighlight">\(R\)</span>.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-a">
<span class="sig-name descname"><span class="pre">+a</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-a" title="Permalink to this definition">¶</a></dt>
<dd><p>Semi-major axis of the ellipsoid, <span class="math notranslate nohighlight">\(a\)</span>.</p>
</dd></dl>
</section>
<section id="ellipsoid-shape-parameters">
<span id="id3"></span><h2>Ellipsoid shape parameters<a class="headerlink" href="#ellipsoid-shape-parameters" title="Permalink to this headline">¶</a></h2>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-rf">
<span class="sig-name descname"><span class="pre">+rf</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-rf" title="Permalink to this definition">¶</a></dt>
<dd><p>Reverse flattening of the ellipsoid, <span class="math notranslate nohighlight">\(1/f\)</span>.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-f">
<span class="sig-name descname"><span class="pre">+f</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-f" title="Permalink to this definition">¶</a></dt>
<dd><p>Flattening of the ellipsoid, <span class="math notranslate nohighlight">\(f\)</span>.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-es">
<span class="sig-name descname"><span class="pre">+es</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-es" title="Permalink to this definition">¶</a></dt>
<dd><p>Eccentricity squared, <span class="math notranslate nohighlight">\(e^2\)</span>.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-e">
<span class="sig-name descname"><span class="pre">+e</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-e" title="Permalink to this definition">¶</a></dt>
<dd><p>Eccentricity, <span class="math notranslate nohighlight">\(e\)</span>.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-b">
<span class="sig-name descname"><span class="pre">+b</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-b" title="Permalink to this definition">¶</a></dt>
<dd><p>Semi-minor axis, <span class="math notranslate nohighlight">\(b\)</span>.</p>
</dd></dl>
<p>The ellipsoid definition may be augmented with a spherification flag, turning
the ellipsoid into a sphere with features defined by the ellipsoid.</p>
</section>
<section id="ellipsoid-spherification-parameters">
<span id="id4"></span><h2>Ellipsoid spherification parameters<a class="headerlink" href="#ellipsoid-spherification-parameters" title="Permalink to this headline">¶</a></h2>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-R_A">
<span id="cmdoption-arg-r-a"></span><span class="sig-name descname"><span class="pre">+R_A</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-R_A" title="Permalink to this definition">¶</a></dt>
<dd><p>A sphere with the same surface area as the ellipsoid.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-R_V">
<span id="cmdoption-arg-r-v"></span><span class="sig-name descname"><span class="pre">+R_V</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-R_V" title="Permalink to this definition">¶</a></dt>
<dd><p>A sphere with the same volume as the ellipsoid.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-R_a">
<span class="sig-name descname"><span class="pre">+R_a</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-R_a" title="Permalink to this definition">¶</a></dt>
<dd><p>A sphere with <span class="math notranslate nohighlight">\(R = (a + b)/2\)</span> (arithmetic mean).</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-R_g">
<span id="cmdoption-arg-r-g"></span><span class="sig-name descname"><span class="pre">+R_g</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-R_g" title="Permalink to this definition">¶</a></dt>
<dd><p>A sphere with <span class="math notranslate nohighlight">\(R = \sqrt{ab}\)</span> (geometric mean).</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-R_h">
<span id="cmdoption-arg-r-h"></span><span class="sig-name descname"><span class="pre">+R_h</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-R_h" title="Permalink to this definition">¶</a></dt>
<dd><p>A sphere with <span class="math notranslate nohighlight">\(R = 2ab/(a+b)\)</span> (harmonic mean).</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-R_lat_a">
<span id="cmdoption-arg-r-lat-a"></span><span class="sig-name descname"><span class="pre">+R_lat_a</span></span><span class="sig-prename descclassname"><span class="pre">=<phi></span></span><a class="headerlink" href="#cmdoption-arg-R_lat_a" title="Permalink to this definition">¶</a></dt>
<dd><p>A sphere with <span class="math notranslate nohighlight">\(R\)</span> being the arithmetic mean of the corresponding
ellipsoid at latitude <span class="math notranslate nohighlight">\(\phi\)</span>.</p>
</dd></dl>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-R_lat_g">
<span id="cmdoption-arg-r-lat-g"></span><span class="sig-name descname"><span class="pre">+R_lat_g</span></span><span class="sig-prename descclassname"><span class="pre">=<phi></span></span><a class="headerlink" href="#cmdoption-arg-R_lat_g" title="Permalink to this definition">¶</a></dt>
<dd><p>A sphere with <span class="math notranslate nohighlight">\(R\)</span> being the geometric mean of the corresponding
ellipsoid at latitude <span class="math notranslate nohighlight">\(\phi\)</span>.</p>
</dd></dl>
<p>If <a class="reference internal" href="#cmdoption-arg-R"><code class="xref std std-option docutils literal notranslate"><span class="pre">+R</span></code></a> is given as size parameter, any shape and spherification
parameters given are ignored.</p>
</section>
<section id="built-in-ellipsoid-definitions">
<span id="ellipsoid-definitions"></span><h2>Built-in ellipsoid definitions<a class="headerlink" href="#built-in-ellipsoid-definitions" title="Permalink to this headline">¶</a></h2>
<p>The <code class="docutils literal notranslate"><span class="pre">ellps=xxx</span></code> parameter provides both size and shape for a number of
built-in ellipsoid definitions.</p>
<blockquote>
<div><table class="docutils align-default">
<colgroup>
<col style="width: 16%" />
<col style="width: 45%" />
<col style="width: 38%" />
</colgroup>
<thead>
<tr class="row-odd"><th class="head"><p>ellps</p></th>
<th class="head"><p>Parameters</p></th>
<th class="head"><p>Datum name</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>GRS80</p></td>
<td><p>a=6378137.0 rf=298.257222101</p></td>
<td><p>GRS 1980(IUGG, 1980)</p></td>
</tr>
<tr class="row-odd"><td><p>airy</p></td>
<td><p>a=6377563.396 b=6356256.910</p></td>
<td><p>Airy 1830</p></td>
</tr>
<tr class="row-even"><td><p>bessel</p></td>
<td><p>a=6377397.155 rf=299.1528128</p></td>
<td><p>Bessel 1841</p></td>
</tr>
<tr class="row-odd"><td><p>clrk66</p></td>
<td><p>a=6378206.4 b=6356583.8</p></td>
<td><p>Clarke 1866</p></td>
</tr>
<tr class="row-even"><td><p>intl</p></td>
<td><p>a=6378388.0 rf=297.</p></td>
<td><p>International 1909 (Hayford)</p></td>
</tr>
<tr class="row-odd"><td><p>WGS60</p></td>
<td><p>a=6378165.0 rf=298.3</p></td>
<td><p>WGS 60</p></td>
</tr>
<tr class="row-even"><td><p>WGS66</p></td>
<td><p>a=6378145.0 rf=298.25</p></td>
<td><p>WGS 66</p></td>
</tr>
<tr class="row-odd"><td><p>WGS72</p></td>
<td><p>a=6378135.0 rf=298.26</p></td>
<td><p>WGS 72</p></td>
</tr>
<tr class="row-even"><td><p>WGS84</p></td>
<td><p>a=6378137.0 rf=298.257223563</p></td>
<td><p>WGS 84</p></td>
</tr>
<tr class="row-odd"><td><p>sphere</p></td>
<td><p>a=6370997.0 b=6370997.0</p></td>
<td><p>Normal Sphere (r=6370997)</p></td>
</tr>
</tbody>
</table>
</div></blockquote>
<p>If size and shape are given as <code class="docutils literal notranslate"><span class="pre">ellps=xxx</span></code>, later shape and size parameters
are are taken into account as modifiers for the built-in ellipsoid definition.</p>
<p>While this may seem strange, it is in accordance with historical PROJ
behavior. It can e.g. be used to define coordinates on the ellipsoid
scaled to unit semimajor axis by specifying <code class="docutils literal notranslate"><span class="pre">+ellps=xxx</span> <span class="pre">+a=1</span></code></p>
</section>
<section id="transformation-examples">
<span id="ellipsoid-transformation-examples"></span><h2>Transformation examples<a class="headerlink" href="#transformation-examples" title="Permalink to this headline">¶</a></h2>
<p>Spherical earth with radius 7000km:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=laton R=7000000
</pre></div>
</div>
<p>Using the GRS80 ellipsoid:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=laton ellps=GRS80
</pre></div>
</div>
<p>Expressing ellipsoid by semi-major axis and reverse flattening (<span class="math notranslate nohighlight">\(1/f\)</span>):</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=laton a=6378137.0 rf=298.25
</pre></div>
</div>
<p>Spherical earth based on volume of ellipsoid</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=laton a=6378137.0 rf=298.25 +R_V
</pre></div>
</div>
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