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<li>Ellipsoids</li> + <li class="wy-breadcrumbs-aside"> + <a href="https://github.com/OSGeo/PROJ/edit/8.2/docs/source/usage/ellipsoids.rst" class="fa fa-github"> Edit on GitHub</a> + </li> + </ul><div class="rst-breadcrumbs-buttons" role="navigation" aria-label="Sequential page navigation"> + <a href="transformation.html" class="btn btn-neutral float-left" title="Geodetic transformation" accesskey="p"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="environmentvars.html" class="btn btn-neutral float-right" title="Environment variables" accesskey="n">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> + </div> + <hr/> +</div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <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> +</section> +</section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="transformation.html" class="btn btn-neutral float-left" title="Geodetic transformation" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="environmentvars.html" class="btn btn-neutral float-right" title="Environment variables" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> + </div> + + <hr/> + + <div role="contentinfo"> + <p>© Copyright 1983-2022. + <span class="lastupdated">Last updated on 22 Mar 2022. + </span></p> + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + +</footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + +</body> +</html>
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