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href="operations/index.html">Coordinate operations</a></li> +<li class="toctree-l1"><a class="reference internal" href="resource_files.html">Resource files</a></li> +<li class="toctree-l1 current"><a class="current reference internal" href="#">Geodesic calculations</a><ul> +<li class="toctree-l2"><a class="reference internal" href="#introduction">Introduction</a></li> +<li class="toctree-l2"><a class="reference internal" href="#solution-of-geodesic-problems">Solution of geodesic problems</a></li> +<li class="toctree-l2"><a class="reference internal" href="#additional-properties">Additional properties</a></li> +<li class="toctree-l2"><a class="reference internal" href="#multiple-shortest-geodesics">Multiple shortest geodesics</a></li> +<li class="toctree-l2"><a class="reference internal" href="#background">Background</a></li> +</ul> +</li> +<li class="toctree-l1"><a class="reference internal" href="development/index.html">Development</a></li> +<li class="toctree-l1"><a class="reference 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class="wy-breadcrumbs-aside"> + <a href="https://github.com/OSGeo/PROJ/edit/8.2/docs/source/geodesic.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="resource_files.html" class="btn btn-neutral float-left" title="Resource files" accesskey="p"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="development/index.html" class="btn btn-neutral float-right" title="Development" 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="geodesic-calculations"> +<span id="geodesic"></span><h1>Geodesic calculations<a class="headerlink" href="#geodesic-calculations" title="Permalink to this headline">¶</a></h1> +<section id="introduction"> +<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2> +<p>Consider an ellipsoid of revolution with equatorial radius <span class="math notranslate nohighlight">\(a\)</span>, polar +semi-axis <span class="math notranslate nohighlight">\(b\)</span>, and flattening <span class="math notranslate nohighlight">\(f=(a-b)/a\)</span>. Points on +the surface of the ellipsoid are characterized by their latitude <span class="math notranslate nohighlight">\(\phi\)</span> +and longitude <span class="math notranslate nohighlight">\(\lambda\)</span>. (Note that latitude here means the +<em>geographical latitude</em>, the angle between the normal to the ellipsoid +and the equatorial plane).</p> +<p>The shortest path between two points on the ellipsoid at +<span class="math notranslate nohighlight">\((\phi_1,\lambda_1)\)</span> and <span class="math notranslate nohighlight">\((\phi_2,\lambda_2)\)</span> +is called the geodesic. Its length is +<span class="math notranslate nohighlight">\(s_{12}\)</span> and the geodesic from point 1 to point 2 has forward +azimuths <span class="math notranslate nohighlight">\(\alpha_1\)</span> and <span class="math notranslate nohighlight">\(\alpha_2\)</span> at the two end +points. In this figure, we have <span class="math notranslate nohighlight">\(\lambda_{12}=\lambda_2-\lambda_1\)</span>.</p> +<blockquote> +<div><center> + <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Geodesic_problem_on_an_ellipsoid.svg/320px-Geodesic_problem_on_an_ellipsoid.svg.png" + alt="Figure from wikipedia" + width="320"> +</center></div></blockquote> +<p>A geodesic can be extended indefinitely by requiring that any +sufficiently small segment is a shortest path; geodesics are also the +straightest curves on the surface.</p> +</section> +<section id="solution-of-geodesic-problems"> +<h2>Solution of geodesic problems<a class="headerlink" href="#solution-of-geodesic-problems" title="Permalink to this headline">¶</a></h2> +<p>Traditionally two geodesic problems are considered:</p> +<ul class="simple"> +<li><p>the direct problem — given <span class="math notranslate nohighlight">\(\phi_1\)</span>, +<span class="math notranslate nohighlight">\(\lambda_1\)</span>, <span class="math notranslate nohighlight">\(\alpha_1\)</span>, <span class="math notranslate nohighlight">\(s_{12}\)</span>, +determine <span class="math notranslate nohighlight">\(\phi_2\)</span>, <span class="math notranslate nohighlight">\(\lambda_2\)</span>, <span class="math notranslate nohighlight">\(\alpha_2\)</span>.</p></li> +<li><p>the inverse problem — given <span class="math notranslate nohighlight">\(\phi_1\)</span>, +<span class="math notranslate nohighlight">\(\lambda_1\)</span>, <span class="math notranslate nohighlight">\(\phi_2\)</span>, <span class="math notranslate nohighlight">\(\lambda_2\)</span>, +determine <span class="math notranslate nohighlight">\(s_{12}\)</span>, <span class="math notranslate nohighlight">\(\alpha_1\)</span>, +<span class="math notranslate nohighlight">\(\alpha_2\)</span>.</p></li> +</ul> +<p>PROJ incorporates <a class="reference external" href="https://geographiclib.sourceforge.io/1.52/C/">C library for Geodesics</a> from <a class="reference external" href="https://geographiclib.sourceforge.io">GeographicLib</a>. This library provides +routines to solve the direct and inverse geodesic problems. Full double +precision accuracy is maintained provided that +<span class="math notranslate nohighlight">\(\lvert f\rvert<\frac1{50}\)</span>. Refer to the <a class="reference external" href="https://geographiclib.sourceforge.io/1.52/C/geodesic_8h.html">application programming interface</a> +for full documentation. A brief summary of the routines is given in +geodesic(3).</p> +<p>The interface to the geodesic routines differ in two respects from the +rest of PROJ:</p> +<ul class="simple"> +<li><p>angles (latitudes, longitudes, and azimuths) are in degrees (instead +of in radians);</p></li> +<li><p>the shape of ellipsoid is specified by the flattening <span class="math notranslate nohighlight">\(f\)</span>; this can +be negative to denote a prolate ellipsoid; setting <span class="math notranslate nohighlight">\(f=0\)</span> corresponds +to a sphere, in which case the geodesic becomes a great circle.</p></li> +</ul> +<p>PROJ also includes a command line tool, <a class="reference internal" href="apps/geod.html#geod"><span class="std std-ref">geod</span></a>(1), for performing +simple geodesic calculations.</p> +</section> +<section id="additional-properties"> +<h2>Additional properties<a class="headerlink" href="#additional-properties" title="Permalink to this headline">¶</a></h2> +<p>The routines also calculate several other quantities of interest</p> +<ul class="simple"> +<li><p><span class="math notranslate nohighlight">\(S_{12}\)</span> is the area between the geodesic from point 1 to +point 2 and the equator; i.e., it is the area, measured +counter-clockwise, of the quadrilateral with corners +<span class="math notranslate nohighlight">\((\phi_1,\lambda_1)\)</span>, <span class="math notranslate nohighlight">\((0,\lambda_1)\)</span>, +<span class="math notranslate nohighlight">\((0,\lambda_2)\)</span>, and +<span class="math notranslate nohighlight">\((\phi_2,\lambda_2)\)</span>. It is given in +meters<sup>2</sup>.</p></li> +<li><p><span class="math notranslate nohighlight">\(m_{12}\)</span>, the reduced length of the geodesic is defined such +that if the initial azimuth is perturbed by <span class="math notranslate nohighlight">\(d\alpha_1\)</span> +(radians) then the second point is displaced by <span class="math notranslate nohighlight">\(m_{12}\,d\alpha_1\)</span> +in the direction perpendicular to the +geodesic. <span class="math notranslate nohighlight">\(m_{12}\)</span> is given in meters. On a curved surface +the reduced length obeys a symmetry relation, <span class="math notranslate nohighlight">\(m_{12}+m_{21}=0\)</span>. +On a flat surface, we have <span class="math notranslate nohighlight">\(m_{12}=s_{12}\)</span>.</p></li> +<li><p><span class="math notranslate nohighlight">\(M_{12}\)</span> and <span class="math notranslate nohighlight">\(M_{21}\)</span> are geodesic scales. If two +geodesics are parallel at point 1 and separated by a small distance +<span class="math notranslate nohighlight">\(dt\)</span>, then they are separated by a distance <span class="math notranslate nohighlight">\(M_{12}\,dt\)</span> at +point 2. <span class="math notranslate nohighlight">\(M_{21}\)</span> is defined similarly (with the geodesics +being parallel to one another at point 2). <span class="math notranslate nohighlight">\(M_{12}\)</span> and +<span class="math notranslate nohighlight">\(M_{21}\)</span> are dimensionless quantities. On a flat surface, +we have <span class="math notranslate nohighlight">\(M_{12}=M_{21}=1\)</span>.</p></li> +<li><p><span class="math notranslate nohighlight">\(\sigma_{12}\)</span> is the arc length on the auxiliary sphere. +This is a construct for converting the problem to one in spherical +trigonometry. The spherical arc length from one equator crossing to +the next is always <span class="math notranslate nohighlight">\(180^\circ\)</span>.</p></li> +</ul> +<p>If points 1, 2, and 3 lie on a single geodesic, then the following +addition rules hold:</p> +<ul class="simple"> +<li><p><span class="math notranslate nohighlight">\(s_{13}=s_{12}+s_{23}\)</span>,</p></li> +<li><p><span class="math notranslate nohighlight">\(\sigma_{13}=\sigma_{12}+\sigma_{23}\)</span>,</p></li> +<li><p><span class="math notranslate nohighlight">\(S_{13}=S_{12}+S_{23}\)</span>,</p></li> +<li><p><span class="math notranslate nohighlight">\(m_{13}=m_{12}M_{23}+m_{23}M_{21}\)</span>,</p></li> +<li><p><span class="math notranslate nohighlight">\(M_{13}=M_{12}M_{23}-(1-M_{12}M_{21})m_{23}/m_{12}\)</span>,</p></li> +<li><p><span class="math notranslate nohighlight">\(M_{31}=M_{32}M_{21}-(1-M_{23}M_{32})m_{12}/m_{23}\)</span>.</p></li> +</ul> +</section> +<section id="multiple-shortest-geodesics"> +<h2>Multiple shortest geodesics<a class="headerlink" href="#multiple-shortest-geodesics" title="Permalink to this headline">¶</a></h2> +<p>The shortest distance found by solving the inverse problem is +(obviously) uniquely defined. However, in a few special cases there are +multiple azimuths which yield the same shortest distance. Here is a +catalog of those cases:</p> +<ul class="simple"> +<li><p><span class="math notranslate nohighlight">\(\phi_1=-\phi_2\)</span> (with neither point at +a pole). If <span class="math notranslate nohighlight">\(\alpha_1=\alpha_2\)</span>, the geodesic +is unique. Otherwise there are two geodesics and the second one is +obtained by setting +<span class="math notranslate nohighlight">\([\alpha_1,\alpha_2]\leftarrow[\alpha_2,\alpha_1]\)</span>, +<span class="math notranslate nohighlight">\([M_{12},M_{21}]\leftarrow[M_{21},M_{12}]\)</span>, +<span class="math notranslate nohighlight">\(S_{12}\leftarrow-S_{12}\)</span>. +(This occurs when the longitude difference is near <span class="math notranslate nohighlight">\(\pm180^\circ\)</span> +for oblate ellipsoids.)</p></li> +<li><p><span class="math notranslate nohighlight">\(\lambda_2=\lambda_1\pm180^\circ\)</span> (with +neither point at a pole). If <span class="math notranslate nohighlight">\(\alpha_1=0^\circ\)</span> or +<span class="math notranslate nohighlight">\(\pm180^\circ\)</span>, the geodesic is unique. Otherwise there are two +geodesics and the second one is obtained by setting +<span class="math notranslate nohighlight">\([\alpha_1,\alpha_2]\leftarrow[-\alpha_1,-\alpha_2]\)</span>, +<span class="math notranslate nohighlight">\(S_{12}\leftarrow-S_{12}\)</span>. (This occurs when +<span class="math notranslate nohighlight">\(\phi_2\)</span> is near <span class="math notranslate nohighlight">\(-\phi_1\)</span> for prolate +ellipsoids.)</p></li> +<li><p>Points 1 and 2 at opposite poles. There are infinitely many +geodesics which can be generated by setting +<span class="math notranslate nohighlight">\([\alpha_1,\alpha_2]\leftarrow[\alpha_1,\alpha_2]+[\delta,-\delta]\)</span>, +for arbitrary <span class="math notranslate nohighlight">\(\delta\)</span>. +(For spheres, this prescription applies when points 1 and 2 are +antipodal.)</p></li> +<li><p><span class="math notranslate nohighlight">\(s_{12}=0\)</span> (coincident points). There are infinitely many +geodesics which can be generated by setting +<span class="math notranslate nohighlight">\([\alpha_1,\alpha_2]\leftarrow[\alpha_1,\alpha_2]+[\delta,\delta]\)</span>, +for arbitrary <span class="math notranslate nohighlight">\(\delta\)</span>.</p></li> +</ul> +</section> +<section id="background"> +<h2>Background<a class="headerlink" href="#background" title="Permalink to this headline">¶</a></h2> +<p>The algorithms implemented by this package are given in <span id="id1">[<a class="reference internal" href="zreferences.html#id20" title="Karney, C. F. F. Algorithms for geodesics. Journal of Geodesy, 87(1):43–55, 2013. doi:10.1007/s00190-012-0578-z.">Karney2013</a>]</span> +(<a class="reference external" href="https://geographiclib.sourceforge.io/geod-addenda.html">addenda</a>) +and are based on <span id="id2">[<a class="reference internal" href="zreferences.html#id4" title="Bessel, F. W. The calculation of longitude and latitude from geodesic measurements. Astronomische Nachrichten, 4(86):241–254, 1825. arXiv:0908.1824.">Bessel1825</a>]</span> and <span id="id3">[<a class="reference internal" href="zreferences.html#id15" title="Helmert, F. R. Mathematical and Physical Theories of Higher Geodesy. Volume 1. Teubner, Leipzig, 1880. doi:10.5281/zenodo.32050.">Helmert1880</a>]</span>; the algorithm for +areas is based on <span id="id4">[<a class="reference internal" href="zreferences.html#id7" title="Danielsen, J. The area under the geodesic. Survey Review, 30(232):61–66, 1989. doi:10.1179/sre.1989.30.232.61.">Danielsen1989</a>]</span>. These improve on the work of +<span id="id5">[<a class="reference internal" href="zreferences.html#id41" title="Vincenty, T. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Survey Review, 23(176):88–93, 1975. doi:10.1179/sre.1975.23.176.88.">Vincenty1975</a>]</span> in the following respects:</p> +<ul class="simple"> +<li><p>The results are accurate to round-off for terrestrial ellipsoids (the +error in the distance is less than 15 nanometers, compared to 0.1 mm +for Vincenty).</p></li> +<li><p>The solution of the inverse problem is always found. (Vincenty’s +method fails to converge for nearly antipodal points.)</p></li> +<li><p>The routines calculate differential and integral properties of a +geodesic. This allows, for example, the area of a geodesic polygon to +be computed.</p></li> +</ul> +<p>Additional background material is provided in GeographicLib’s <a class="reference external" href="https://geographiclib.sourceforge.io/geodesic-papers/biblio.html">geodesic +bibliography</a>, +Wikipedia’s article “<a class="reference external" href="https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid">Geodesics on an ellipsoid</a>”, and <span id="id6">[<a class="reference internal" href="zreferences.html#id22" title="Karney, C. F. F. Geodesics on an ellipsoid of revolution. ArXiv e-prints, 2011. arXiv:1102.1215.">Karney2011</a>]</span> +(<a class="reference external" href="https://geographiclib.sourceforge.io/geod-addenda.html#geod-errata">errata</a>).</p> +</section> +</section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="resource_files.html" class="btn btn-neutral float-left" title="Resource files" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="development/index.html" class="btn btn-neutral float-right" title="Development" 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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