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<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>
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