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Test set of geodesics on a triaxial ellipsoid

<p>This is a set of 500000 shortest geodesics on a triaxial ellipsoid. &nbsp;The ellipsoid is defined by</p> <p>$$\frac{X^2}{a^2} + \frac{Y^2}{b^2} + \frac{Z^2}{c^2} - 1 = 0,$$</p> <p>with \(a = \sqrt2\), \(b = 1\), \(c = 1/\sqrt2\) (measured in arbitrary units). &nbsp;(This ellipsoid was studied by A. Cayley, <em>On the geodesic lines on an ellipsoid</em>, Mem. Roy. Astron. Soc. <strong>39</strong>, 31-53, 1872.) &nbsp;Each line of the test set consists of 10 space-delimited numbers</p> <ul> <li>the latitude at point 1, \(\beta_1\) (\(^\circ\), exact)</li> <li>the longitude at point 1, \(\omega_1\) (\(^\circ\), exact)</li> <li>the azimuth at point 1, \(\alpha_1\) (\(^\circ\), accurate to \(10^{-18}{}^\circ\))</li> <li>the latitude at point 2, \(\beta_2\) (\(^\circ\), exact)</li> <li>the longitude at point 2, \(\omega_2\) (\(^\circ\), exact)</li> <li>the azimuth at point 2, \(\alpha_2\) (\(^\circ\), accurate to \(10^{-18}{}^\circ\))</li> <li>the geodesic distance from 1 to 2, \(s_{12}\) (units, accurate to \(10^{-20}\))</li> <li>the reduced length of the geodesic, \(m_{12}\) (units, accurate to \(10^{-20}\))</li> <li>the geodesic scale, \(M_{12}\) (accurate to \(10^{-20}\))</li> <li>the geodesic scale, \(M_{21}\) (accurate to \(10^{-20}\))</li> </ul> <p>Here \(\beta\), \(\omega\), and \(\alpha\), are the&nbsp;<em>ellipsoidal</em> latitude, longitude, and azimuth. &nbsp;For a given \((\beta, \omega)\), the Cartesian coordinates of a point are</p> <p>$$\begin{align}<br>&nbsp; X &amp;= a \cos\omega<br>&nbsp; &nbsp; &nbsp; \frac{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}}<br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;{\sqrt{a^2 - c^2}}, \\<br>&nbsp; Y &amp;= b \cos\beta \sin\omega, \\<br>&nbsp; Z &amp;= c \sin\beta<br>&nbsp; &nbsp; &nbsp; \frac{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}}<br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;{\sqrt{a^2 - c^2}}.<br>\end{align}$$</p> <p>Lines of constant \(\beta\) and \(\omega\) are orthogonal. &nbsp;The azimuth \(\alpha\) of a geodesic is the direction measured clockwise from North (defined as \(\beta\) increasing at constant \(\omega\)). &nbsp;The coordinates are singular at the four <em>umbilical</em> points \(\cos\beta = \sin\omega = 0\). &nbsp;The azimuth of a geodesic jumps by \(\pm\frac12\pi\) on passage through such points and the value for such points is the azimuth on <em>leaving</em> the umbilical point.</p> <p>The geodesics are computed using high-precision inverse calculations with the exact&nbsp;<em>integer</em> values for \((\beta_1, \omega_1)\) and \((\beta_2, \omega_1)\). &nbsp;Any of the other entries reported as an integer is also exact.</p> <p>For most pairs of points, there is a unique shortest geodesic. &nbsp;However</p> <ul> <li>for opposite umbilical points, \(\alpha_1\) and \(\alpha_2\) can take on arbitrary values provided that the ratio \(\tan\alpha_1/\tan\alpha_2\) is maintained;</li> <li>if \(\beta_1 + \beta_2 = 0\) and if \(\cos\alpha_1\) and \(\cos\alpha_2\) have opposite signs, then there is another shortest geodesic with azimuths \(\pi - \alpha_1\) and \(\pi - \alpha_2\).</li> </ul> <p>For a particular \((\beta_1, \omega_1)\) and \((\beta_2, \omega_2)\), additional geodesics of the same length can be trivially generated by swapping the points or by reflecting them in any of the coordinate planes. &nbsp;A non-trivial symmetry is given by swapping <em>just</em> the longitude coordinates; this also results in a geodesic of the same length. &nbsp;The data set has had any such redundant geodesics removed.</p> <p>The data set is sorted according to whether either point</p> <ul> <li>is an umbilical point</li> <li>lies on the middle principal ellipse, with \(\sin\omega = 0\)</li> <li>lies on the middle principal ellipse, with \(\cos\beta = 0\)</li> <li>lies on the major principal ellipse, \(Z = 0\)</li> <li>lies on the minor principal ellipse, \(X = 0\)</li> <li>is near an umbilical point</li> <li>is general (all other points)</li> </ul> <p>Approximately 85% of the entries are with two general points. &nbsp;If only a small set of random test cases is needed, select a random subset with, e.g.,</p> <p>&nbsp; <code>shuf Geod3Test.txt | head -1000 &gt; Geod3Test-samp.txt</code></p> <p>&nbsp;</p>