zenodoopen

Test set of geodesics on a triaxial ellipsoid

This is a set of 500000 shortest geodesics on a triaxial ellipsoid.  The ellipsoid is defined by $$\frac{X^2}{a^2} + \frac{Y^2}{b^2} + \frac{Z^2}{c^2} - 1 = 0,$$ with $a = \sqrt2$, $b = 1$, $c = 1/\sqrt2$ (measured in arbitrary units).  (This ellipsoid was studied by A. Cayley, On the geodesic lines on an ellipsoid, Mem. Roy. Astron. Soc. 39, 31-53, 1872.)  Each line of the test set consists of 10 space-delimited numbers <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> Here $\beta$, $\omega$, and $\alpha$, are the ellipsoidal latitude, longitude, and azimuth.  For a given $(\beta, \omega)$, the Cartesian coordinates of a point are $$\begin{align}   X &= a \cos\omega       \frac{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}}            {\sqrt{a^2 - c^2}}, \\   Y &= b \cos\beta \sin\omega, \\   Z &= c \sin\beta       \frac{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}}            {\sqrt{a^2 - c^2}}. \end{align}$$ Lines of constant $\beta$ and $\omega$ are orthogonal.  The azimuth $\alpha$ of a geodesic is the direction measured clockwise from North (defined as $\beta$ increasing at constant $\omega$).  The coordinates are singular at the four umbilical points $\cos\beta = \sin\omega = 0$.  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 leaving the umbilical point. The geodesics are computed using high-precision inverse calculations with the exact integer values for $(\beta_1, \omega_1)$ and $(\beta_2, \omega_1)$.  Any of the other entries reported as an integer is also exact. For most pairs of points, there is a unique shortest geodesic.  However <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> 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.  A non-trivial symmetry is given by swapping just the longitude coordinates; this also results in a geodesic of the same length.  The data set has had any such redundant geodesics removed. The data set is sorted according to whether either point <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> Approximately 85% of the entries are with two general points.  If only a small set of random test cases is needed, select a random subset with, e.g.,   <code>shuf Geod3Test.txt | head -1000 > Geod3Test-samp.txt</code>

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44/100

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Score breakdown

These five areas show where the dataset supports — or may limit — practical reuse.

Stewardship: 8
Harmonization: 4
Access: 20
Reuse readiness: 8
Engagement: 4

Topics

geodesic triaxial ellipsoid