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

• the latitude at point 1, \(\beta_1\) (\(^\circ\), exact)
• the longitude at point 1, \(\omega_1\) (\(^\circ\), exact)
• the azimuth at point 1, \(\alpha_1\) (\(^\circ\), accurate to \(10^{-18}{}^\circ\))
• the latitude at point 2, \(\beta_2\) (\(^\circ\), exact)
• the longitude at point 2, \(\omega_2\) (\(^\circ\), exact)
• the azimuth at point 2, \(\alpha_2\) (\(^\circ\), accurate to \(10^{-18}{}^\circ\))
• the geodesic distance from 1 to 2, \(s_{12}\) (units, accurate to \(10^{-20}\))
• the reduced length of the geodesic, \(m_{12}\) (units, accurate to \(10^{-20}\))
• the geodesic scale, \(M_{12}\) (accurate to \(10^{-20}\))
• the geodesic scale, \(M_{21}\) (accurate to \(10^{-20}\))

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

• 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;
• 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 \(\pi - \alpha_2\).

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

• is an umbilical point
• lies on the middle principal ellipse, with \(\sin\omega = 0\)
• lies on the middle principal ellipse, with \(\cos\beta = 0\)
• lies on the major principal ellipse, \(Z = 0\)
• lies on the minor principal ellipse, \(X = 0\)
• is near an umbilical point
• is general (all other points)

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.,

  shuf Geod3Test.txt | head -1000 > Geod3Test-samp.txt