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本数据集包含500000条三轴椭球（triaxial ellipsoid）上的最短测地线（geodesic）。该椭球由以下方程定义： $$frac{X^2}{a^2} + frac{Y^2}{b^2} + frac{Z^2}{c^2} - 1 = 0,$$ 其中$a = sqrt{2}$、$b = 1$、$c = 1/sqrt{2}$（单位任意）。该椭球曾由A. Cayley在1872年发表于《论椭球上的测地线》（On the geodesic lines on an ellipsoid, Mem. Roy. Astron. Soc. 39, 31-53, 1872）一文展开研究。 测试集的每一行由10个以空格分隔的数值组成，依次为： 1. 点1的纬度$eta_1$（单位：度，精确值） 2. 点1的经度$omega_1$（单位：度，精确值） 3. 点1的方位角$alpha_1$（单位：度，精度达$10^{-18}$度） 4. 点2的纬度$eta_2$（单位：度，精确值） 5. 点2的经度$omega_2$（单位：度，精确值） 6. 点2的方位角$alpha_2$（单位：度，精度达$10^{-18}$度） 7. 点1到点2的测地线距离$s_{12}$（单位，精度达$10^{-20}$） 8. 测地线的约化长度$m_{12}$（单位，精度达$10^{-20}$） 9. 测地线尺度$M_{12}$（精度达$10^{-20}$） 10. 测地线尺度$M_{21}$（精度达$10^{-20}$） 其中$eta$、$omega$与$alpha$分别为椭球纬度（ellipsoidal latitude）、椭球经度（ellipsoidal longitude）与方位角。给定$(eta, omega)$时，点的笛卡尔坐标可由下式计算： $$egin{align} X &= a cosomega frac{sqrt{a^2 - b^2sin^2eta - c^2cos^2eta}}{sqrt{a^2 - c^2}}, \ Y &= b coseta sinomega, \ Z &= c sineta frac{sqrt{a^2sin^2omega + b^2cos^2omega - c^2}}{sqrt{a^2 - c^2}}. end{align}$$ 恒定$eta$与$omega$的曲线相互正交。测地线的方位角$alpha$指以正北（定义为$omega$固定时$eta$增大的方向）为基准顺时针测量的方向。该坐标在四个脐点（umbilical point）$coseta = sinomega = 0$处存在奇点。测地线经过此类点时，方位角会发生$pmfrac{1}{2}pi$的跳变，此时记录的方位角为离开脐点后的数值。 本数据集的测地线通过高精度反演计算得到，其中$(eta_1, omega_1)$与$(eta_2, omega_2)$的取值为精确整数。其余以整数形式报告的数值同样为精确值。 对于绝大多数点对，仅存在唯一的最短测地线。但存在以下例外： 1. 对于对跖脐点，$alpha_1$与$alpha_2$可取任意值，仅需满足比值$ analpha_1/ analpha_2$固定； 2. 若$eta_1 + eta_2 = 0$且$cosalpha_1$与$cosalpha_2$符号相反，则存在另一条最短测地线，其方位角分别为$pi - alpha_1$与$pi - alpha_2$。 对于特定的$(eta_1, omega_1)$与$(eta_2, omega_2)$，可通过交换两点或关于任意坐标平面对称的方式，轻松生成等长的额外测地线。另一种非平凡对称操作是仅交换经度坐标，同样可得到等长的测地线。本数据集已移除所有此类冗余测地线。 数据集按照两点的类型进行排序，分类标准如下： 1. 任意一点为脐点 2. 任意一点位于中间主椭圆且$sinomega = 0$ 3. 任意一点位于中间主椭圆且$coseta = 0$ 4. 任意一点位于长轴主椭圆且$Z = 0$ 5. 任意一点位于短轴主椭圆且$X = 0$ 6. 任意一点靠近脐点 7. 一般点（其余所有点） 约85%的条目对应两个一般点。若仅需少量随机测试用例，可随机选取子集，例如执行以下命令： shuf Geod3Test.txt | head -1000 > Geod3Test-samp.txt