## 5 Concluding Remarks  
In the present study, the relationship between the curvature of a point \(P\) of the orbit with the main curvatures of an ellipsoid of revolution generated by the rotation of the orbital ellipse around its semiminor axis has been addressed from a strictly geometric point of view. Geometric geodesy studies in depth, as a preliminary topic, the geometry of the ellipsoid of revolution, resulting that at a point \(P\), the directions of the main curvature are given by the curvature of the meridian ellipse that passes through the point \(\kappa_{1}\) and the normal direction of the parallel to the ellipsoid that passes through the given point. Since the parallel is not normal to the surface of the ellipsoid, the second principal curvature \(\kappa_{2}\) is determined by the intersection curve between the first vertical that passes through the point \(P\) and the ellipsoid. The inverses of the principal curvatures determine the radii of the main curvatures \(\rho_{1}\) and \(\rho_{2}\). From these curvatures and given a direction in \(P\) of geodesic azimuth \(A\), we have, without further ado than applying classical theory, the curvatures of the normal sections in \(P\), \(\kappa_{A}\) according to the directions corresponding to the different azimuths, as well as the corresponding radii of normal curvature \(\rho_{A}\). The expressions for the curvatures and radii of curvature above are somewhat complicated, as they contain fractional exponents. Next, the mean curvature is obtained as an integral mean of the normal curvature with respect to the different azimuths and the average radius of curvature, as well as an integral mean with respect to the different azimuths. The mean curvature and the mean radius of curvature are not inverse, so for simplicity and convenience, \(R\) is denoted by the mean radius of curvature, whose \(R^{2}\) turns out to be the inverse of the Gaussian curvature \(\kappa=\kappa_{1}\kappa_{2}\), an expression in which fractional powers no longer appear in its latitude-dependent part. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **M** & \(g\) & \(f\) & \(\tau^{*}\) & \(s^{*}\) & \(w\) & \(f^{\prime}\) & \(\Psi\) \\ \hline \(\alpha\) & \(0\) & \(1\) & \(2\) & \(1.5\) & \(0.5\) & \(1.5\) & \(1\) & \(2\) \\ \(\beta\) & \(0\) & \(0\) & \(0\) & \(0\) & \(-0.5\) & \(0.5\) & \(1\) & \(1\) \\ \hline \(||\Delta f||\) & \(9.54\times 10^{-40}\) & \(1.12\times 10^{-65}\) & \(9.49\times 10^{-10}\) & \(2.86\times 10^{-68}\) & \(4.51\times 10^{-64}\) & \(1.07\times 10^{-67}\) & \(2.60\times 10^{-40}\) & \(8.03\times 10^{-66}\) \\ \(||\Delta f||\) & \(7.71\times 10^{-30}\) & \(9.01\times 10^{-30}\) & \(3.56\times 10^{-11}\) & \(2.41\times 10^{-11}\) & \(3.64\times 10^{-67}\) & \(4.41\times 10^{-11}\) & \(2.10\times 10^{-40}\) & \(6.51\times 10^{-90}\) \\ \hline \(\gamma\) & \(0\) & \(1\) & \(2\) & \(1.5\) & \(1\) & \(1\) & \(0\) & \(1\) \\ \(\delta\) & \(0\) & \(0\) & \(0\) & \(0\) & \(-0.5\) & \(0.5\) & \(1\) & \(1\) \\ \hline \end{tabular}
\end{table} 
Table 1: Integration errors in position (Km) and velocity (Km/s) for different anomalies in the biparametric family \(\Psi_{\alpha,\beta}\) and in their equivalents in the representation \(\Phi_{\gamma,\delta}\). The calculation of the product of the radius vectors \(r\,r^{\prime}\) leads to an expression directly proportional to \(R\), so that the partition function of the biparametric family of anomalies \(Q(r)=K_{\alpha,\beta}r^{\alpha}r^{\prime\beta}\) can be written as \(Q(r)=C_{\gamma,\delta}r^{\prime\gamma}R^{\delta}\), where \(\gamma=\alpha-1\) and \(\delta=\beta\), which appears perfectly factored with respect to the radius vector of the secondary with respect to the primary \(r\) and the average radius of curvature of the ellipsoid at point \(P\).