## 4 Numerical Examples  
To illustrate the previously obtained results with respect to the new geometric interpretation that transforms the anomaly \(\Psi_{\alpha,\beta}\) into its equivalent \(\Phi_{\gamma,\delta}\) we will first proceed to represent a set of 20 equally spaced points \(\Phi_{\gamma,\delta}\). For this, we will take as an example a satellite with eccentricity \(e=0.7\) and \(M\) corresponding to \(\gamma=0\), \(\delta=0\); \(g\) corresponding to \(\gamma=1\), \(\delta=0\); \(\tau\) corresponding to \(\gamma=1.5\), \(\delta=0\); and \(f\) corresponding to \(\gamma=1.5\), \(\delta=0\). In Figure 4, it can be seen that the factor \(r^{\gamma}\) causes a displacement of points from the region corresponding to the apoapsis, that is, from the left part of the ellipse to the region of the periapsis, the right part of the ellipse. This fact implies a distribution of points more in line with the dynamics, which is reflected in better integration values. Secondly, if the value of \(\gamma\) is fixed, we will focus on the example in Figure 5 in the case \(\gamma=1\) corresponding to symmetric anomalies. This choice is to show the action of \(\gamma\) more clearly. Also, in Figure 6, twenty uniformly distributed points for the eccentric anomaly and the central anomaly are presented on the same graph, and it can be seen that the big black points are more concentrated towards the periapsis and apoapsis regions than those corresponding to the central anomaly. The smaller red dots are more concentrated in the regions corresponding to the intersection of the orbit with respect to the semiminor axis, so it is expected that for large eccentricities, the central anomaly will provide worse precision in the study of orbital motion than the eccentric anomaly. To interpret the effect on the errors in numerical integration, we proceed to integrate the old artificial satellite Heos II, which is performed in order to compare our results with those of other authors cited above who used this satellite as a test. The satellite's orbital elements are given by \(a=118,363.47\) km, \(e=0.942572319\), \(i=28^{\circ}.16096\), \(\Omega=185^{\circ}.07554\), \(\omega=270^{\circ}.07151\), \(M0=0^{\circ}\). The period of the satellite is \(4.69\) days and for the Earth the spaceflight constant \(\mu=GM=3.986005\cdot 10^{5}\) km\({}^{3}\)-s\({}^{-2}\). Finally, we will analyze the integration errors for various anomalies in the biparametric family according to the new interpretation based on \(\Phi_{\gamma,\delta}=\Psi_{\alpha,\beta}\). The results obtained are listed in Table 1, which is, in essence, the same Table 1 that appears in [16] corrected, to which the column corresponding to the semifocal anomaly  
Figure 5: Point distribution for \(e=0.7\), \(g,s,w,\Psi\). Figure 6: Point distribution for \(e=0.7\); \(g\), big black points; \(\Phi\), small red points. has been added, and the one that has been added in the lower part the values \((\gamma,\delta)\) corresponding to the respective \((\alpha,\beta)\). Table 1 shows the errors regarding the position and initial velocity made in the numerical integration using \(10,000\) equally spaced points, taking different anomalies as integration variables. If the table with value \(\gamma=0\) is observed, the mean anomaly \(M\) and the antifocal anomaly \(f^{\prime}\) appear, which are differentiated by the value of \(\delta\), which in the case of the mean anomaly takes a value of \(0\) and in the case of the antifocal anomaly takes a value of \(1\). It can be seen in Figure 6 that the distribution of points differs in both cases, concentrating more points in the periapsis region, and correspondingly in the apoapsis region in the case of the antifocal anomaly, which is reflected in the results. In the case of \(\gamma=1\), which corresponds to symmetric anomalies, if the eccentric anomaly \(g\) is taken as a basis, the regularized arc length \(s^{*}\)\(\delta\) takes the value \(-0.5\), which moves the points away from the apsidal regions, causing worse results.