## 5 Concluding Remarks
On the other hand, in geodesy, three latitudes are used: the parametric latitude \(u\), the geodetic latitude \(\Psi\), and the geocentric latitude \(\Psi\), which can be modified for a meridian ellipse by making its value oscillate between \([0,2\pi]\), thus making \(u\) coincide with the eccentric anomaly \(g\). The geodetic latitude is shown to coincide with the semifocal anomaly, and the modified geocentric latitude \(\Phi\) does not coincide with any of the known anomalies and is defined as a central anomaly. Once a sufficiently complete study of this variable has been carried out, we obtain the relationships between \(\Phi\) and the eccentric anomaly \(g\) in closed form. The partition function that relates the anomaly \(\Phi\) to the mean anomaly \(M\) is also obtained. One of the first results observed is that said anomaly, which is obviously symmetrical, does not belong to the biparametric family of anomalies. Despite this, it is possible to factorize the partition function as a function proportional to \(r\cdot h(R)\), which allows an interpretation similar to that of the anomalies of the biparametric family, which allows the method to be extended to anomalies whose function of partition is proportional to \(r^{\alpha}h(R)\), provided that they constitute an admissible reparameterization of the ellipse. In the numerical examples section, the effect of \(\gamma\) and \(\delta\) is observed for the case of the biparametric family. The factor \(r^{\gamma}\) causes a translation of the points from the apoapsis region to the periapsis region when a set of equally spaced points is taken in the corresponding anomaly. This translation is more significant the larger \(\gamma\) is. The factor \(R^{\delta}\) causes a symmetric distribution of the points with respect to the axes, bringing them closer to the apsidal region when \(\gamma>0\) and to the region of the orbit that cuts the semiminor axis when \(\delta<0\). Finally, a case of integration over one revolution of a highly eccentric artificial satellite is studied, interpreting the precision obtained following these guidelines, which is totally coherent. In this way, the main objective of the present work is achieved, which consists of finding a geometric explanation of why, among similar anomalies \(\alpha-\beta=cte\) some behave better than others, not only through empirical knowledge such as that currently developed by numerous authors, ourselves included, but through an analytical explanation that allows us to interpret, for example, why the antifocal anomaly produces less error in integration than the average anomaly; why the eccentric anomaly \(e\), the regularized arc length \(s^{*}\), the elliptical anomaly \(w\), and the geodesic anomaly \(\Psi\), all of which are symmetrical, produce different errors in the integration, etc. In this study, we provide a theoretical explanation for this, which considerably increases the consistency of the method by going from a simple empirical description of a biparametric family to being able to provide a causal explanation. As we have seen previously, the result can be extended to other families, although this should be addressed in the future. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript. This work has been partially supported by grant 16I358.01/1 of University Jaume I of Castellon. No new data were created or analyzed in this study. Data sharing is not applicable to this article. The authors declare that they have no conflicts of interest regarding this work.