## 3 Other Symmetric Variables Not Belonging to the Biparametric Family
From the previous equation, we have:  
\[\rho=a\sqrt{1-e^{2}\sin^{2}g}. \tag{22}\]  
On the other hand, from the equation of the ellipse \(\frac{X^{2}}{a^{2}}+\frac{Y^{2}}{b^{2}}=1\), \(\rho\) is obtained as a function of \(\Phi\) as:  
\[\rho=\frac{a\sqrt{1-e^{2}}}{\sqrt{1-e^{2}\sin^{2}\Phi}}. \tag{23}\]  
Figure 3: Relation between semifocal and geodetic anomalies. From (20), (21), and (22), it follows that:  
\[\cos\Phi=\frac{\cos g}{\sqrt{1-e^{2}\sin^{2}g}},\quad\sin\Phi=\frac{\sqrt{1-e^{2} }\sin g}{\sqrt{1-e^{2}\sin^{2}g}}. \tag{24}\]  
From this equation, and after some elemental calculations, we obtain:  
\[\cos g=\frac{\sqrt{1-e^{2}}\cos\Phi}{\sqrt{1-e^{2}}\cos^{2}\Phi},\quad\sin g= \frac{\sin\Phi}{\sqrt{1-e^{2}}\cos^{2}\Phi}. \tag{25}\]  
From (24) or (25), we immediately have:  
\[\tan g=\frac{\tan\Phi}{\sqrt{1-e^{2}}}. \tag{26}\]  
Deriving (26), and after some calculations, we have:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]0  
and taking into account (22), the following relationship is obtained:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]1  
and considering [PERSON]'s equation for the eccentric anomaly:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]2  
where \(M\) is the mean anomaly. It is obtained by derivation:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]3  
Finally, from (30), (28), and (4), we obtain:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]4  
where, from (22) and (31), the partition function \(Q(r)=\frac{r\rho^{2}}{a^{3}\sqrt{1-e^{2}}}\), and after some calculations, we can express \(Q(r)\) as:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]5  
It is evident that the central anomaly \(\Phi\) is not in the biparametric family of anomalies. This is not new, since the intermediate radials of Cid [17], the natural family of anomalies [19], and the geometric family of anomalies [20] are not in general in the biparametric family, but they are in the symmetrical case since they coincide, respectively, with the semifocal anomaly and with the eccentric anomaly. Therefore, we have a symmetrical anomaly that does not belong to the biparametric family. However, it is possible to represent (32) in the form:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]6  
from which we have \(Q(r)=r\,h(R)\), where \(h(R)\) is a function of \(R\) given by \(h(R)=\frac{2-e^{2}}{a\sqrt{1-e^{2}}}-\frac{R}{a^{2}}\). This fact can be generalized to \(\Theta\) reparameterizations:  
\[\xi=a(e-\cos g),\quad\eta=a\sqrt{1-e^{2}}\sin g, \tag{19}\]7  
where the partition function can be written as the product of a power of \(r\) times a function of the mean radius of curvature of the ellipsoid of revolution generated by the orbital ellipse. Note that, on the one hand, the normalization factor has been included in the function \(h\) and, on the other hand, \(dM=Q(r)d\Theta\) must be an admissible reparametrization. As a final consideration in this section, it should be noted that \(r^{\delta}\) plays the same role as in the case of symmetric anomalies and \(h(R)\) plays a role similar to that of \(R^{\delta}\). However, in the latter case, there are more types of symmetrical motions of the points with respect to the semi-axes than those given by the term \(R^{\delta}\), which appears in the case of the biparametric family.