## 2 The Biparametric Family of Anomalies as a Function of Vector Radius and Curvature
To construct the corresponding normal section, the first vertical must intersect with the ellipsoid, and from their curvatures [4; 21], it turns out that:  
\[\kappa_{2}=\frac{\sqrt{1-e^{2}\sin^{2}\Psi}}{a},\quad\rho_{2}=\frac{a}{\sqrt {1-e^{2}\sin^{2}\Psi}}. \tag{7}\]  
For a normal section whose direction in \(P\) has azimuth \(A\) measured from the north, it is verified that its curvature and its radius of curvature are given [21] by:  
\[\kappa_{A}=\kappa_{1}\cos^{2}A+\kappa_{2}\sin^{2}A,\quad\rho_{A}=\frac{\kappa _{1}\kappa_{2}}{\kappa_{1}\sin^{2}A+\kappa_{2}\cos^{2}A}. \tag{8}\]  
It is also of interest to calculate the average curvature \(\kappa_{m}\) and the average radius of curvature \(\rho_{m}\), which leads to the quantities [21]  
\[\kappa_{m}=\frac{4}{2\pi}\int_{0}^{\pi/2}\kappa_{A}\,dA=\frac{1}{2}(\kappa_{1 }+\kappa_{2}). \tag{9}\]  
If, in addition, we average the radii of normal curvatures following directions of different azimuths \(A\) at a point \(P\)[21], we have that:  
\[\rho_{m}=\frac{4}{2\pi}\int_{0}^{\pi/2}\rho_{A}\,dA=\frac{4}{2\pi}\int_{0}^{ \pi/2}\frac{\kappa_{1}\kappa_{2}}{\kappa_{1}\sin^{2}A+\kappa_{2}\cos^{2}A}\, dA=\sqrt{\rho_{1}\rho_{2}}\,. \tag{10}\]  
Notice that the relation \(\rho_{m}\kappa_{m}=1\) is not satisfied. For this reason and to avoid confusion, we will use the notation \(R=\rho_{m}\), since, as we have seen, it is not the inverse of the mean curvature. Taking into consideration the Gaussian curvature of a surface \(K((P)=\kappa_{1}(P)\kappa_{2}(P)\), which is an intrinsic magnitude, a result known as [PERSON]'s theorem egregium [22], we have that \(R\sqrt{K(P)}=1\) and, therefore, defining \(\kappa\) as  
\[\kappa=\sqrt{K(P)} \tag{11}\]  
it is verified that  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]0  
To obtain the mean radius of curvature \(R\) as a function of the geodetic latitude, just replace (6) and (7) in (10), from which we obtain:  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]1  
On the other hand, as will be seen in the next section, the geodetic latitude coincides with the semifocal anomaly. Therefore, the sine of the latitude \(\Psi\) is expressed as a function of the eccentric anomaly \(g\)[14] as:  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]2Replacing (14) in (13), we have:  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]3  
Finally, we obtain that the product of the radius vectors \(r\) and \(r^{\prime}\) given by (4) is given by:  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]4  
Comparing this last expression with (15), we obtain the fundamental relation:  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]5  
Given an anomaly in the biparametric family (2) \(\Psi_{a,\beta}\), we can define an equivalent anomaly \(\Phi_{\gamma,\delta}\) whose relationship to the mean anomaly is given by:  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]6  
where \(\gamma=\alpha-1\), \(\delta=\beta\), \(C_{\gamma,\delta}=a^{\beta}(1-e^{2})^{\beta/2}K_{a,\beta}\). In this way, we have the anomaly \(\Phi_{\gamma,\delta}=\Psi_{a,\beta}\) but whose connection with the mean anomaly is given by (18), where a normalization constant appears. The radius vector \(r\) is raised to an exponent \(\gamma\) times the mean radius of curvature of the ellipsoid of revolution that contains the orbit as a meridian ellipse \(R\) raised to \(\delta\). This fact is of great importance because of the effect that the factor \(r^{\gamma}\) causes when we work with a constant step number, namely, a translation of the points from the apoapsis region, in the case of working with \(M\), to more concentrated points in the periapsis region; it is more remarkable as \(\gamma\) increases. In the case of \(\gamma=1\), we are faced with a symmetrical anomaly with respect to the axes of the orbital ellipse.