## 2 The Biparametric Family of Anomalies as a Function of Vector Radius and Curvature  
This section studies the relationship between the product of the radii vectors \(r\) and \(r^{\prime}\) of a point \(P\), which represents the secondary point in the elliptic case of the two-body problem, where \(r\) represents the distance to the main focus where the primary is located and \(r^{\prime}\) the distance to the secondary focus of the ellipse. These magnitudes are reflected in Figure 1, where the true anomaly \(f\) and the antifocal anomaly \(f^{\prime}\) also appear. The radii vectors \(r\), \(r^{\prime}\), since the orbit is an ellipse, are related by  
\[r+r^{\prime}=2a. \tag{3}\]  
Taking into account the relationship between the radius vectors as a function of the eccentric anomaly in the two-body problem and (3), we immediately have:  
\[r=a(1-e\cos g)\,,\quad r^{\prime}=a(1+e\cos g)\,. \tag{4}\]  
In order to find simple geometric relationships, we will generate an ellipsoid of revolution from the ellipse by rotating the orbit around its semiminor axis. In this ellipsoid, we will take the meridian of zero longitude as the one containing the periapsis of the orbit. It is enough to obtain its longitude and latitude to determine the position of a point \(P\) on the ellipsoid. In the case of points located on the orbit whose rotation generates the ellipsoid and the origin of whose length is fixed, it is more convenient to dispense with the length and take a modified length that takes values between \([0,2\pi]\), which does not present any difficulty and characterizes all the points \(P\) on the orbit. Figure 2 shows the variables \(u\), \(\Psi\), and \(\Phi\)[4; 21], called in geodesy the parametric or reduced latitude, geodesic latitude, and geocentric latitude. When we restrict ourselves to the orbit generated by the ellipsoid, as stated above, we will call them the eccentric anomaly \(g\) in the case of \(u\), the geodesic anomaly in the case of \(\Psi\), and the geocentric anomaly in the case of \(\Phi\). In the next section, it will be shown that the geodetic anomaly coincides with the semifocal anomaly in elliptical motion. It will be enough to study the main curvature and the mean and Gaussian curvature in the ellipse's first quadrant for reasons of symmetry. Firstly, given a non-umbilical point \(P\)[22] on the first quadrant, we will consider a coordinate system \(OXYZ\) so that \(O\) is located in the center of the orbit and therefore of the ellipsoid, the axis \(OX\) runs to the peripsis, the axis \(OY\) is in the direction of the semiminor axis of the ellipse, and the axis \(OZ\) forms a direct trihedron with \(O\) and \(OY\). This notation, which is different from that used in geodesy, is used in celestial mechanics when we take the orbital plane as the \(OXY\) plane and place \(O\) in the center of the orbit. The relationship between the different latitudes [4; 21] is determined by:  
\[\frac{1}{\sqrt{1-e^{2}}}\tan\Psi=\tan u=\sqrt{1-e^{2}}\tan\Phi\,. \tag{5}\]  
In the ellipse that defines the equation of the orbit, the main directions of curvature are given by the direction of the meridian and that of the parallel. In the first case, the ellipse  
Figure 2: Section of revolution ellipsoid containing the orbit. Parametric \(u\), geodetic \(\Psi\), and geocentric \(\Phi\) latitudes. itself is normal to the surface of the ellipse; its curvature will be the main curvature \(\kappa_{1}\), and the corresponding radius, which we will denote by \(\rho_{1}\), is given by [4; 21]:  
\[\kappa_{1}=\frac{(1-e^{2}\sin^{2}\Psi)^{3/2}}{a(1-e^{2})},\quad\rho_{1}=\frac{a (1-e^{2})}{(1-e^{2}\sin^{2}\Psi)^{3/2}}\,. \tag{6}\]  
On the other hand, the parallel is not a normal section of the ellipsoid.