8). Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the z axis is aligned with the angular momentum vector L and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the y axis. Since the motion is planar, and p and L are perpendicular, pz = ηz = 0 and attention may be restricted to the three-dimensional vector . The family of Apollonian circles of momentum hodographs (Figure 7) correspond to a family of great circles on the three-dimensional sphere, all of which intersect the ηx axis at the two foci , corresponding to the momentum hodograph foci at px = ±p0. These great circles are related by a simple rotation about the ηx-axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another; however, such a rotation is orthogonal to the usual three-dimensional rotations, since it transforms the fourth dimension ηw. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector. An elegant action-angle variables solution for the Kepler problem can be obtained by eliminating the redundant four-dimensional coordinates in favor of elliptic cylindrical coordinates where sn, cn and dn are Jacobi's elliptic functions. Generalizations to other potentials and relativity The Laplace–Runge–Lenz vector can also be generalized to identify conserved quantities that apply to other situations. In the presence of a uniform electric field E, the generalized Laplace–Runge–Lenz vector is where q is the charge of the orbiting particle. Although is not conserved, it gives rise to a conserved quantity, namely . Further generalizing the Laplace–Runge–Lenz vector to other potentials and special relativity, the most general form can be written as where and , with the angle θ defined by and γ is the Lorentz factor. As before, we may obtain a conserved binormal vector B by taking the cross product with the conserved angular momentum vector These two vectors may likewise be combined into a conserved dyadic tensor W, In illustration, the LRL vector for a non-relativistic, isotropic harmonic oscillator can be calculated. Since the force is central, the angular momentum vector is conserved and the motion lies in a plane. The conserved dyadic tensor can be written in a simple form although p and r are not necessarily perpendicular. The corresponding Runge–Lenz vector is