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, p z = η z = 0 and attention may be restricted to the three-dimensional vector η η = ( η η w , η η x , η η y ) . The family of Apollonian circles of momentum hodographs (Figure 7) correspond to a family of great circles on the three-dimensional η η sphere, all of which intersect the η x axis at the two foci η x = ±1, corresponding to the momentum hodograph foci at p x = ± p 0. These great circles are related by a simple rotation about the η x -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 η w. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.