In this chapter we return to the Kepler problem, which we are now going to study with a variety of geometric techniques. We shall encounter geometric transformations such as inversions and polar reciprocation, new types of spaces such as hyperbolic space or the projective plane, and differential geometric concepts such as geodesics and curvature.

In Section 8.1 we prove a theorem due to Hamilton, which says that the velocity vector of a Kepler solution traces out a circle (or an arc of a circle).

A remarkable consequence of this observation will be described in Sections 8.3 and 8.5 (for the elliptic case and the hyperbolic/parabolic case, respectively). By a geometric transformation known as inversion, one can set up a one-to-one correspondence between the velocity curves of Kepler solutions and geodesics, i.e. locally shortest curves, in spherical, hyperbolic or euclidean geometry. Not only does this permit a unified view of all solutions of the Kepler problem, including the regularised collision solutions, it also gives a natural interpretation of the eccentric anomaly as the arc length parameter of the corresponding geodesic. Sections 8.2 on inversion and 8.4 on hyperbolic geometry provide the necessary geometric background.

In Section 8.6, Hamilton's theorem is taken as the basis for an alternative proof of Kepler's first law. The geometric concepts behind this proof include polar reciprocals and the curvature of planar curves. In Section 8.6.5 I introduce the projective plane, which is the apposite setting for polar reciprocation.

In Section 8.7 it is shown that suitable holomorphic transformations of the complex plane lead to a duality between Kepler solutions on the one hand and solutions of Hooke's law for springs on the other. Since the latter are much easier to determine, this leads to yet another proof of Kepler's first law. A more geometric interpretation of this duality will follow in Section 9.1.