  It is argued that, for motion in a central force field, polar reciprocals of
trajectories are an elegant alternative to hodographs. The principal advantage
of polar reciprocals is that the transformation from a trajectory to its polar
reciprocal is its own inverse. The form of polar reciprocals $k_*$ of Kepler
problem orbits is established, and then the orbits $k$ themselves are shown to
be conic sections using the fact that $k$ is the polar reciprocal of $k_*$. A
geometrical construction is presented for the orbits of the Kepler problem
starting from their polar reciprocals. No obscure knowledge of conics is
required to demonstrate the validity of the method. Unlike a graphical
procedure suggested by Feynman (and amended by Derbes), the algorithm based on
polar reciprocals works without alteration for all three kinds of trajectories
in the Kepler problem (elliptical, parabolic, and hyperbolic).
