Orbits of the Kepler problem via polar reciprocals

Polar reciprocals of trajectories are an elegant alternative to hodographs for motion in a central force field. Their principal advantage is that the transformation from a trajectory to its polar reciprocal is its own inverse. The form of the polar reciprocals of Kepler orbits is established, and 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 extended by Derbes, the method based on polar reciprocals works without changes for elliptical, parabolic, and hyperbolic trajectories.