Orbits of the Kepler problem via polar reciprocals

TL;DR

This paper introduces polar reciprocals as an elegant and inverse transformation method to analyze Kepler orbits, providing a geometric construction that works for all orbit types without complex conic knowledge.

Contribution

It establishes the form of polar reciprocals for Kepler orbits and demonstrates a universal geometric method for deriving these orbits from their reciprocals.

Findings

Polar reciprocals of Kepler orbits are conic sections.

The method works for elliptical, parabolic, and hyperbolic trajectories.

The geometric construction is simple and does not require advanced conic knowledge.

Abstract

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).