Reinventing the wheel: Hodographic solutions to the Kepler problems

There are two Kepler problems: given the inverse-square law, find the trajectories; or, given Kepler’s laws, find the inverse-square law. Traditionally these problems are solved in the classroom via calculus, but the amount of calculus needed may be prohibitively high for a first-year course. Alternative solutions to the Kepler problems have been discovered, forgotten, and rediscovered for centuries. Many of these employ Hamilton’s hodograph, a graphical representation of an object’s velocity. This article demonstrates hodographic solutions to the Kepler problems, including an algorithm for the construction of parabolic trajectories.