Rotor coordinates, vector trigonometry and Kepler-Newton orbits

Abstract: 

Rotor coordinates provide a new algebraic alternative to polar coordinates for describing kinematic motion. Here we provide an introduction to this 'vector trigonometry', relating it to the usual 'tan of half the angle' substitution in calculus, and using it to derive some basic facts about motion in fixed and moving reference frames.

These ideas are then used for an alternative approach to one of the key derivations of modern science: Newton's explanation of Kepler's elliptic orbits for a body moving in a central inverse square law force field.