A simple geometric proof of Kepler's first Law

Abstract

We present a simple geometric proof of Kepler's first law. The core of the argument rests on a kinematic invariant that we call the velocity projection ratio - the ratio between the radial projection of the velocity and its projection onto a fixed direction parallel to the periapsis line. We show that the constancy of this ratio is an immediate consequence of Newton's laws, and that it leads directly to the focus-directrix property of the orbit. By establishing a direct link between planetary dynamics and the classical geometric definition of conics, our proof reveals the conic nature of the orbit as an inevitable consequence of the inverse-square law.