Hodograph
The hodograph is a curve traced by the tip of the velocity vector as a body moves. For Keplerian orbits this hodograph is always a perfect circle — a beautiful hidden symmetry discovered by Hamilton in 1847, related to the Laplace-Runge-Lenz vector.
About the Hodograph
Hamilton's discovery (1847): For any inverse-square law (gravity, electrostatics), the hodograph — the curve of velocity vectors — is a perfect circle. This is equivalent to the conservation of the Laplace-Runge-Lenz vector, which points from the focus to the periapsis and has constant magnitude e (the eccentricity).
The hodograph circle has radius k/(L) where k = GMm and L is angular momentum. Its center is offset from the origin by e·k/L. For a circular orbit (e = 0) the hodograph is centered at the origin; for elliptical orbits the center shifts, and the particle's velocity vector traces the circle at varying rates.
Left panel: the elliptical orbit in position space. The star sits at one focus. Right panel: velocity space. Watch the velocity vector tip trace a perfect circle — even though the orbit is an ellipse, not a circle.