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A small variation of the circular shape of the hodograph theorem states thatfor every elliptical solution of the two-body problem, it is possible to findan appropriate inertial frame such that the speed of the bodies is constant. Weuse this result and data from the NASA JPL Horizon Web Interface to find thebest fitting ellipse for the trajectory of Mercury, Venus, Earth, Mars, andJupiter. The process requires us to find procedures to obtain the plane andellipse that best fit a collection of points in space. We show that if we aimfor the plane that minimizes the sum of the square distances from the givenpoints to the unknown plane, we obtain three planes that appear to divide theset of points equally into octants, one of these being our desired plane ofbest fit. We provide a detailed proof of the hodograph theorem.

A Small Variation of the Circular Hodograph Theorem and the Best Elliptical Trajectory of the Planets