Solving the Gibbs Problem with Algebraic Projective Geometry

Abstract: Orbit determination (OD) from three position vectors is one of the classical problems in astrodynamics. Early contributions to this problem were made by J. Willard Gibbs in the late 1800s and OD of this type is known today as ``Gibbs Problem''. There are a variety of popular solutions to the Gibbs problem. While some authors solve for the orbital elements directly, most contemporary discussions are based on a vector analysis approach inspired by Gibbs himself. This work presents a completely different solution to those just described. Although there is nothing wrong with the vector analysis approach, some interesting insights may be gained by considering the problem from the perspective of algebraic projective geometry. Such an algebraic solution is presented here. The OD procedure is based upon a novel and geometrically meaningful solution to the algebraic fitting of an ellipse with a focus at the origin using only three points. Although the final OD result is identical to the classical vector analysis approach pioneered by Gibbs, this new algebraic solution is interesting in its own right.