  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.
