Projective quadrics and orbital elements

Abstract

Due to the fact that Keplerian orbits are conic sections, projective geometry gives a description of orbits based on projective hyperquadric properties. Using matrix algebra to describe hyperquadrics allows the construction of a new set of orbital elements using the eigenvalues and eigenvectors of the improper hyperquadric matrix and the construction of reference systems associated with the orbit. It is possible to work directly with the orbit in Cartesian coordinates in the three-dimensional space as a 4x4 matrix, constructing the quadric associated with the orbital conic.