Initial Orbit Determination (IOD) is the classical problem of estimating the orbit of a body in space without any presumed information about the orbit. The geometric formulation of the “angles-only” IOD in three-dimensional space: find a conic curve with a given focal point meeting the given lines of sight (LOS). We provide an algebraic reformulation of this problem and confirm that five is the minimal number of lines necessary to have a finite number of solutions in a non-special case, and the number of complex solutions is 66. We construct a subdivision method to search for the normal direction to the orbital plane as a point on the real projective plane. The resulting algorithm is fast as it discovers only a handful of the solutions that are real and physically meaningful.