Hypercomplex eccentric anomaly in the unified solution to the relative orbital motion

The present work offers an approach to the relative orbital motion by using hypercomplex algebra. An extension to this notion is used for vectors, by introducing the hypercomplex vector in the same way as hypercomplex numbers are defined. The solution to the relative orbital motion is offered in all possible situations (it stands for any Keplerian reference or targeted trajectories). A unified view on the relative orbital motion is suggested, by generalizing the previous approaches. The solution is offered to the nonlinear model of the relative motion and it is expressed in a coordinate-free hypercomplex vectorial closed form. The key element of this approach is the hypercomplex eccentric anomaly, introduced via a Sundman-like vectorial regularization.