In this paper, the relative orbit elements (OREs) are strictly defined using the spherical geometry method. The accurate transformation between relative orbit elements and absolute orbit elements (AOEs) is deduced. Under the elliptical and circular reference orbit relative motion conditions, the approximate transformation equations between relative orbit elements and absolute orbit elements are derived to meet the needs of different missions. Some characteristics of the relative orbit elements are also analyzed for the case of close relative motion. A new relative linear model is derived based on the relative orbit elements with no singularity, which is suitable for close relative motion under elliptical or circular reference orbit conditions. The characteristics of elliptical relative motion are analyzed. Then an error analysis is carried out through a comparison of near-circular relative motion equations with the results of exact solutions. The proposed method and conclusions are validated by simulation through some typical examples. The study improves the basic theory of relative orbit elements, and unifies the expressions of elliptical and near-circular close relative motion.