 For representing horizontal positions on Earth, the ellipsoid is a reference ellipsoid and the vector is decomposed in an Earth-centered Earth-fixed coordinate system.It behaves smoothly at all Earth positions, and it holds the mathematical one-to-one property.More in general, the concept can be applied to representing positions on the boundary of a strictly convex bounded subset of "k"-dimensional Euclidean space, provided that that boundary is a differentiable manifold.