  Penrose's idea of asymptotic flatness provides a framework for understanding
the asymptotic structure of gravitational fields of isolated systems at null
infinity. However, the studies of the asymptotic behaviour of fields near
spatial infinity are more challenging due to the singular nature of spatial
infinity in a regular point compactification for spacetimes with non-vanishing
ADM mass. Two different frameworks that address this challenge are Friedrich's
cylinder at spatial infinity and Ashtekar's definition of asymptotically
Minkowskian spacetimes at spatial infinity that give rise to the 3-dimensional
asymptote at spatial infinity $\mathcal{H}$. Both frameworks address the
singularity at spatial infinity although the link between the two approaches
had not been investigated in the literature. This article aims to show the
relation between Friedrich's cylinder and the asymptote as spatial infinity. To
do so, we initially consider this relation for Minkowski spacetime. It can be
shown that the solution to the conformal geodesic equations provides a
conformal factor that links the cylinder and the asymptote. For general
spacetimes satisfying Ashtekar's definition, the conformal factor cannot be
determined explicitly. However, proof of the existence of this conformal factor
is provided in this article. Additionally, the conditions satisfied by physical
fields on the asymptote $\mathcal{H}$ are derived systematically using the
conformal constraint equations. Finally, it is shown that a solution to the
conformal geodesic equations on the asymptote can be extended to a small
neighbourhood of spatial infinity by making use of the stability theorem for
ordinary differential equations. This solution can be used to construct a
conformal Gaussian system in a neighbourhood of $\mathcal{H}$.
