A common way to put coordinates on $\mathbb P^k\mathbb R$ is to choose $k+2$ points (such that no one of them lies on the hyperplane generated by any $k$ of the others) and interpret them as the origin $(0, 0, \dots, 0, 1)$, the unit versors $(1, 0, \dots, 0, 1)$, $(0, 1, \dots, 0, 1)$, ..., $(0, 0, \dots, 1, 1)$ and the unity point $(1, 1, \dots, 1, 1)$ of an Euclidean space $\mathbb R^k \subset \mathbb P^k \mathbb R$ (with the immersion $(x_1, \dots, x_k) \mapsto (x_1, \dots, x_k, 1)$).
Another way to specify a projective reference is to take the points corresponding to the origin and the unit versors of $\mathbb R^k$ and specify furthermore the coordinate infinity points $(1, 0, \dots, 0, 0)$, $(0, 1, \dots, 0, 0)$, ..., $(0, 0, \dots, 1, 0)$, with of course the prescription that each of them lies on the line connecting the origin with the appropriate unit versor (and does not coincide with any of them).
Suppose now that you want to pass from one system to the other using only synthetic methods (that is, just by computing spans and intersections of projective subspaces). This means that you want to recover the unity point knowing the coordinate infinity points (and the origin and unit versors) and viceversa.
It is easy to build the unity point: you take the intersection of all the $k$ planes spanned by $k-1$ coordinate infinity points and the unit versor corresponding to the missing direction.
However, I could not devise a synthetic construction to obtain the coordinate infinity points from the unity point. Is anybody able to provide one? (or provide some proof that there cannot be one, although that would appear really strange to me)
Remark. If you have the coordnate infinity points and the unit versor, the origin is redundant (since all the axes can be recovered directly and then intersected). Here I am not bothering with optimality, uniqueness or things like that, I am just curious of how to construct the coordinate infinity points.