  Parreau compactified the Hitchin component of a closed surface $S$ of
negative Euler characteristic in such a way that a boundary point corresponds
to the projectivized length spectrum of an action of $\pi_1(S)$ on an $\mathbb
R$-Euclidean building. In this paper, we use the positivity properties of
Hitchin representations introduced by Fock and Goncharov to explicitly describe
the geometry of a preferred collection of apartments in the limiting building.
