  We numerically study the geometry of a driven elastic string at its
sample-dependent depinning threshold in random-periodic media. We find that the
anisotropic finite-size scaling of the average square width $\bar{w^2}$ and of
its associated probability distribution are both controlled by the ratio
$k=M/L^{\zeta_{\mathrm{dep}}}$, where $\zeta_{\mathrm{dep}}$ is the
random-manifold depinning roughness exponent, $L$ is the longitudinal size of
the string and $M$ the transverse periodicity of the random medium. The
rescaled average square width $\bar{w^2}/L^{2\zeta_{\mathrm{dep}}}$ displays a
non-trivial single minimum for a finite value of $k$. We show that the initial
decrease for small $k$ reflects the crossover at $k \sim 1$ from the
random-periodic to the random-manifold roughness. The increase for very large
$k$ implies that the increasingly rare critical configurations, accompanying
the crossover to Gumbel critical-force statistics, display anomalous roughness
properties: a transverse-periodicity scaling in spite that $\bar{w^2} \ll M$,
and subleading corrections to the standard random-manifold longitudinal-size
scaling. Our results are relevant to understanding the dimensional crossover
from interface to particle depinning.
