  Given a point set $P$ in the Euclidean plane and a parameter $t$, we define
an \emph{oriented $t$-spanner} $G$ as an oriented subgraph of the complete
bi-directed graph such that for every pair of points, the shortest closed walk
in $G$ through those points is at most a factor $t$ longer than the shortest
cycle in the complete graph on $P$. We investigate the problem of computing
sparse graphs with small oriented dilation. As we can show that minimising
oriented dilation for a given number of edges is NP-hard in the plane, we first
consider one-dimensional point sets. While obtaining a $1$-spanner in this
setting is straightforward, already for five points such a spanner has no plane
embedding with the leftmost and rightmost point on the outer face. This leads
to restricting to oriented graphs with a one-page book embedding on the
one-dimensional point set. For this case we present a dynamic program to
compute the graph of minimum oriented dilation that runs in $O(n^7)$ time for
$n$ points, and a greedy algorithm that computes a $5$-spanner in $O(n\log n)$
time. Expanding these results finally gives us a result for two-dimensional
point sets: we prove that for convex point sets the greedy triangulation
results in a plane oriented $t$-spanner with $t=19 \cdot t_g$, where $t_g$ is a
upper bound on the dilation of the greedy triangulation.
