  For a family $\mathcal{C}$ of properly embedded curves in the 2-dimensional
disk $\mathbb{D}^{2}$ satisfying certain uniqueness properties, we consider
convex polygons $P\subset \mathbb{D}^{2}$ and define a metric $d$ on $P$ such
that $(P,d)$ is a geodesically complete metric space whose geodesics are
precisely the curves $\left\{ c\cap P\bigm\vert c\in \mathcal{C}\right\}.$
  Moreover, in the special case $\mathcal{C}$ consists of all Euclidean lines,
it is shown that $P$ with this new metric is not isometric to any convex domain
in $\mathbb{R} ^{2}$ equipped with its Hilbert metric.
  We generalize this construction to certain classes of uniquely geodesic
metric spaces homeomorphic to $\mathbb{R}^{2}.$
