Geometries on Polygons in the unit disc

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 \(\mathbb{R}^{2}.\)