We propose a new framework for the generation and adaptation of unit curvilinear P2 meshes in dimension 2. In this approach, curvature is not only used to match curved boundaries but also to capture features of the interpolated solutions, and it results in meshes that would not have been achievable by simply curving a posteriori a straight-sided mesh. We proceed as follows. Starting with a smooth function f(x,y), a metric field, based on f and its derivatives up to order 3, is constructed. A unit P2 mesh is then generated, with edges within an adimensional length range of [0.7,1.4] with respect to this metric field. Points are then spawned in such a way that their geodesic distance corresponds to edges of unit size, and these points are then connected in a standard isotropic fashion. A curvilinear mesh quality criterion is then proposed to drive the mesh optimization process. The triangulation is subsequently modified using straight-sided edge swap, straight-sided edge curving, curvilinear edge swap and Curvilinear Small Polygon Reconnection (CSPR) to form the desired unit mesh. A unit curvilinear mesh containing only valid “Geodesic Delaunay triangles” is obtained this way. A number of application examples are presented in order to demonstrate the capabilities of the mesh adaptation procedure. The resulting adapted meshes allow, most of the times, a significant reduction of the approximation error compared with straight-sided P2 meshes of the same density.