International audience; Parametric surface mesh generation is one of the crucial step of the computational pipeline. Standard techniques, that are now mature, control the deviation to the tangent plane by using intrinsic quantities as the minimum and maximum curvatures. However, for high-order meshes, deriving intrinsic quantities that have the ability to control the mesh generation process is much more challenging. Indeed, those provided by the first and second fundamental forms of a surface are not sufficient when high order curved meshes are employed. In this paper, we introduce a new set of error estimates for high-order surface mesh generation. It is based on performing a Taylor expansion of the underlying surface in the tangent plane. The independence to the parametric space is obtained by using an inversion formula. High-order terms of this expansion are then used to derive an optimal metric by using the log-simplex approach. Examples are shown to prove the efficiency of the method.