Let Γ be a closed surface in ℝ3. We assume Γ is given by the local parameter representation (1.1) $$f_i :\Omega _i \to \mathbb{R}^3 ,i = 1 \ldots p,$$ as a two-dimensional topological manifold, whereΏ i⊂ℝ2 are open domains of the parameter plane. Our aim is the construction of anh parameter-dependent familyΓ h of approximations toΓ such that everyΓ h becomes a triangulated two-dimensional manifold. The familyΓ h is required to converge in a certain way toΓ, where everyΓ h will allow a triangulation in the parameter plain and of Γ by using curved triangles. This problem always comes up when a closed surface Γ is to be approximated by finite elements such that no intersections or overlappings occur. This is a standard problem in all algorithms for boundary element methods (or panel methods) for the numerical solution of three-dimensional boundary value problems. If one gives up the aim to choose the grid in the parameter domains as far as possible and allows instead that these grids on the surface are generated by special finite element meshes, one can use the procedure of Allgower and Gnutzmann. If, however, one wants to choose the grid in the parameter domain, as for example quasi-regular grids with special mesh grading, which is necessary near singularities, one needs a different strategy. In many special cases of given surfaces Γ, this can be done by choosing an atlas with polygonal parameter domains and their triangulations. Nedelec assumes this situation. If Γ is the boundary surface of a convex regionΏ i⊂ℝ3, Γ=∂Ω, then indeed such polygonal domains and representations of the surface can always be found on which the approximationsΓ h can be constructed by using triangulations and Lagrange elements. For a general, closed surface this construction is still an open problem. Even when the surface is piecewise analytic and non-convex, we do not know whether one can proceed as above. Here we are able to close this gap by allowing appropriate polygonal approximations of the parameter domains. As we will see, the desired procedure is very simple and rather straightforward. At the same time we will also show how the triangulations can be mapped onto each other in such a way that no overlappings of the triangulations appear.