A method to construct continuous order k surfaces of arbitrary topological type by tracing a mesh that represents a 2-manifold is presented. The presented approach constructs the surface from a collection of pieces which overlap, using the known technology of manifolds. The surface is built from the given mesh and two arbitrary input parameters, k and n. The value of k gives the continuity of the resulting surface and n controls the local influence of the vertices of the mesh. The presented scheme generalizes the B-spline approach in terms of manifolds. The surface that arises from regular zones of the mesh is a tensorial product B-spline surface. For the rest, that is, the irregular zones, the surface has the following good properties: it is k continuous, affine invariant, the convex hull property is guaranteed, and it has local control. The algorithm is simple and efficient in time and space.