Uniform approximation of near-singular surfaces

We consider the problem of a piecewise-polynomial (spline) approximation of algebraic surfaces of a given degree. Conventional accuracy estimates in such approximations include bounds on the high-order derivatives, or on the surface curvature. As an algebraic surface degenerates to a singular one, the curvature blows up. Consequently, the same happens with the complexity of the approximation: to keep the required accuracy, we need more and more patches at near-singular (high curvature) areas. This indeed happens in any conventional “triangulation” algorithm.Nevertheless, using “Ck-reparametrization” theorem (which originally appeared in dynamical applications) we show in this paper that for such near-singular families of surfaces there exist approximations of any fixed accuracy and of a uniformly bounded complexity.To give an example of a situation, where our construction becomes explicit, we consider one of the possible models for singular (and near-singular) surfaces. In this model, developed in [Y. Yomdin, Generic singularities of surfaces, in: D. Cheniot, N. Dutertre, C. Murolo, D. Trotman, A. Pichon, (Eds), Singularity Theory, Dedicated to J.-P. Brasselet on his 60-th Birthday (Proceedings of the 2005 Marseille Singularity School and Conference, Marseille, France), 2005, pp. 24.1–25.2; D. Haviv, MA Thesis, Weizmann Institute, 2006] a surface appears as a part of the level surface F=c, with F=F1F2…Fm having a product form. The advantage of this approach is that “edges” and “corners” appear in a generic and stable way, as well as “near-edges” and “near-corners”. Using special model-based representation developed in [Y. Yomdin, Generic singularities of surfaces, in: D. Cheniot, N. Dutertre, C. Murolo, D. Trotman, A. Pichon, (Eds), Singularity Theory, Dedicated to J.-P. Brasselet on his 60-th Birthday (Proceedings of the 2005 Marseille Singularity School and Conference, Marseille, France), 2005, pp. 24.1–25.2; D. Haviv, MA Thesis, Weizmann Institute, 2006], we produce explicit formulae for the uniform C2-reparametrization of some of these near-singularities.