Nonlinear subdivision schemes on irregular meshes

The present article deals with convergence and smoothness analysis of geometric, nonlinear, subdivision schemes in the presence of extraordinary points. We discuss when the existence of a proximity condition between a linear scheme and its nonlinear analogue implies convergence of the nonlinear scheme (for dense enough input data). Furthermore, we obtain C1 smoothness of the nonlinear limit function in the vicinity of an extraordinary point over Reif's characteristic parametrisation. The results apply to the geometric analogues of well known subdivision schemes like Doo-Sabin or Catmull-Clark schemes. keywords: nonlinear subdivision, extraordinary point, irregular mesh. 2000 MSC. Primary 65D17, Secondary 68U05, 53A99, 58C07.