Nonlinear subdivision processes in irregular meshes

The present article deals with convergence and smoothness analysis of geometric, nonlinear, subdivision schemes in the presence of extraordinary points. For a certain class of stationary linear subdivision schemes we can show that if a convergent linear scheme and its nonlinear analogue meet a proximity condition, then the nonlinear scheme converges for dense enough input data. Furthermore, we obtain C1 smoothness of the nonlinear limit function in the vincinity 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.