Nonlinear subdivision schemes on irregular meshes (Q975343)

The convergence and the \(C^{1}\) smoothness of nonlinear subdivision rules for irregular meshes are studied. First of all, a class of nonlinear schemes \(T\) related to a class of linear schemes is defined by identifying a proximity condition. The case when the existence of such a proximity condition implies the convergence of the nonlinear scheme for dense enough input data is emphasized. The \(C^{1}\) smoothness of the nonlinear limit function in the vicinity of an extraordinary point over \textit{U. Reif}'s [Comput. Aided Geom. Des. 12, No.~2, 153--174 (1995; Zbl 0872.65007)] characteristic parametrization is obtained. The manner in which subdivision in the geometric setting can be used to generate manifold-valued smooth functions on smooth 2D manifolds comes as an application.