We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q∗-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: We would like to thank Wolfgang Schief and Jan Techter for many fruitful discussions. This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics.” – www.discretization.de. Helmut Pottmann was supported through Grant I 2978-N35 of the Austrian Science Fund (FWF).