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We investigate the common underlying discrete structures for various smoothand discrete nets. The main idea is to impose the characteristic properties ofthe nets not only on elementary quadrilaterals but also on larger parameterrectangles. For discrete planar quadrilateral nets, circular nets, $Q^*$-netsand conical nets we obtain a characterization of the corresponding discretemulti-nets. In the limit these discrete nets lead to some classical classes ofsmooth surfaces. Furthermore, we propose to use the characterized discrete netsas discrete extensions for the nets to obtain structure preserving subdivisionschemes.

Multi-Nets. Classification of discrete and smooth surfaces with characteristic properties on arbitrary parameter rectangles