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

TL;DR

This paper explores the underlying discrete structures of various smooth and discrete nets, characterizing them on larger parameter rectangles and connecting them to classical smooth surfaces, with applications in structure-preserving subdivision schemes.

Contribution

It introduces a unified characterization of multiple types of nets on larger parameter rectangles, bridging discrete and smooth surface classes, and proposes their use in subdivision schemes.

Findings

Characterization of discrete nets on larger parameter rectangles.

Connection between discrete nets and classical smooth surfaces.

Application in structure-preserving subdivision schemes.

Abstract

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 larger 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.