On $G^1$ stitched bi-cubic B\'ezier patches with arbitrary topology

TL;DR

This paper investigates the conditions under which $G^1$ smooth bi-cubic Bézier patches can be constructed over arbitrary topology meshes, analyzing the effectiveness of Doo-Sabin subdivision in enabling such constructions.

Contribution

It provides a theoretical analysis of the limitations and possibilities for $G^1$ smooth bi-cubic Bézier patches on arbitrary meshes, especially considering mesh refinement techniques.

Findings

Lower bounds restrict $G^1$ smoothness on arbitrary topology meshes.

Doo-Sabin subdivision does not generally enable $G^1$ continuity with bi-degree 3 patches.

The layout constraints are fundamental and not overcome by simple mesh refinement.

Abstract

Lower bounds on the generation of smooth bi-cubic surfaces imply that geometrically smooth ($G^1$) constructions need to satisfy conditions on the connectivity and layout. In particular, quadrilateral meshes of arbitrary topology can not in general be covered with $G^1$ -connected B\'ezier patches of bi-degree 3 using the layout proposed in [ASC17]. This paper analyzes whether the pre-refinement of the input mesh by repeated Doo-Sabin subdivision proposed in that paper yields an exception.