On $G^1$ stitched bi-cubic Bézier patches with arbitrary topology
For researchers in geometric modeling, this resolves a conjecture about the feasibility of smooth surface constructions on arbitrary meshes, showing a fundamental limitation.
The paper proves that pre-refining an arbitrary topology quadrilateral mesh via repeated Doo-Sabin subdivision does not enable a $G^1$ bi-cubic Bézier construction, contradicting a prior claim. It establishes a lower bound showing such a layout cannot achieve geometric smoothness in general.
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 Bézier 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.