Today I read a paper titled “On the Complexity of Smooth Spline Surfaces from Quad Meshes”
The abstract is:
This paper derives strong relations that boundary curves of a smooth complex of patches have to obey when the patches are computed by local averaging
These relations restrict the choice of reparameterizations for geometric continuity
In particular, when one bicubic tensor-product B-spline patch is associated with each facet of a quadrilateral mesh with n-valent vertices and we do not want segments of the boundary curves forced to be linear, then the relations dictate the minimal number and multiplicity of knots: For general data, the tensor-product spline patches must have at least two internal double knots per edge to be able to model a G^1-conneced complex of C^1 splines
This lower bound on the complexity of any construction is proven to be sharp by suitably interpreting an existing surface construction
That is, we have a tight bound on the complexity of smoothing quad meshes with bicubic tensor-product B-spline patches