  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.
