On the complexity of smooth spline surfaces from quad meshes

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 G1-connected complex of C1 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.