At issue is the question of smoothness. Previously, tangent plane continuity was used as the smoothness criteria. Piecewise polynomial (or rational polynomial) surface patches were fit to the data so that adjacent patches were tangent plane continous along their common boundary. Unfortunately, surfaces constructed in this fashion have poor visual shape despite meeting the mathematical smoothness constraints. Curvature plots show that curvature in unequally distributed across these patches, resulting in surfaces that look like smooth polyhedra (at best).
One class of techniques that have been used to construct better looking surfaces are variational techniques. These techniques are global in the sense that all the data is considered when constructing each patch (this is in constrast to the more common definition of global optimization meaning finding global minimums instead of local minimums).
While variational techniques provide far better surfaces, this improvement comes at a high computational price. Even for small numbers of surface patches, these techniques require hours of CPU time to compute the surface. I am currently looking at local techniques that create surfaces with better shape. There are several thrusts to my research, but all seem to indicate that lower degree surface patches with better settings of their degrees of freedom give high quality surfaces.
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