Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes

We solve the following problem: given a polynomial of order \(n\) and the corresponding \(Bézier\) tensor product patches over an unstructured regular quadrilateral mesh of any valence, find a solution to the \(G^{1}1\) or \(C^{1}1\) approximation (resp. interpolation) problem ! Constraints defining regularity conditions across patches have to be satisfied. The resulting number of free degrees of freedom must be such that for instance the interpolation problem has a solution. This is similar to studying the minimal determining set (MDS) for a \(C^{1}\) continuity construction. The givenunstructured quadrilateral mesh can include a cubic boundary curve. The final surface approximation or PDE solution is obtained by energy methods. We completely solve the problem and show that there is always a solution for \(n\ge 5\) and under some mesh restrictions for \(n=4\). From a practical point of view, the present paper provides a way to build first order smooth interpolation/approximation and solutions to partial differential equations for arbitrary structures of quadrilateral meshes.