Approximation by cubic \(C^ 1\)-splines on arbitrary triangulations
We present an efficient global representation for bivariate piecewise cubic splines of class \(C^ 1\) on arbitrary triangulations. The polynomial pieces of the splines on the different triangles are uniquely determined by a minimal number of real-valued parameters. The \(C^ 1\)- smoothness of the splines is ensured by imposing a linear equation on the parameters for every interior edge in the triangulation. The considerable saving in the amount of stored information makes this representation attractive for practical computations. A numerical method is discussed for computing the dimension of the space of these splines. Furthermore, we consider subspaces of splines satisfying certain boundary conditions. Some applications are given where piecewise cubic \(C^ 1\)-functions are used to solve practical approximation problems, such as the interpolation to data in combination with the minimization of the thin plate functional and the least-squares approximation of scattered data.
least-squares approximation of scattered data
scientific article; zbMATH DE number 3957865
scientific article