Local Hermite interpolation by bivariate \(C^1\) cubic splines on checkerboard triangulations (Q2912526)

The paper starts with a general convex quadrangulation with all interior vertices of degree four and considers the triangulation obtained by drawing in both diagonals of each quadrilateral. Imposing extra smoothness conditions across interior edges, the paper constructs a \(C^1\)-spline surface which is piecewise cubic on this triangulation and performs a Hermite interpolation of function and gradient values at the vertices of the original quadrangulation. NEWLINENEWLINEReviewer's remark: Two results of this paper appear to be incomplete, namely the claim in the abstract and the introduction that the proposed Hermite interpolation scheme is proved to have optimal approximation order; but the error bound presented in Theorem 5 is only obtained for the Lagrange scheme interpolating function-values at all the Bernstein-Bézier domain-points of a minimal determining set. Also, it is not specified what the points \((x_i,y_i)\) represent in Theorem 4.