Lagrange interpolation by bivariate splines on cross-cut partitions

The piecewise algebraic curve is a generalization of the classical algebraic curve. In this paper, we give the Nِther-type theorem of piecewise algebraic curves on cross-cut partitions. By using interpolation along a piecewise algebraic curve, we also present a method to construct Lagrange interpolation sets for bivariate spline spaces on cross-cut partitions.