A Note on the Upper Bound of Dimension of Bivariate Spline Space over Triangulation

It is known that there is not a natural generalization for the dimension of multivariate spline spaces, since the dimension of multivariate spline spaces depends not only on the topological property of partition but also on the geometric property of partition. The aim of this paper is to improve the upper bound of the dimension of bivariate spline space for degree k and smoothness mu over arbitrary triangulation by using a new index of vertex coding. A new upper bound of the spline space over triangulation is obtained, which improves the known upper bound of the dimension in in the paper. The advantages of the improved result can be seen from the consequences and examples in the end of the paper.