On structure of bivariate spline space of lower degree over arbitrary triangulation

The structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property. A new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces. The upper bound of the dimension of spline spaces over triangulation given by L.L. Schumaker is slightly improved via the new vertex coding method. The structure of multivariate spline spaces S 2 1 ( ▵ ) and S 3 1 ( ▵ ) over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by Luo and Wang. A kind of sufficient conditions on judging non-singularity of the S 2 1 ( ▵ ) and S 3 1 ( ▵ ) spaces over arbitrary triangulation is given, which only depends on the topological property of the triangulation. From the sufficient conditions, a triangulation strategy is presented at the end of the paper. The strategy ensures that the constructed triangulation is non-singular (or generic) for S 2 1 and S 3 1 .