Research On Dimensions Of Spline Spaces Over T-meshes

Spline function is an important approximation tool in computational geometry,and it is widely used in many engineering fields. In order to better understand splinespace and to apply it, the first problem is to explicit its algebraic structure and itsdimension. However, the dimension of multivariate spline space depends not onlyits topological property of its partition, but also sometimes heavily on the geometricproperty of its partition. T-mesh spline space is firstly presented by Sederberg,etal.[1]in 2003, which is defined in general T-mesh. Using B-nets method, JiansongDeng, etal. [2] derived the dimension on T-mesh spline space when the smoothness isless than half of the degree of the spline functions, which depends on the topologicalproperty of T-mesh. But when the smoothness is close to the degree, the dimensionformula hasn't gotten.In this paper, firstly, a general formula of dimension of spline space S(m, n,α,β,τ)over T-mesh is given by means of ordering skill to inner edges of T-mesh under theframe of Smoothing Cofactor-Conformality method. The result of the first partimproves the corresponding results in [2] and [17]. Secondly, based on the theoryof Generator Basis of Module and Smoothing Cofactor-Conformality method, wederive the dimension of spline space S_k~μ(Г)over T-mesh.