New Proof of Dimensicn Form of Splme galcge New Proof of Dimension Formula of Spline Spaces over T-meshes via Smoothing Cofactors 3 a b Figure 2: Examples of non-T-mesh b1 b2 b3 b4 b5 b8 b7 b6 b9 b10 b11 v1 v2 v3 v4 v5 v6 v7 ￾ ￾ ￾￾￾ ￾ ￾￾ Figure 3: A T-mesh with notations other vertices vi, i = 1,..., 5 are interior vertices. Interior vertices have two types. One is crossing, for example, v2 in Figure 3; and the other is T-junctional, for example, v1 in Figure 3. We call them crossing vertices and T-vertices respectively. The line segment connecting two adjacent vertices on a grid line is called an edge of the T-mesh. If an edge is on the boundary of the T-mesh, then it is called a boundary edge; otherwise it is called an interior edge. For example, in Figure 3, b11v1 and v1v2 are interior edges while b1b2 is a boundary edge. Except the boundary grid lines, there are three types of grid lines. We call a grid line a cross-cut or a ray, if both or only one of its endpoints lies on the boundaries, respectively. For example, in Figure 3, b5b10 and b4b11 are cross-cuts, while v5b2 , v4b7 and v7b9 are rays. Now we define the third type of grid lines. A grid line is called a in-line, if none of its endpoints lie on the boundaries. For example, in Figure 3, v1v6 is a in-line. For any grid line, it consists of one or several edges. We define its valence as the number of edges on the grid line. Two cells are called adjacent if they share a common edge as part of their boundaries. If one cell is above(below) the other, then they are called adjacent vertically. If one cell is on the left(right) of the other, then they are called adjacent horizontally. A cell is called adjacent to a grid line (an edge or composition of several edges) if some boundary line of the cell is part of the grid line. As in [2], we consider only T-meshes whose boundary grid lines form a rectangle, see Figure 1(b). We call this type of T-meshes regular T-meshes. 2.2 The spline space Given a T-mesh T , we use F to denote all the cells in T and Ω to denote the region occupied