此新e )-)一r-0° 2) 新e书一w-e=ae0外aeo ,-,》-g,0g-j4 a(y)across 红》-入s,-工P1+4红，别-红8jg-)+1= f与T-vert(Fgre5(以thn the6 tdf,功at4i ((红，-A红w〔z-x+1于工，g-)41三0， 5 +Ae-+a在-e，g-P=0 New Proof of Dimension Formula of Spline Spaces over T-meshes via Smoothing Cofactors 5 According to the smoothing cofactor theory, if the common edge e is on the straight line x − x0 = 0(see Figure 4(a)), then there exists λ(x, y) ∈ Pm−α−1,n such that s2(x, y) − s1(x, y) = λ(x, y)(x − x0) α+1. (2) If the common edge e is on the straight line y − y0 = 0(see Figure 4(b)), then there exists μ(x, y) ∈ Pm,n−β−1 such that s2(x, y) − s1(x, y) = μ(x, y)(y − y0) β+1. (3) Here λ(x, y) and μ(x, y) are called the smoothing cofactors of s(x, y) across the corresponding edges, respectively. ￾ vi λi1 λi2 μi1 μi2 a ￾ ￾ vi λi1 λi2 μi vi λi1 λi2 μi b ￾ ￾ μi1 vi μi2 λi μi1 vi μi2 λi c Figure 5: Conformality conditions We use λ(x, y) to denote the smoothing cofactor across a vertical interior edge from left to right, and μ(x, y) to denote the smoothing cofactor across a horizontal interior edge from bottom to top. Let V = {ν1,...,νV }, Eh = {εh 1 ,...,εh Eh }, Ev = {εv 1,...,εv Ev }. be the set of interior vertices, interior horizontal edges and interior vertical edges, respectively. Hence the set of interior edges E = Eh ∪ Ev. Suppose νi = (xi, yi) is a interior vertex of T , which is a crossing vertex or a T-vertex(see Figure 5). The horizontal T-vertex on the left of Figure 5(b) is called a right T-vertex, while the one on the right of Figure 5(b) is called a left T-vertex. The vertical T-vertex on the top of Figure 5(c) is called a down T-vertex, while the one on the bottom of Figure 5(c) is called an up T-vertex. If νi is a crossing vertex(Figure 5(a)), then the conformality condition of s(x, y) at νi is (λi1(x, y) − λi2(x, y))(x − xi) α+1 + (μi1(x, y) − μi2(x, y))(y − yi) β+1 ≡ 0; (4) if νi is a horizontal T-vertex (Figure 5(b)), then the conformality condition of s(x, y) at νi is (λi1(x, y) − λi2(x, y))(x − xi) α+1 ∓ μi(x, y)(y − yi) β+1 ≡ 0, (5) for a right T-vertex, the sign before μi(x, y) is ‘−’, otherwise ‘+’; if νi is a vertical T-vertex (Figure 5(c)), then the conformality condition of s(x, y) at νi is ∓λi(x, y)(x − xi) α+1 + (μi1(x, y) − μi2(x, y))(y − yi) β+1 ≡ 0, (6)