6 ZJ.HUANG,J.S.DENG.Y.Y.FENG AND F.L.CHEN eA队A如红以 （,o同)间.aam for some,y》,y,y》∈Pm-a-1.n-s-1 Putting thes bock into Eu4()giw (E.yl+h4工，z-z+1y-)+1=D. 皮怎ak）e m+10ea-朗+m-a0n+1)+a, we have e folkwing dimmsin of the spline dimS(m,n.0.1.T)-(mw+1K(n+1)+5(m+1(n-+(m-0)(+1)+o (10) =vm-ajn-助 a7ata-+moo++-oe-网 6 Z.J. HUANG, J.S. DENG, Y.Y. FENG AND F.L. CHEN for a down T-vertex, the sign before λi(x, y) is ‘−’, otherwise ‘+’. Here λi(x, y), λi1(x, y), λi2(x, y) ∈ Pm−α−1,n, and μi(x, y), μi1(x, y), μi2(x, y) ∈ Pm,n−β−1. We analysis further the conformality conditions in Equation (4), (5) or (6). For Equation (4), since (x − xi)α+1 and (y − yi)β+1 are prime, we have λi1(x, y) − λi2(x, y) = vi(x, y)(y − yi) β+1, μi1(x, y) − μi2(x, y) = hi(x, y)(x − xi) α+1 for some hi(x, y), vi(x, y) ∈ Pm−α−1,n−β−1. Putting these back into Equation (4) gives (vi(x, y) + hi(x, y))(x − xi) α+1(y − yi) β+1 ≡ 0. Hence, vi(x, y) = −hi(x, y) = ci(x, y) ∈ Pm−α−1,n−β−1. We call ci(x, y) the vertex cofactor of s(x, y) corresponding to an interior vertex νi. Then the conformality condition at a crossing vertex in Equation (4) can be rewritten as λi1(x, y) − λi2(x, y) = ci(x, y)(y − yi) β+1, μi1(x, y) − μi2(x, y) = −ci(x, y)(x − xi) α+1. (7) Similarly, the conformality condition at a horizontal T-vertex in Equation (5) can be rewrit￾ten as λi1(x, y) − λi2(x, y) = ci(x, y)(y − yi) β+1, ∓μi(x, y) = −ci(x, y)(x − xi) α+1; (8) and the conformality condition at a vertical T-vertex in Equation (6) can be rewritten as ∓λi(x, y) = ci(x, y)(y − yi) β+1, μi1(x, y) − μi2(x, y) = −ci(x, y)(x − xi) α+1. (9) We note that, even if all the vertex cofactors ci(x, y), i = 1,...,V are determined, for each cross-cut, there is still one interior edge whose smoothing cofactor is completely free. Let Ec h and Ec v be the number of horizontal cross-cuts and the number of vertical cross-cuts respectively, and σ be the number of independent free coefficients of the vertex cofactors. Then the number of free coefficients of the smoothing cofactors of all interior edges is Ec h(m + 1)(n − β) + Ec v(m − α)(n + 1) + σ, hence, we have the following dimension formula of the spline space over a T-mesh: dim S(m, n, α, β, T )=(m + 1)(n + 1) + Ec h(m + 1)(n − β) + Ec v(m − α)(n + 1) + σ (10) 3.2 In-line conformality conditions If there are no in-lines in T , namely, T is a simple quasi-cross-cut partition, the vertex cofactors are relatively independent. Then we have σ = V (m − α)(n − β), and the dimension formula (10) becomes dim S(m, n, α, β, T )=(m+ 1)(n+ 1)+Ec h(m+ 1)(n−β)+Ec v(m−α)(n+ 1)+V (m−α)(n−β)