9.2 Blending surface approximation in terms of B-splines et us consider two parametric surface patches rmim,(, u), Imana(a, y) and linkage curves Ri(t)= u(t), u(t, R2(t)= r(t),y(t)] defined in the parameter spaces of the two patches respectively The unit normal to Ri that is tangent to the patch(see Figure 9.2) 02=N2 S1|a=a() (9.3) where N=rux r is the normal to the patch, and S1=i(tru(u(t), v(t))+i(tr,(t), v(t)) where r=rmin1(u, v). In general, N and Si are high order polynomials. For example, when the patch is bicubic and the linkage curve is cubic in the parameter space, we have niU?vuv2Nu5uSNt3o s,ir. t2t6t9n tI7 Next, we introduce bias functions b1(t),b2(t)to control the shape of the blending surface and define q1(t)=b1(t)r1(t (9.4) q2(t)=b2(t)r2(t) (9.5) to be used as parametric derivatives of the patch in the direction between the linkage curves For details, see Bardis and Patrikalakis [1 R1(t) Figure 9.2: N, SI rI frame9.2 Blending surface approximation in terms of B-splines Let us consider two parametric surface patches rm1n1 (u, v), rm2n2 (x, y) and linkage curves R1(t) = [u(t), v(t)], R2(t) = [x(t), y(t)] defined in the parameter spaces of the two patches, respectively. The unit normal to R1 that is tangent to the patch (see Figure 9.2): r1(t) = N × S1 |N × S1|          u = u(t) v = v(t) (9.3) where N = ru × rv is the normal to the patch, and S1 = u˙(t)ru(u(t), v(t)) + v˙(t)rv(u(t), v(t)) where r = rm1n1 (u, v). In general, N and S1 are high order polynomials. For example, when the patch is bicubic and the linkage curve is cubic in the parameter space, we have N ∼ ru × rv ∼ u 2 v 3u 3 v 2 ∼ u 5 v 5 ∼ t 30 S1 ∼ u˙ru ∼ t 2u 2 v 3 ∼ t 2 t 6 t 9 ∼ t 17 Next, we introduce bias functions b1(t), b2(t) to control the shape of the blending surface and define: q1(t) = b1(t)r1(t) (9.4) q2(t) = b2(t)r2(t) (9.5) to be used as parametric derivatives of the patch in the direction between the linkage curves. For details, see Bardis and Patrikalakis [1]. N 1 R (t) r 1 S 1 Figure 9.2: N, S1 r1 frame 4