Example Join bicubic patches along arbitrary cubic linkage curves in parametric space(see Figure 9.1) Q( w,s) R(U,v) v=v(t) (t3) s=s(+3) Figure 9. 1: Bicubic patches joined along arbitrary cubic linkage curves Linkage curve 1 is: R1(t=R(t)=R(u,v)=R(tS), if u=u(t), =v(t) Similarly curve 2 is R2(t)=Q(t)=Q(3,s3)=Q(t28),i=(3,s=s(t2) So position continuity alone requires a high degree surface in the t parameter direction (of degree 18 in this example). High degree surfaces are expensive to evaluate (e.g. the de Casteljau or Cox-de Boor algorithms have quadratic complexity in the degree of the curve or surface), may lead to greater inaccuracy of evaluation(as the degree increases), and are difficult to process in a solid modeling environment(e. g. through intersections ) Consequently researchers have developed Approximations of blending surfaces with low order B-spline surfaces Procedural definitions of blending surfaces(e.g. "lofted"surfaces, generalized cylinders) in order to reduce some of these problemsExample Join bicubic patches along arbitrary cubic linkage curves in parametric space (see Figure 9.1). R(u , v ) 3 3 Q (w , s ) 3 3 v = v (t )3 u = u (t )3 w = w (t ) 3 s = s (t )3 Figure 9.1: Bicubic patches joined along arbitrary cubic linkage curves. Linkage curve 1 is: R1(t) ≡ R(t) = R(u 3 , v 3 ) ≡ R(t 18), if u = u(t 3 ), v = v(t 3 ) (9.1) Similarly, curve 2 is: R2(t) ≡ Q(t) = Q(w 3 , s 3 ) ≡ Q(t 18), if w = w(t 3 ), s = s(t 3 ) (9.2) So position continuity alone requires a high degree surface in the t parameter direction (of degree 18 in this example). High degree surfaces are expensive to evaluate (e.g. the de Casteljau or Cox-de Boor algorithms have quadratic complexity in the degree of the curve or surface), may lead to greater inaccuracy of evaluation (as the degree increases), and are difficult to process in a solid modeling environment (e.g. through intersections). Consequently, researchers have developed: • Approximations of blending surfaces with low order B-spline surfaces • Procedural definitions of blending surfaces (e.g. “lofted” surfaces, generalized cylinders) in order to reduce some of these problems. 3