Minimum of all local minima of boundary "curves"eg. F(0, U)(i.e. Fu=0) Values of F(u, u) at corners, ie. F(0, 0), F(0, 1), F(1,0), F(1,1); and then choose uo, to from above solutions where F(uo, Uo)=0 The disadvantages of a minimization method are. (a) Initial approximation is required; (b) Possibility of divergence (c) No guarantee that all minima are located. (We need to enhance confidence by (d)First and second derivativers of F(a, u) are required Tote: When R= R(u, u) is a polynomial parametric surface patch, it is helpful to reformulate F(u, v)to F(u,0)=∑∑B1k(u)B3m(m) (10.7) If wi>0 for all i, j then there is no solution. We could use wi to construct initial approximations for the various local minima to be computed by usual descent numerical methods. These initial approximation may be obtained by discrete sampling or subdivi- SIc Let F(u, u) be expressed in the Bernstein basis, as in equation(10.7). Then, let us als press Fu, Fu in the Bernstein basis F(u,v)=∑∑AB1k-1(u)B3mn(v) F(u,0)=∑∑BBk(u)B1m-1()=0 (10.8) The equations Fu(u, v)=0 and F,(u, u)=0 represent planar algebraic curves illustrated in Figure 10. 11. Their intersection are the required extrema from which the minima can be selected using elementary calculus a geometrically motivated solution of the system 10.8 is possible using the convex hull property and subdivision to isolate an area where convex hulls intersect Taking G(u, u)= Fu(u, a) and n=k-l for example, we can write G(u,0)=∑∑A3B1n(u)B31m(v) i=0j=0 We can reformulate this "height" function into a parametric surface as follows w={u,,=∑∑LB,m(u)Bn() 17• Minimum of all local minima of boundary “curves” eg. F(0, v) (i.e. Fv = 0); • Values of F(u, v) at corners, ie. F(0, 0), F(0, 1), F(1, 0), F(1, 1); and then choose u0, v0 from above solutions where F(u0, v0) = 0. The disadvantages of a minimization method are: (a) Initial approximation is required; (b) Possibility of divergence; (c) No guarantee that all minima are located. (We need to enhance confidence by subdivision.) (d) First and second derivativers of F(u, v) are required. Note: When R = R(u, v) is a polynomial parametric surface patch, it is helpful to reformulate F(u, v) to F(u, v) = X k i=0 Xm j=0 wijBi,k(u)Bj,m(n) (10.7) If wij > 0 for all i, j then there is no solution. We could use wij to construct initial approximations for the various local minima to be computed by usual descent numerical methods. These initial approximation may be obtained by discrete sampling or subdivi￾sion. Let F(u, v) be expressed in the Bernstein basis, as in equation (10.7). Then, let us also express Fu, Fv in the Bernstein basis: Fu(u, v) = k X−1 i=0 Xm j=0 AijBi,k−1(u)Bj,m(v) = 0 Fv(u, v) = X k i=0 mX−1 j=0 BijBi,k(u)Bj,m−1(v) = 0 (10.8) The equations Fu(u, v) = 0 and Fv(u, v) = 0 represent planar algebraic curves illustrated in Figure 10.11. Their intersection are the required extrema from which the minima can be selected using elementary calculus. A geometrically motivated solution of the system 10.8 is possible using the convex hull property and subdivision to isolate an area where convex hulls intersect. Taking G(u, v) = Fu(u, v) and n = k − 1 for example, we can write w = G(u, v) = Xn i=0 Xm j=0 AijBi,n(u)Bj,m(v) We can reformulate this “height” function into a parametric surface as follows: w = [u, v, G] = XXLijBi,m(u)Bj,n(v) Lij = [ i m , i n , Aij ] 17