The computed values of y(B)will not, in general, agree with the corresponding boundary condition at s= B Consequently, we need to adjust the initial values and try again The process is repeated until the computed values at the final point agree with the boundary conditions and referred as shooting method Formulation: Using the first fundamental form, given pa we can obtain gA from FpA±VFmA-G(EnA-1) Thus we assume pA and solve the differential equation as IvP using, say runge- Kutta method. Here we also have to assume the entire arc length of the geodesic path s to stop the integration. Thus the unknowns can be considered as pA and s If we denote the computed value of(uB, UB) as(uB, UB), the difference can be given as (ub -uB, UB We need to adjust pa and s to make the difference zero This can be done by employing the Newtons method B n+1 dub uB(PA+△pA)-ubB(PA) dus uB(s+△s)-uB(s) ap B UR(PA+ ApA)-uR(pa) auk u*(s+As)-uB(s) e Relaxation method The second method is based on a finite difference approximation to a on a mesh of points in the interval [A, B This method starts with an initial guess and improves the solution iteratively and referred as. direct method. relaxation method or finite difference method The shooting method is often very sensitive to the unknown initial angles at point A and unless a good initial guess is provided, the integrated path will never reach the other point B, while the relaxation method starts with two end points fixed and relaxes to the true solution and hence it is much more stable Let us consider a mesh of points satisfying A=S1<S2 <sm=B. We approximate the n first order differential equations by the trapezoidal rule [8 -11 =Gk+Gk-l],k=2,3 (20.24)
� � � (�) (�) – The computed values of y(B) will not, in general, agree with the corresponding boundary condition at s = B. – Consequently, we need to adjust the initial values and try again. – The process is repeated until the computed values at the final point agree with the boundary conditions and referred as shooting method. – Formulation: Using the first fundamental form, given pA we can obtain qA from − 2 F pA ± F2pA − G(Ep2 q A − 1) A = . G Thus we assume pA and solve the differential equation as IVP using, say RungeKutta method. Here we also have to assume the entire arc length of the geodesic path s to stop the integration. Thus the unknowns can be considered as pA and s. � B), the difference can be given � If we denote the computed value of (uB, vB) as (uB, v as (u This can be done by employing the Newton’s method � T B − vB) � B − uB, v . We need to adjust pA and s to make the difference zero. ⎣ � �−1 B �s � B ⎤ ⎦ ⎤ ⎦ �u �u ⎤ ⎦ pA pA �pA uB − uB = − � B �s � B �v �v s n+1 s �p vB − vB n A n where � B � B(pA) � B(pA + �pA) − u � B � B(s) � πu u πu uB(s + �s) − u = , = πPA �pA πs �s � B v � B(pA) � B(pA + �pA) − v � B � B(s) � πv πv vB(s + �s) − v = , = πPA �pA πs �s • Relaxation method dy – The second method is based on a finite difference approximation to ds on a mesh of points in the interval [A, B]. – This method starts with an initial guess and improves the solution iteratively and referred as, direct method, relaxation method or finite difference method. – The shooting method is often very sensitive to the unknown initial angles at point A and unless a good initial guess is provided, the integrated path will never reach the other point B, while the relaxation method starts with two end points fixed and relaxes to the true solution and hence it is much more stable. – Let us consider a mesh of points satisfying A = s1 < s2 < . . . < sm = B. We approximate the n first order differential equations by the trapezoidal rule [8]. Yk − Yk−1 1 = [Gk + Gk−1], k = 2, 3, . . . , m (20.24) sk − sk−1 2 6
where the n-vectors Yk, Gk are meant to approximate y(sk)and g(sk) with bound ary conditions (uA, UA, P1, q1), Ym=B=(u Yi has n1=2 known components, while Ym has n2=n-n1=4-2=2 known components Boundary Conditions Unknown S1 4 SM-1 SM Figure 20.2: Mesh points This discrete approximation will be accurate to the order of h2(h= maxk sk Sk-1)). Equation(20.24)forms a system of(m-1)n nonlinear algebraic equations with(m-1)n unknowns Let us refer to equation(20. 24)as FR=(FLk, F2 F and the boundary conditions(20.25)as F1=(F1 F m+1 (F1 12.n Ym-B then we h F
������������������������������������� where the n-vectors Yk, Gk are meant to approximate y(sk) and g(sk) with boundary conditions Y1 = β = (uA, vA, p1, q1) T , Ym = � = (uB, vB, pm, qm) T (20.25) – Y1 has n1 = 2 known components, while Ym has n2 = n − n1 = 4 − 2 = 2 known components. �������������������������������������✥ u1 u2 u3 u4 uM−2 uM−1 uM �������������������������������������✥ Boundary Conditions Unknowns ������������������������������������� v1 v2 v3 v4 vM−2 vM−1 vM �������������������������������������✥ ������������������������������������� p1 p2 p3 p4 pM−2 pM−1 pM �������������������������������������✥ ������������������������������������� q1 q2 q3 q4 . . . . . . . . . qM−2 qM−1 qM s1 s2 s3 s4 sM−2 sM−1 sM Figure 20.2: Mesh points. s – This discrete approximation will be accurate to the order of h2 (h = maxk{sk − k−1}). Equation (20.24) forms a system of (m − 1)n nonlinear algebraic equations with (m − 1)n unknowns. – Let us refer to equation (20.24) as 1 Fk = (F1,k, F2,k, . . . , Fn,k) T = Yk − Yk−1 [Gk + Gk−1] = 0 (20.26) sk − sk−1 − 2 – and the boundary conditions (20.25) as F F1 = (F1,1, F2,1, . . . , Fn1,1) T = Y1 − β = 0, (20.27) m+1 = (F1,m+1, F2,m+1, . . . , Fn2,m+1) T = Ym − � = 0 – then we have mn nonlinear algebraic equations F = (FT 2 , . . . , FT 1 , FT m+1) T = 0 (20.28) 7
oints Tolerance Iterations Geodesic Distance 0E310110|1:6611.8651.661 Table 20.1: Numerical conditions and results for the computation of the geodesic path between corner points of the wave-like surface and can be computed by quadratically convergent Newton iteration, if a sufficiently accurate starting vector Y(O)=(YT, Y2, .. Ym) is provided. The Newton iter- ation scheme is given by (+1) Y+△Y Jo]△Y0 (20.30) where jJo] is the mn by mn Jacobian matrix of FO) with respect to Yo Figure 20.3: Geodesic paths on the wave-like bicubic B-spline surface between points of two corners 20.1.5 Example Bilinear surface r(u, v)=(u, v, uv) Eu=0, F E= 2 G2=0
Points Tolerance Iterations Geodesic Distance m κN L M R L M R 101 1.0E-3 10 1 10 1.661 1.865 1.661 Table 20.1: Numerical conditions and results for the computation of the geodesic path between corner points of the wave-like surface. – and can be computed by quadratically convergent Newton iteration, if a sufficiently accurate starting vector Y(0) = (YT 2 , . . . , YT 1 , YT m)T is provided. The Newton iteration scheme is given by Y(i+1) [J = Y(i) + �Y(i) (20.29) (i) ]�Y(i) = −F(i) (20.30) where [J(i) ] is the mn by mn Jacobian matrix of F(i) with respect to Y(i) . x y z Figure 20.3: Geodesic paths on the wave-like bicubic B-spline surface between points of two corners. 20.1.5 Example Bilinear surface r(u, v) = (u, v, uv). E E E = 1 + v2, F = uv, G = 1 + u2 u = 0, Fu = v, Gu = 2u v = 2v, Fv = u, Gv = 0 8
科人 Figure 20.4: Convergence of the right geodesic path in Figure 20.3 2()v+ut:(2 2(1+02)(1+2)-2=0 2 u2+u2+1 Finally the differential equations are given by 山一凼 d山 ds +u2+1 dq
x y z Figure 20.4: Convergence of the right geodesic path in Figure 20.3. �1 −2(uv) · v + uv · (2v) = = 0 11 2 2[(1 + v2)(1 + u2) − u v2] �2 = �1 22 = �22 = 0 2 � 11 1 v 12 = � u2 + v2 + 1 2 u 12 = u2 + v2 + 1 Finally the differential equations are given by du = p ds dv = q ds dp −2u = ds u2 + v2 + 1 pq dq −2v = ds u2 + v2 + 1 pq. 9