,,.,..., NURBS. : 2, B PDE. 3, PDE B., PDE. 2, Laplace-Beltrami Giaquinta-Hildebrandt. B PDE [1]). S = {x(u 1, u 2 ) R 3 : (u 1, u 2 ) D R 2 } g αβ =

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1 G 1 B 1, 1 ( 1 LSEC,,,, ) B. Laplace-Beltrami Giaquinta-Hildebrandt, G 1 B. G 1. : B,,,. : TP391.7 :A 1, Bloor Wilson[3] (PDE),,., PDE., (MCF) PDE, ( [2, 7, 14])., MCF G 1, PDE( [6, 19, 22]),., B. B Riesenfeld [15]., Versprille[18] Riesenfeld B, NURBS., NURBS. B NURBS.,.,., B PDE Bloor Wilson [4], B. Qin Terzopoulos NURBS (D-NURBS)[17]. D-NURBS., NURBS G 1. [1], B PDE, G 1 B., PDE B,. PDE B., PDE,. B,,,. [1], ( ) (2004CB318000).. : xuguo@lsec.cc.ac.cn. 1

2 ,,.,..., NURBS. : 2, B PDE. 3, PDE B., PDE. 2, Laplace-Beltrami Giaquinta-Hildebrandt. B PDE [1]). S = {x(u 1, u 2 ) R 3 : (u 1, u 2 ) D R 2 } g αβ = x u α, x u β b αβ = n, x u α uβ S, x u α = x u α, x u α u β = 2 x u α u β, α, β = 1, 2, n = (x u x v )/ x u x v, (u, v) := (u 1, u 2 ),, R 3. [ g αβ ] = [ g αβ ] 1, [ b αβ ] = [ b αβ ] 1, g = det[ g αβ ], b = det[ b αβ ]. S H K H = 1 2 [ gαβ ]:[ b αβ ] and K = b g, (1) A:B A T B. H = Hn, K = Kn.. f S C 1, f s ( [11] 102 ) s f = [x u, x v ][ g αβ ][ f u, f v ] T R 3 (2) = g u f u + g v f v, (3) g u = 1 g (g 22x u g 12 x v ), g v = 1 g (g 11x v g 12 x u ).. f C 1 (S), f ( [20]) f = [x u, x v ][h αβ ][ f u, f v ] T R 3 (4) = 1 g (b 22 f u x u + b 11 f v x v b 12 f u x v b 12 f v x u ) (5) = g u f u + g v f v (6) 2

3 [h αβ ] := 1 g b 22 b 12 b 12 b 11, (7) g u = 1 g (b 22x u b 12 x v ), g v = 1 g (b 11x v b 12 x u ).. v S C 1, v div s (v) = 1 g [ u, v] [ g [ g αβ ] [x u, x v ] T v ]. (8) Laplace-Beltrami (LBO). f C 2 (S), f Laplace-Beltrami ( [12] 83 ) s f = div s ( s f ). (2) (8) s f = [ ] 1 g u, [ g [ g αβ ] [ ] f u, f T ] v v (9) = g u f u + g v f v + g uu f uu + g uv f uv + g vv f vv, (10) g u = ( g 11 (g 22 g 122 g 12 g 222 )+2g 12 (g 12 g 212 g 22 g 112 )+g 22 (g 22 g 111 g 12 g 211 ) ) /g 2, g v = ( g 11 (g 11 g 222 g 12 g 122 )+2g 12 (g 12 g 112 g 11 g 212 )+g 22 (g 11 g 211 g 12 g 111 ) ) /g 2, g uu = g 22 /g, g uv = 2g 12 /g, g vv = g 11 /g. s, s x = 2H. Giaquinta-Hildebrandt (GHO). f C 2 (S), f Giaquinta-Hildebrandt ( [13]) f = div s ( f ). div s, f = [ 1 g u, ] [ g [ hαβ ] [ ] f u, f T ] v v (11) = g u f u + g v f v + g uu f uu + g uv f uv + g vv f vv, (12) g u = [b 11 (g 22 g 122 g 12 g 222 )+2b 12 (g 12 g 212 g 22 g 112 )+b 22 (g 22 g 111 g 12 g 211 )]/g 2, g v = [b 11 (g 11 g 222 g 12 g 122 )+2b 12 (g 12 g 112 g 11 g 212 )+b 22 (g 11 g 211 g 12 g 111 )]/g 2, g uu = b 22 /g, g uv = 2b 12 /g, g vv = b 11 /g. 3

4 2.1 B B, [9, 16], blossoming [5] [8, 10].,.. m, k U = {u 0,, u m+2k } u 0 u i u i+1 u i+2 u m+2k. B N i,0 (u) = N i,k (u) = 1, u [u i, u i+1 ), 0,, i = 0, 1,, m + 2k 1, u u i N i,k 1 (u) + u i+k+1 u N i+1,k 1 (u), i = 0, 1,, m + k 1, u i+k u i u i+k+1 u i+1 (13) 0 0 = 0, i N i,k (u), k. B. m, n k, k B x(u, v) = U = {u 0,, u m+2k }, V = {v 0,, v n+2k }, m+k 1 i=0 n+k 1 j=0 p N i,k (u)n j,k (v), (u, v) Ω := [0, 1] 2,, p R 3 x(u, v). i = 0 m + k 1, j = 0 n + k 1, p,,,. C 2, k PDE G 1 B, PDE : x t + ( f Kn) s( f H n) f H x + (2H f H + 2K f K f ) s x +2 f H s H f H + 2H s f H s ( f 2K f K ) = 0, S(0) = S 0, S(t) = Γ. (14) S L 2 ( [21]) F (S) = f (K, H)dA = f (K, H) g dudv. S f (H, K) C 1 (R R) Lagrange. f H f K f H = 2αH + 2βK + µ, f K = 2γH + 2δK + ν, (15) Ω (14) x t + (γ s x + δ x + νn) s(α s x + β x + µn) 4

5 q q q x 02 x 12 x 22 q 01 q 11 q 21 x 01 x 11 x 21 q 00 q 10 q 20 x00 x 10 x f H x + (2H f H + 2K f K f ) s x + 2 f H s H f H + 2H s f H s ( f 2K f K ) = 0. (16) 3 PDE B 3.1 B, B,. (i) :, ( 3.2). (ii) :,., ( [1]). (iii) : PDE,. PDE( 3.3). 3.2 ( [1]).,. x 11, x 00, x 10, x 20, x 01, x 21, x 02, x 12 x 22 x 11 ( 1)., q 00 = ( 1, 1), q 10 = (0, 1), q 20 = (1, 1), q 01 = ( 1, 0), q 11 = (0, 0), q 21 = (1, 0), q 02 = ( 1, 1), q 12 = (0, 1), q 22 = (1, 1), : x(u, v) = 2 2 c u i v j, c R 3, i=0 j=0 x(q ) = x, i, j = 0, 1, 2. c 10 = 1 2 (x 21 x 01 ), c 01 = 1 2 (x 12 x 10 ), 5

6 c 20 = 1 2 (x 21 + x 01 2x 11 ), c 02 = 1 2 (x 12 + x 10 2x 11 ), c 11 = 1 4 (x 00 + x 22 x 20 x 02 ). (0, 0) x u = c 10, x v = c 01, x uu = 2c 20, x uv = c 11, x vv = 2c 02. f x, ( 2 ), : s f 1 2 g u ( f 21 f 01 ) g v ( f 12 f 10 ), f 1 2 g u ( f 21 f 01 ) g v ( f 12 f 10 ), s f 1 2 g u ( f 21 f 01 ) g v ( f 12 f 10 ) + g uu( f 21 + f 01 2 f 11 ) g uv( f 00 + f 22 f 20 f 02 ) + g vv( f 12 + f 10 2 f 11 ), f 1 2 g u ( f 21 f 01 ) g v ( f 12 f 10 ) + g uu( f 21 + f 01 2 f 11 ) g uv( f 00 + f 22 f 20 f 02 ) + g vv( f 12 + f 10 2 f 11 ), f x u, x v, x uu, x uv x vv x ii 4, x i 1,i, x i+1,i, x i,i 1, x i,i+1 ; x i 1,i 1, x i+1,i 1, x i 1,i+1, x i+1,i+1 x ii. F(u, v) R 3, q i 1,i, q i+1,i, q i,i 1, q i,i+1 q i 1,i 1, q i+1,i 1, q i 1,i+1, q i+1,i+1 x i+s,i+t = F(q i+s,i+t ), s, t = 1, 0, 1,. :, q i 1,i 1 = ( h, h), q i,i 1 = (0, h), q i+1,i 1 = (h, h), q i 1,i = ( h, 0), q i,i = (0, 0), q i+1,i = (h, 0), q i 1,i+1 = ( h, h), q i,i+1 = (0, h), q i+1,i+1 = (h, h), : x(u, v) = 2 2 c u i v j, c R 3. i=0 j=0 c 10 = 1 2h (x i+1,i x i 1,i ). (0, 0), x u = c 10 = 1 2h (F(q i+1,i) F(q i 1,i )) = 1 2h {[F(q ii) + F u h + 1 2! F uuh 2 + O(h 3 )] [F(q ii ) F u h + 1 2! F uuh 2 O(h 3 )]} = F u + O(h 2 ), 6

7 x u., x v, x uu, x uv, x vv. g u, g v, g u, g v, g v, g uu, g uv, g vv, g u, g v, g uu, g uv g vv x u, x v, x uu, x uv x vv,.,, PDE,. u v Hermite, B,,, p (u) p (0) = 1 2 (p(u) p (v) ( [1]) p (v) ), i = 2,, m + k 3, j = 2,, n + k 3, Ω = [0, 1] 2. [ ] [ i ui, v j = M, j ] T, i = 0, 1,, M, j = 0, 1,, N, N, M m + k 1, N n + k 1. x(u, v) B. [ ui, v j ] T, i = 2,, M 2, j = 2,, N 2, (17) PDE, x(u, v). x (k) (u, v) = m+k 1 i=0 n+k 1 j=0 p (k) N i,k(u)n j,k (v) t = kτ PDE, t = (k + 1)τ x (k+1) (u, v). x (k) = x (k) (u i, v j ), (16). (16) : (16) 2 (γ s x + δ x + νn) x t x (k,l) N 1 (i, j) (k+1) x (k), (18) τ w kl (γ kl s x kl + δ kl x kl + ν kl n kl ), (19), w kl, γ kl, δ kl, ν kl n kl. s x kl x kl s x kl w µνx (k+1) µν, (20) x kl (µ,ν) N 1 (k,l) (µ,ν) N 1 (k,l) 7 w µνx (k+1) µν, (21)

8 , w µν w µν. (20) (21) (19), (16) x (k+1) µν. (16) f H x f H x (k) (16) 6 k,l) N 1 (i, j) (2H f H + 2K f K f ) x kl (2H f H + 2K f K f ) f H s x f H x (k) w kl x(k+1) kl, (22) x (k) k,l) N(i, j) k,l) N 1 (i, j).. (18) (23) x (k+1) αβ x (k+1) αβ = m+k 1 i=0 n+k 1 j=0 w kl x(k+1) kl, (23) w kl H(x(k) kl ), (24) p (k+1) N i,k (u αβ )N j,k (v αβ ). [ ] T u i, v j, (16) p (k+1) kl c kl p (k+1) kl + b = 0, i = 2,, M 2, j = 2,, N 2. (25) (25) c kl. B x (k) (u, v), x (k) (u, v). (25), M > m + k 1 N > n + k 1, p (k+1) p (k+2), p (k+3),, p (l) max p (l 1). ɛ ɛ = p (l) τ,.,.., V 0 j u = 0, v = j. x(0, j) x(1, j) V 0 j.. < ɛ x(u, v) = [acos(uπ), asin(uπ), bv] T, [u, v] [0, 1] 2, a b. u = 0, v = j [0, 0, b] T, b x(0, j) x(1, j).,, PDE,. 2 Willmore.,. 8.

9 (a) (b) (c) (d) 2 (a). (b). (c). (d) PDE PDE, 5 PDE,. (a) (b) f = 1 (c) f = H 2 (d) f = K 2 (e) f = H 2 + K 2 (f) f = H 2 K PDE. 3 (b). 3 (c) (e). 3 (d) 3 (f) PDE (a), 5(a) 6(a). 4(b), 5(b) 6(b). G 1. 4(c), 5(c) 6(c). 5 6 G 1. (a) (b) (c) 4 (a). (b). (c) WF. 9

10 (a) (b) (c) 5 (a). (b). (c) WF. (a) (b) (c) 6 (a). (b). (c) WF. (a) (b) (c) 7 (a). (b). (c) WF MCF. 7 (a).. 7 (b). 7 (c) WF MCF. 5, PDE, PDE B, B,., G 1. [1]..,,

11 [2] C. Bajaj and G. Xu. Anisotropic Diffusion of Subdivision Surfaces and Functions on Surfaces. ACM Transactions on Graphics, 22(1):4 32, [3] M. I. G. Bloor and M. J. Wilson. Generating blend surfaces using partial differential equations. Computer Aided Design, 21(3): , [4] M.I.G. Bloor and M.J.Wilson. Representing PDE surfaces in terms of B-splines. Computer Aided Design, 22(6): , [5] Ramshaw L. Blossoming. a connect-the-dots approach to spline. Report 19, Digital,, System Research Center, Palo Alto, CA, [6] U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf, and R. Rusu. A finite element method for surface restoration with smooth boundary conditions. Computer Aided Geometric Design, 21(5): , [7] U. Clarenz, U. Diewald, and M. Rumpf. Anisotropic geometric diffusion in surface processing. In VIS 00: Proceedings of the Conference on Visualization, pages , Los Alamitos, CA, USA, [8] M. G. Cox. The numerical evaluation of B-splines. Jour. Inst. Math. Applic., 10: , [9] H. B. Curry and I. J. Schoenberg. On spline distribution and their limits: the Pólya distribution functions. Bull. Amer. Math. Soc., 53:109, [10] C. de Boor. On calculating with B-splines. Journal of Approximation Theory, 6:50 62, [11] M. P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs, NJ, [12] M. P. do Carmo. Riemannian Geometry. Birkhäuser, Boston, Basel, Berlin, [13] M. Giaquinta and S. Hildebrandt. Calculus of Variations, Vol. I. Number 310 in A Series of Comprehensive Studies in Mathematics. Springer-Verlag, Berlin, [14] Y. Ohtake, A. G. Belyaev, and I. A. Bogaevski. Polyhedral surface smoothing with simultaneous mesh regularization. In Geometric Modeling and Processing 2000 Proceedings, pages [15] R.F. Riesenfeld. Applications of B-spline Approximation to Geometric Problems of Computer-Aided Design. PhD thesis, Syracuse University, [16] I. J. Schoenberg. Contributions to the problem of approximation of equidistance data by analytic functions. Quart. Appl. Math., 4:45 99, [17] D. Terzopoulos and H. Qin. Dynamic NURBS with geometric constraints for interactive sculpting. ACM Transactions on Graphics, 13(2): , [18] K.J. Versprille. Computer-Aided Design Applications of the Rational B-Spline Approximation form. PhD thesis, Syracuse University, [19] G. Xu and Q. Pan. G 1 surface modelling using fourth order geometric flows. Computer-Aided Design, 38(4): , [20] G. Xu and Q. Zhang. Construction of geometric partial differential equations in computational geometry. Mathematica Numerica Sinica, 28(4): , [21] G. Xu and Q. Zhang. A general framework for surface modeling using geometric partial differential equations. Computer Aided Geometric Design, 25(3): , [22] S. Yoshizawa and A. G. Belyaev. Fair triangle mesh generation with discrete elastica. In Geometric Modeling and Processing, pages , Saitama, Japan,

普通高等学校本科专业设置管理规定

普通高等学校本科专业设置管理规定普通高等学校本科专业设置申请表 ( 备案专业适用 ) 学校名称 ( 盖章 ): 学校主管部门 : 专业名称 : 浙江外国语学院浙江省教育厅金融工程专业代码 : 020302 所属学科门类及专业类 : 金融学 / 金融工程类学位授予门类 : 修业年限 :