1. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Spline Methods in CAGD byung-gook lee Dongseo Univ. lbg@dongseo.ac.kr http://kowon.dongseo.ac.kr/~lbg/

2. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Contents Affine combination Lagrange Bezier Spline B-spline E-spline Box-spline Refinement relation Cross-sectional volumes Subdivision schemes Reference “ Spline Methods Draft ” Spline Methods Draft Tom Lyche and Knut Morken “ Subdivision Methods for Geometric Design: A Constructive Approach ” Joe Warren and Henrik Weimer Reference “ Spline Methods Draft ” Spline Methods Draft Tom Lyche and Knut Morken “ Subdivision Methods for Geometric Design: A Constructive Approach ” Joe Warren and Henrik Weimer http://www.uio.no/studier/emner/matnat/ifi/INF-MAT5340/v04/litteratur.xml http://www.subdivision.org/subdivision/index.jsp http://www.microsoft.com/typography/ClearTypeInfo.mspx http://www.cgal.org/

3. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Affine combination Linear combinations Affine(Barycentric) combinations Convex combinations Barycentric coordinates

4. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Affine combination Euclidean coordinate system Coordinate-free system

5. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Polynomial interpolation

6. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Polynomial interpolation Lagrange polynomials

7. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Examples of cubic interpolation

8. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Bezier

9. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Representation Bezier

10. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Bezier Paul de Faget de Casteljau, Citroen, 1959 Pierre Bezier, Renault, UNISUF system, 1962 A.R. Forrest, Cambridge, 1968

11. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Properties of Bezier Affine invariance Convex hull property Endpoint interpolation Symmetry Linear precision Pseudo-local control Variation Diminishing Property

12. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Spline curves J. Ferguson, Boeing Co., 1963 C. de Boor, W. Gordon, General Motors, 1963 to interpolate given data piecewise polynomial curves with certain differentiability constraints not to design free form curves

13. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Piecewise cubic hermite interpolation

14. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Cubic spline interpolation

15. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Linear B-spline

16. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Quadratic B-spline

17. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Quadratic B-spline

18. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline Recurrence Relation Bernstein polynomial

19. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline

20. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline Smoothness=Degree-Multiplicity

21. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Representation B-spline

22. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Cubic B-spline

23. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline space

24. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline C. de Boor, 1972 W. Gordon, Richard F. Riesenfeld, 1974 Larry L. Schumaker Tom Lyche Nira Dyn

25. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline basis functions B 0 ( x ) = ( 1 ; 0 · x < 1 ; 0 ; o t h erw i se B 1 ( x ) = 8 > < > : x ; 0 · x < 1 ; 2 ¡ x ; 1 · x < 2 ; 0 ; o t h erw i se B 2 ( x ) = 8 > > > < > > > : 1 2 x 2 ; 0 · x < 1 ; ¡ 3 2 + 3 x ¡ x 2 ; 1 · x < 2 ; 1 2 ( ¡ 3 + x ) 2 ; 2 · x < 3 ; 0 ; o t h erw i se B ( x: 0 ; 1 ) B ( x: 0 ; 1 ; 2 ) B ( x: 0 ; 1 ; 2 ; 3 )

26. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement relation for B-spline B 0 ( x ) = B 0 ( 2 x ) + B 0 ( 2 x ¡ 1 ) B 1 ( x ) = 1 2 B 1 ( 2 x ) + B 1 ( 2 x ¡ 1 ) + 1 2 B 1 ( 2 x ¡ 2 ) B 2 ( x ) = 1 4 B 2 ( 2 x ) + 3 4 B 2 ( 2 x ¡ 1 ) + 3 4 B 2 ( 2 x ¡ 2 ) + 1 4 B 2 ( 2 x ¡ 3 ) B n ( x ) = 1 2 n n + 1 X i = 0 µ n + 1 i ¶ B n ( 2 x ¡ i )

27. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Repeated integration for B-spline B 0 ( x ) = ( 1 ; 0 · x < 1 ; 0 ; o t h erw i se B n ( x ) = Z 1 0 B n ¡ 1 ( x ¡ t ) d t = B 0 ( x ) ­ B n ¡ 1 ( x ) p ( x ) ­ q ( x ) = Z 1 ¡ 1 p ( t ) q ( x ¡ t ) d t

28. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr E-splines ½ ® ( t ) = 1 + ( t ) e ® t ½ ¡ ! ® ( t ) = P N d m = 1 P n ( m ) n = 1 c m ; n t n ¡ 1 + ( n ¡ 1 ) ! e ® ( m ) ( t ) ½ ¡ ! ® ( t ) = ( ½ ® 1 ­ ½ ® 2 ­ ¢¢¢ ­ ½ ® n )( t ) w h ere ¡ ! ® = ( ® 1 ;:::; ® N ¡ 1 ; ® N ) ¡ ! ® = ( ® n ( 1 ) ( 1 ) ; ® n ( 2 ) ( 2 ) ;:::; ® n ( N d ) ( N d ) ) w h ere P N d m = 1 n ( m ) = N

29. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr E-splines ¯ ® ( t ) = ½ ® ( t ) ¡ e ® ½ ® ( t ¡ 1 ) = ( e ® t ; 0 · t < 1 ; 0 ; o t h erw i se ¯ ¡ ! ® ( t ) = ( ¯ ® 1 ­ ¯ ® 2 ­ ¢¢¢ ­ ¯ ® n )( t )

30. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Truncated powers for B-spline B 1 ( x ) = c 1 ( x ) ¡ 2 c 1 ( x ¡ 1 ) + c 1 ( x ¡ 2 ) c n ( x ) = ( 1 n ! x n ; x ¸ 0 ; 0 ; o t h erw i se B 2 ( x ) = c 2 ( x ) ¡ 3 c 2 ( x ¡ 1 ) + 3 c 2 ( x ¡ 2 ) ¡ c 2 ( x ¡ 3 ) B n ( x ) = n + 1 X i = 0 ( ¡ 1 ) i µ n + 1 i ¶ c n ( x ¡ i )

31. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Cross-sectional Volumes c n ( x ) = 1 p n + 1 vo l n "( f t 0 ; t 1 ;:::; t n g 2 ( R + ) n + 1 j n X i = 0 t i = x )#

32. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline basis as Cross-sectional Volumes B n ( x ) = 1 p n + 1 vo l n "( f t 0 ; t 1 ;:::; t n g 2 H n + 1 j n X i = 0 t i = x )#

33. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Cross-sectional Volumes for subcubes B n ( x ) = 1 p n + 1 vo l n "( f t 0 ; t 1 ;:::; t n g 2 H n + 1 j n X i = 0 t i = x )#

34. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Bivariate Tensor Product B-spline B n ; m ( x ; y ) = B n ( x ) B m ( y )

35. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Box-spline as Cross-sectional Volumes X = ff a i ; b i g 2 z 2 j i = 0 ; 1 ;:::; n g B n ( x ) = 1 p n + 1 vo l n "( f t 0 ; t 1 ;:::; t n g 2 H n + 1 j n X i = 0 t i = x )# B P n ( x ; y ) = vo l n "( f t 0 ; t 1 ;:::; t n g 2 H n + 1 j n X i = 0 f a i ; b i g t i = f x ; y g )#

36. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Bivariate Box spline over triangular grid

37. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Shadows of boxes

38. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement Relation for B-spline S n ( x ) = 1 2 ( 1 + x ) S n ¡ 1 ( x ) S 0 ( x ) = 1 + x S 1 ( x ) = 1 2 ( 1 + 2 x + x 2 ) S 2 ( x ) = 1 4 ( 1 + 3 x + 3 x 2 + x 3 )

39. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement Relation for Box-spline B P ( x ; y ) = ( 1 ; 0 · x ; y < 1 ; 0 ; o t h erw i se X = f( 1 ; 0 ) ; ( 0 ; 1 )g S ~ P ( x ; y ) = 1 2 ( 1 + x a y b ) S P ( x ; y ) ~ X = X [ f( a ; b )g S P ( x ; y ) = 1 2 ( 1 + x )( 1 + y ) B P ( x ; y ) = B P ( 2 x ; 2 y ) + B P ( 2 x ¡ 1 ; 2 y ) + B P ( 2 x ; 2 y ¡ 1 ) + B P ( 2 x ¡ 1 ; 2 y ¡ 1 )

40. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement Relation for Box-spline ~ X = f( 1 ; 0 ) ; ( 0 ; 1 ) ; ( 1 ; 1 )g S ~ P ( x ; y ) = 1 2 ( 1 + x )( 1 + y )( 1 + xy ) = £ 1 xx 2 ¤ 1 2 2 4 110 121 011 3 5 2 4 1 y y 2 3 5

41. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Subdivision for Box-spline · 1 2 0 0 1 2 ¸ £ 1 2 1 2 ¤ £ 1 ¤ · 1 2 1 2 ¸ ~ X = f( 1 ; 0 ) ; ( 0 ; 1 ) ; ( 1 ; 1 )g

42. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement Relation for Box-spline ~ X = f( 1 ; 0 ) ; ( 0 ; 1 ) ; ( 1 ; 1 ) ; ( ¡ 1 ; 1 )g S ~ P ( x ; y ) = 1 4 ( 1 + x )( 1 + y )( 1 + xy )( 1 + x ¡ 1 y ) = £ x ¡ 1 1 xx 2 ¤ 1 4 2 6 6 4 0110 1221 1221 0110 3 7 7 5 2 6 6 4 1 y y 2 y 3 3 7 7 5

43. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Subdivision for Box-spline ~ X = f( 1 ; 0 ) ; ( 0 ; 1 ) ; ( 1 ; 1 ) ; ( ¡ 1 ; 1 )g · 1 4 0 1 2 1 4 ¸· 0 1 4 1 4 1 2 ¸· 1 4 1 2 0 1 4 ¸· 1 2 1 4 1 4 0 ¸

44. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement Relation for Box-spline ~ X = f( 1 ; 0 ) ; ( 1 ; 0 ) ; ( 0 ; 1 ) ; ( 0 ; 1 )g = £ 1 xx 2 ¤ 1 4 2 4 121 242 121 3 5 2 4 1 y y 2 3 5 S ~ P ( x ; y ) = 1 4 ( 1 + x ) 2 ( 1 + y ) 2

45. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement Relation for Box-spline ~ X = f( 1 ; 0 ) ; ( 1 ; 0 ) ; ( 1 ; 0 ) ; ( 0 ; 1 ) ; ( 0 ; 1 ) ; ( 0 ; 1 )g = £ 1 xx 2 x 3 ¤ 1 16 2 6 6 4 1331 3991 3991 1331 3 7 7 5 2 6 6 4 1 y y 2 y 3 3 7 7 5 S ~ P ( x ; y ) = 1 16 ( 1 + x ) 3 ( 1 + y ) 3

46. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Refinement Relation for Box-spline ~ X = f( 1 ; 0 ) ; ( 1 ; 0 ) ; ( 0 ; 1 ) ; ( 0 ; 1 ) ; ( 1 ; 1 ) ; ( 1 ; 1 )g S ~ P ( x ; y ) = 1 16 ( 1 + x ) 2 ( 1 + y ) 2 ( 1 + xy ) 2 = £ 1 xx 2 x 3 x 4 ¤ 1 16 2 6 6 6 6 4 12100 26620 161061 02662 00121 3 7 7 7 7 5 2 6 6 6 6 4 1 y y 2 y 3 y 4 3 7 7 7 7 5

47. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Condition number

48. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Condition number of B-spline basis Tom Lyche and Karl Scherer, On the p-norm condition number of the multivariate triangular Bernstein basis, Journal of Computational and Applied Mathematics 119(2000) 259-273

49. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Stability

50. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Blossom

51. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Blossom

52. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr B-spline problems Degree Elevation Degree Reduction Knot Insertion Knot Deletion Gerald Farin, Curves and Surfaces for Computer Aided Geometric Design, 4 th ed, Academic Press (1996) Ronald N. Goldman, Tom Lyche, editors, Knot Insertion and Deletion Algorithms for B- Spline Curves and Surfaces, SIAM (1993)

53. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Bezier Degree Reduction

54. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Bezier Degree Reduction Least square method Legendre-Bernstein basis transformations Rida T. Farouki, Legendre-Bernstein basis transformations, Journal of Computational and Applied Mathematics 119(2000) 145-160. Byung-Gook Lee, Yunbeom Park and Jaechil Yoo, Application of Legendre-Bernstein basis transformations to degree elevation and degree reduction, Computer Aided Geometric Design 19(2002) 709-718.

55. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Bezier Degree Reduction with constrained

56. KMMCS, Jan. 2006, Spline Methods in CAGD, lbg@dongseo.ac.kr Contents Affine combination Bezier curves Spline curves B-spline curves Condition number L1-norm spline Quasi-interpolant Box-spline Cross-sectional volumes Refinement relation Subdivision Reference “ Spline Methods Draft ” Tom Lyche and Knut Morken “ Subdivision Methods for Geometric Design: A Constructive Approach ” Joe Warren and Henrik Weimer Reference “ Spline Methods Draft ” Tom Lyche and Knut Morken “ Subdivision Methods for Geometric Design: A Constructive Approach ” Joe Warren and Henrik Weimer