1. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Spline Methods in CAGD Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/

2. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Affine combination Linear combinations Affine(Barycentric) combinations Convex combinations Barycentric coordinates

3. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Affine combination Euclidean coordinate system Coordinate-free system

4. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Polynomial interpolation

5. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Polynomial interpolation Lagrange polynomials

6. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Examples of cubic interpolation

7. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Bezier

8. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Representation Bezier

9. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Properties of Bezier Affine invariance Convex hull property Endpoint interpolation Symmetry Linear precision Pseudo-local control Variation Diminishing Property

10. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Linear splines

11. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quadratic splines

12. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quadratic splines

13. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Representation splines

14. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline Recurrence Relation Bernstein polynomial

15. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline

16. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline Smoothness=Degree-Multiplicity

17. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Spline space

18. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Univariate spline

19. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic splines

20. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Bezier Paul de Faget de Casteljau, Citroen, 1959 Pierre Bezier, Renault, UNISUF system, 1962 A.R. Forrest, Cambridge, 1970

21. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail: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

22. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline C. de Boor, 1972 W. Gordon, Richard F. Riesenfeld, 1974 Larry L. Schumaker Tom Lyche Nira Dyn

23. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Piecewise cubic hermite interpolation

24. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic spline interpolation

25. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic spline interpolation

26. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Spline interpolation based on the 1-norm Cubic Spline Interpolation with Natural boundary condition

27. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Condition number

28. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail: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

29. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Stability

30. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Blossom

31. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Blossom

32. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail: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)

33. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Bezier Degree Reduction

34. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail: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.

35. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Bezier Degree Reduction with constrained

36. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quasi-interpolants

37. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Reproduce spline space

38. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr A cubic quasi-interpolant

39. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quasi-interpolants local property the same order as the best spline approximation can be computed directly without solving systems of equations Lyche, T. and L. L. Schumaker, Local spline approximation methods, Journal of Approximation Theory 15(1975) 294-325. Lyche, T.,L. L. Schumaker and S. Stanley, Quasi-interpolants based on trogonometric splines, Journal of Approximation Theory 95(1998) 280-309.

40. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Contents Affine combination Bezier curves Spline curves B-spline curves Condition number L1-norm spline Quasi-interpolant Reference “ Spline Methods Draft ” Tom Lyche and Knut Morken Reference “ Spline Methods Draft ” Tom Lyche and Knut Morken http://kowon.dongseo.ac.kr/~lbg/cagd/