1. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Graphics Programming Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/

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

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

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

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

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

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

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

9. Graphics Programming 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. Graphics Programming 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

11. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Piecewise cubic hermite interpolation

12. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic spline interpolation

13. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic spline interpolation

14. Graphics Programming 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

15. Graphics Programming 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

16. Graphics Programming 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

17. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Linear splines

18. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quadratic splines

19. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quadratic splines

20. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Representation splines

21. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline Recurrence Relation Bernstein polynomial

22. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline

23. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline Smoothness=Degree-Multiplicity

24. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline basis functions

25. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Refinement relation for B-spline

26. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Repeated integration for B-spline

27. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Truncated powers for B-spline

28. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cross-sectional Volumes

29. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cross-sectional Volumes for subcubes

30. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Box-spline as Cross-sectional Volumes

31. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Bivariate Box spline over triangular grid

32. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Spline space

33. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Univariate spline

34. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic splines

35. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Condition number

36. Graphics Programming 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

37. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Stability

38. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Blossom

39. Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Blossom

40. Graphics Programming 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)