1. Degree reduction of Bézier curves Lizheng Lu lulz_zju@yahoo.com.cn Mar. 8, 2006

2. Outline • Overview • Recent developments • Our work

3. Problem formulation Problem I: Given a curve of degree n in , to find a curve of degree m, such that,

4. An example Degree from 7 to 4

5. Two kinds of methods • Component-wise • Vector decomposition • Degree reduction at each decomposition • Combining all the components • Euclidean [Brunnett et al., 1996] • Consider all the components together

6. Unconstrained and constrained degree reduction

7. Constrained degree reduction Problem II: Given a curve of degree n in , to find a curve of degree m, such that, I) II)

8. Metric choice • -norms on C[0,1] • Weighted -norms • Others • Control points perturbing

9. Approximation Theory • L1-norm • Chebyshev polynomials of second kind • L2-norm • Legendre polynomials • L∞-norm • Chebyshev polynomials of first kind

10. Lp-norms • L1-norm • Kim and Moon, 1997 • L2-norm • Ahn et al., 2004; Chen and Wang, 2002; Eck, 1995; Zheng and Wang, 2003; Zhang and Wang, 2005; • L∞-norm • Eck, 1993; Ahn, 2003

11. Present status • Unconstrained • Solved and very mature • Constrained (Optimal approximation) • Solved for L2-norm • Unsolved for L1-norm and L∞-norm • Some methods have been proposed, but not optimal

12. Outline • Overview • Recent developments • Our work

13. Multiple degree reduction and elevation of Bézier curves using Jacobi-Bernstein basis transformations Rababah, A., Lee, B.G., Yoo, J. Submitted to Applied Numerical Mathematics

14. Main contribution • Unified matrix representations • An unconstrained component-wise method • Explicit • Simple and efficient • Explicit approximating error • Include three previous methods • Lp-norm, p =1, 2, ∞

15. Jacobi polynomials [Szegö, 1975] Orthonormal on [0,1] with

16. Special kinds of orthonormal polynomials • α=β=-1/2 • Chebyshev polynomials of second kind • α=β=0 • Legendre polynomials • α=β=1/2 • Chebyshev polynomials of first kind

17. Jacobi-Bernstein basis transformation [Rababah, 2004] Jacobi Bernstein

18. μ,ν= 0,1, … ,n

19. Lemma

21. Degree elevation (1) (2) (3) (4)

22. Theorem for degree elevation

23. Jacobi-weighted L2-norm

24. Jacobi-weighted L2-norm

25. Special cases forJacobi-weighted L2-norm • α=β=-1/2 • L∞ -approximation • α=β=0 • L2 -approximation • α=β=1/2 • L1 -approximation

26. Degree reduction by degree elevation ?

27. Degree reduction by degree elevation ? = 0

28. Degree reduction by Jacobi polynomials ?

29. Degree reduction by Jacobi polynomials ? (1)

30. Degree reduction by Jacobi polynomials ? (1) (2)

31. Degree reduction by Jacobi polynomials ? (1) (2) (3)

32. Degree reduction by Jacobi polynomials (1) (2) (3) (4)

33. Theorem for degree reduction

34. Error estimation

35. Example

36. Example

37. Summary:Advantages • Unified matrix representations • Explicit approximating error • Optimal approximation under unconstrained degree reduction • Include three previous methods • Lp-norm, p=1, 2, ∞

38. Summary:Disadvantages • Constrained degree reduction • Unsolved • Challenge to unified representation ?

39. Outline • Overview • Recent developments • Our work

40. Optimal multi-degree reduction of Bézier curves with G1-continuity Lizheng Lu and Guozhao Wang To be published in JZUS

41. Motivation:parametric representations Optimal approximation not unique

42. Motivation:Geometric Hermite Interpolation • Theorem. [Boor et al., 1987] If the curvature at one endpoint is not vanished, a planar curve can be interpolated by cubic spline with G 2-continuity and that the approximation order is 6. BHS method. More methods about GHI. [Degen, 2005]

43. Main contributions • Multi-degree reduction • G1: position and tangent direction • Minimize Euclidean distance between control points • Optimal approximation

44. An example Ahn et al., 2004 Ours

45. Problem Problem III: Given a curve of degree n in , to find a curve of degree m, such that, I) Gk-continuous: II)

46. Special cases • G0-continuity • Endpoint interpolation • G1-continuity • Position and tangent direction • G2-continuity • G1 + curvature

47. Main challenges • Consider all the components together • Control points free moving • How to be optimal? • Error estimating • Numerical problems • Convergence, uniqueness, stability, etc.

48. Algorithm overview • G1 condition • Discrete coefficient norm • Through degree elevation • Solution and improvement • Numerical methods

49. G1-continuous Goal Given

50. G1-continuous Goal Given G1 continuous at both endpoints