1 / 92

Degree reduction of Bézier curves

Degree reduction of Bézier curves. Lizheng Lu lulz_zju@yahoo.com.cn Mar. 8, 2006. Outline. Overview Recent developments Our work. Problem formulation. Problem I: Given a curve of degree n in , to find a curve of degree m , such that,. An example.

alize
Télécharger la présentation

Degree reduction of Bézier curves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

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

  21. Theorem for degree elevation

  22. Jacobi-weighted L2-norm

  23. Jacobi-weighted L2-norm

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

  25. Degree reduction by degree elevation ?

  26. Degree reduction by degree elevation ? = 0

  27. Degree reduction by Jacobi polynomials ?

  28. Degree reduction by Jacobi polynomials ? (1)

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

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

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

  32. Theorem for degree reduction

  33. Error estimation

  34. Example

  35. Example

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

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

  38. Outline • Overview • Recent developments • Our work

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

  40. Motivation:parametric representations Optimal approximation not unique

  41. 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]

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

  43. An example Ahn et al., 2004 Ours

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

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

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

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

  48. G1-continuous Goal Given

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

More Related