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Degree reduction of Bézier curve/surface

Degree reduction of Bézier curve/surface. Lian Zhou zhoulia5729@yahoo.com.cn Dec. 14, 2006. Outline. Introduction of degree reduction in CAGD Related work Degree reduction of curves Degree reduction of tensor product Bézier surfaces Degree reduction of triangular Bézier surfaces

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Degree reduction of Bézier curve/surface

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  1. Degree reduction of Bézier curve/surface Lian Zhou zhoulia5729@yahoo.com.cn Dec. 14, 2006

  2. Outline • Introduction of degree reduction in CAGD • Related work • Degree reduction of curves • Degree reduction of tensor product Bézier surfaces • Degree reduction of triangular Bézier surfaces • Our work and future work

  3. Problem Statement Degree from to Input: control points of Output: control points of Objective function:

  4. Applications • Data transfer and exchange • Data compression • Data comparison • Surface intersection • Curve smoothness • Boolean operations and rendering • Michael S. Floater, High order approximation of rational curves by polynomial curves, Computer Aided Geometric Design 23 (2006) 621–628 • CONSURF BUILD UNISURF CATIA COMPAC Geomod PADL GEMS

  5. Early work • Based on the control points approaching • Inverse of elevation • Forrest, A.R., Interactive interpolation and approximation by Bézier curve, The Computer Journal, 15(1972), 71-79. • G. Farin, Algorithms for rational Bezier curves, Computer Aided Design 15 (1983) 73–77. • Approximate conversion • Danneberg, L., and Nowacki, H., Approximate conversion of surface representations with polynomial bases, Computer Aided Geometric Design, 2(1985), 123-132. • Hoschek, J., Approximation of spline curves, Computer Aided Geometric Design, 4(1987), 59-66.

  6. Early work • Constrained optimization • Moore, D. and Warren, J., Least-square approximation to Bezier curves and surfaces in James Arvo eds. Computer Gemes (II), Academic Press, New York, 1991. • Lodha, S. and Warren, J., Degree reduction of Bezier simplexes, Computer Aided Design, 26(1994), 735-746. • Perturbing control points • 胡事民，CAD系统数据通讯中若干问题的研究 : [博士学位论文], 杭州, 浙江大学数学系, 1996. • Hu, S.M., Sun, J.G., Jin T.G., et al., Approximate degree reduction of Bezier curves, Tsinghua Science and Technology, 3(1998), 997-1000.

  7. Early work • Based on the basis transformation • Watkins, M. and Worsey, A., Degree reduction for Bézier curves, Computer Aided Design, 20(1988), 398-405. • Lachance, M.A., Chebyshev economization for parametric surfaces. Computer Aided Geometric Design, 5(1988), 195-208. • Eck, M., Degree reduction of Bézier curves, Computer Aided Geometric Design, 10(1993), 237-257.69 • Bogacki, P., Weinstein, S. and Xu, Y., Degree reduction of Bézier curves by uniform approximation with endpoint interpolation, Computer Aided Design, 27(1995), 651-661. • Eck, M., Least squares degree reduction of Bézier curves, Computer Aided Design, 27(1995), 845-851.48

  8. Recent work • Optimal multi-degree reduction • Chen Guodong, Wang Guojin, Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity. Computer Aided Geometric Design, 2002,19: 365-377 • Zheng, J., Wang, G., Perturbing Bézier coefficients for best constrained degree reduction in the -norm. Graphical Models 2003, 65, 351–368. • Zhang Renjiang and Wang Guojin, Constrained Bézier curves’ best multi-degree reduction in the -norm, Progress in Natural Science, 2005, 15(9): 843-850 • Others

  9. Key progress

  10. Key progress B--J D Jacobi

  11. Strength • Optimal • Multi-degree reduction • Explicit expression • Precise error • Less time consuming

  12. Idea • Jacobi polynomial • Basis transformation

  13. Key progress D

  14. Jacobi polynomial • . • . • . D

  15. Others • Lutterkort, D., Peters, J., Reif, U., 1999. Polynomial degree reduction in the -norm equals best Euclidean approximation of Bézier coefficients. Computer Aided Geometric Design 16, 607–612. • Ahn, Y.J., Lee, B.G., Park, Y., Yoo, J., 2004. Constrained polynomial degree reduction in the -norm equals best weighted Euclidean approximation of Bézier coefficients. Computer Aided Geometric Design 21, 181–191.

  16. Optimal multi-degree reduction of Bézier curves with -continuity Lizheng Lu , Guozhao Wang Computer Aided Geometric Design 23 (2006) 673–683

  17. Problem statement

  18. Motivation

  19. condition

  20. Algorithm for -constrained degree reduction

  21. Least square method

  22. Influence of the parameters ,

  23. A weakness • The approximation curve will be singular at the endpoint when or is nearly equal to 0.

  24. Regularization

  25. Conjugate gradient method

  26. Algorithm 1

  27. Improvement of the singularities

  28. Remark

  29. -constrained degree reduction

  30. Example 1

  31. Example 2

  32. Example 3

  33. Degree reduction of tensor product Bézier surfaces

  34. Related work • 陈发来,丁友东, 矩形域上参数曲面的插值降阶逼近, 高等学校计算数学学报(计算几何专辑),1993,7,22-32 • Hu Shimin, Zheng Guoqin, Sun Jiaguang. Approximate degree reduction of rectangular Bézier surfaces, Journal of Software, 1997, 4(4): 353-361 • 周登文, 刘芳, 居涛, 孙家广, 张量积Bézier曲面降阶逼近的新方法, 计算机辅助设计与图形学学报, 2002 14(6), 553-556 • Chen Guodong and Wang Guojin, Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations, SCIENCE IN CHINA, Series F,2002, 45(1): 51~58 • 郭清伟, 朱功勤, 张量积Bézier曲面降多阶逼近的方法,计算机辅助设计与图形学学报, 2004,16(6) • 章仁江, CAGD中曲线曲面的降阶与离散技术的理论研究: [博士学位论文],杭州,浙江大学数学系，2004.

  35. Our work Best Better Best locally

  36. Fruit 1 • Control points • Approximate error

  37. Fruit 2 • Control points are • Error bound is

  38. Example 1 Bézier Original surface

  39. Error surface

  40. Bézier Example 2 Original surface

  41. Error surface

  42. Example 3 Bézier Original surface

  43. Error surface D

  44. Key progress Jacobi

  45. Degree reduction of triangular Bézier surfaces • Refer to the report of Lizheng Lu in the Ph.D student seminar on Sep. 13

  46. Related work • Hu SM, Zuo Z, Sun JG. Approximate degree reduction of triangular Bézier surface. Tsinghua Science and Technology 1998;3(2):1001–4 • Rababah A. degree reduction of triangular Bézier surfaces with common tangent planes at vertices. International Journal of Computational Geometry & Applications 2005;15(5):477–90. • 郭清伟, 陶长虹, 三角Bézier曲面的降多阶逼近.复旦学报(自然科学版) 2006 Vol.45 No.2 P.270-276 • Lizheng Lu, Guozhao Wang, Multi-degree reduction of triangular Bézier surfaces with boundary constraints. Computer-Aided Design 38 (2006) 1215–1223

  47. Future work • Optimal approximation in various norm • Geometry continuous • Reduce the degree of a Bézier surface composed of some small Bézier surface holistically

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