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## Degree reduction of Bézier curve/surface

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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

Problem Statement

Degree from to

Input: control points of

Output: control points of

Objective function:

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

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.

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.

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

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

Strength

- Optimal
- Multi-degree reduction
- Explicit expression
- Precise error
- Less time consuming

Idea

- Jacobi polynomial
- Basis transformation

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.

Optimal multi-degree reduction of Bézier

curves with -continuity

Lizheng Lu , Guozhao Wang

Computer Aided Geometric Design 23 (2006) 673–683

A weakness

- The approximation curve will be singular at the endpoint when or is nearly equal to 0.

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.

Fruit 1

- Control points
- Approximate error

Fruit 2

- Control points are
- Error bound is

Key progress

Jacobi

Degree reduction of triangular Bézier surfaces

- Refer to the report of Lizheng Lu in the Ph.D student seminar on Sep. 13

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

Future work

- Optimal approximation in various norm
- Geometry continuous
- Reduce the degree of a Bézier surface composed of some small Bézier surface holistically

Gerald Farin

- Degree: Ph.DUniversity of Braunschweig, 1979
- Biography:

Gerald Farin joined ASU in 1987. He has also worked at the University of Utah and spent four years in CAD/CAM development at Mercedes-Benz, Stuttgart, Germany. He has taught CAGD tutorials worldwide and has given more than 100 invited lectures worldwide.

- Research:- Computer Aided Geometric Design

- NURBS

- Modeling 3D

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