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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 b zier curve surface

Degree reduction of Bézier curve/surface

Lian Zhou

zhoulia5729@yahoo.com.cn

Dec. 14, 2006


Outline

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

Problem Statement

Degree from to

Input: control points of

Output: control points of

Objective function:


Applications

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

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 work1

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 work2

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

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


Key progress

Key progress


Key progress1

Key progress

B--J

D

Jacobi


Strength

Strength

  • Optimal

  • Multi-degree reduction

  • Explicit expression

  • Precise error

  • Less time consuming


Degree reduction of b zier curve surface

Idea

  • Jacobi polynomial

  • Basis transformation


Key progress2

Key progress

D


Jacobi polynomial

Jacobi polynomial

  • .

  • .

  • .

D


Others

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.


Degree reduction of b zier curve surface

Optimal multi-degree reduction of Bézier

curves with -continuity

Lizheng Lu , Guozhao Wang

Computer Aided Geometric Design 23 (2006) 673–683


Problem statement1

Problem statement


Motivation

Motivation


Condition

condition


Algorithm for constrained degree reduction

Algorithm for -constrained degree reduction


Least square method

Least square method


Influence of the parameters

Influence of the parameters ,


A weakness

A weakness

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


Regularization

Regularization


Conjugate gradient method

Conjugate gradient method


Algorithm 1

Algorithm 1


Improvement of the singularities

Improvement of the singularities


Remark

Remark


Constrained degree reduction

-constrained degree reduction


Example 1

Example 1


Example 2

Example 2


Example 3

Example 3


Degree reduction of tensor product b zier surfaces

Degree reduction of tensor product Bézier surfaces


Related work

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.


Our work

Our work

Best

Better

Best locally


Fruit 1

Fruit 1

  • Control points

  • Approximate error


Fruit 2

Fruit 2

  • Control points are

  • Error bound is


Example 11

Example 1

Bézier

Original surface


Error surface

Error surface


Example 21

Bézier

Example 2

Original surface


Error surface1

Error surface


Example 31

Example 3

Bézier

Original surface


Error surface2

Error surface

D


Key progress3

Key progress

Jacobi


Degree reduction of triangular b zier surfaces

Degree reduction of triangular Bézier surfaces

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


Related work1

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

Future work

  • Optimal approximation in various norm

  • Geometry continuous

  • Reduce the degree of a Bézier surface composed of some small Bézier surface holistically


Degree reduction of b zier curve surface

Thanks!


A lemma

A lemma


A lemma1

A lemma

D


Gerald farin

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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