Full Text:   <3173>

CLC number: TP391.72

On-line Access: 

Received: 2008-04-21

Revision Accepted: 2008-08-29

Crosschecked: 2009-02-09

Cited: 5

Clicked: 5067

Citations:  Bibtex RefMan EndNote GB/T7714

-   Go to

Article info.
1. Reference List
Open peer comments

Journal of Zhejiang University SCIENCE A 2009 Vol.10 No.4 P.554-561


Optimal approximate merging of a pair of Bézier curves with G2-continuity

Author(s):  Ping ZHU, Guo-zhao WANG

Affiliation(s):  Institute of Computer Graphics and Image Processing; more

Corresponding email(s):   gumpforrest1982@yahoo.com.cn, wanggz@zju.edu.cn

Key Words:  Approximate merging, G1-continuity, G2-continuity, Discrete subdivision, Point constraints

Ping ZHU, Guo-zhao WANG. Optimal approximate merging of a pair of Bézier curves with G2-continuity[J]. Journal of Zhejiang University Science A, 2009, 10(4): 554-561.

@article{title="Optimal approximate merging of a pair of Bézier curves with G2-continuity",
author="Ping ZHU, Guo-zhao WANG",
journal="Journal of Zhejiang University Science A",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T Optimal approximate merging of a pair of Bézier curves with G2-continuity
%A Ping ZHU
%A Guo-zhao WANG
%J Journal of Zhejiang University SCIENCE A
%V 10
%N 4
%P 554-561
%@ 1673-565X
%D 2009
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A0820301

T1 - Optimal approximate merging of a pair of Bézier curves with G2-continuity
A1 - Ping ZHU
A1 - Guo-zhao WANG
J0 - Journal of Zhejiang University Science A
VL - 10
IS - 4
SP - 554
EP - 561
%@ 1673-565X
Y1 - 2009
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.A0820301

We present a novel approach for dealing with optimal approximate merging of two adjacent Bézier curves with g2-continuity. Instead of moving the control points, we minimize the distance between the original curves and the merged curve by taking advantage of matrix representation of Bézier curve’s discrete structure, where the approximation error is measured by L2-norm. We use geometric information about the curves to generate the merged curve, and the approximation error is smaller. We can obtain control points of the merged curve regardless of the degrees of the two original curves. We also discuss the merged curve with point constraints. Numerical examples are provided to demonstrate the effectiveness of our algorithms.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1] Ahn, Y.J., Lee, B.G., Park, Y., Yoo, J., 2004. Constrained polynomial degree reduction in the L2-norm equals best weighted Euclidean approximation of Bézier coefficients. Computer Aided Geometric Design, 21(2):181-191.

[2] Chen, F.L., Wu, Y., 2004. Degree reduction of disk Bézier curves. Computer Aided Geometric Design, 21(3):263-280.

[3] Hoschek, J., 1987. Approximate conversion of spline curves. Computer Aided Geometric Design, 4(1-2):59-66.

[4] Hu, Q.Q., Wang, G.J., 2008. Optimal multi-degree reduction of triangular Bézier surfaces with corners continuity in the norm L2. J. Comput. Appl. Math., 215(1):114-126.

[5] Hu, S.M., Tong, R.F., Ju, T., Sun, J.G., 2001. Approximate merging of a pair of Bézier curves. Computer-Aided Design, 33(2):125-136.

[6] Pottmann, H., Leopoldseder, S., Hofer, M., 2002. Approximation with Active B-spline Curves and Surfaces. Proc. Pacific Graphics, p.8-25.

[7] Sunwoo, H., 2005. Matrix representation for multi-degree reduction of Bézier curves. Computer Aided Geometric Design, 22(3):261-273.

[8] Wang, G.J., Wang, G.Z., Zheng, J.M., 2001. Computer Aided Geometric Design (1st Ed.). China Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, p.10-11 (in Chinese).

[9] Wu, Y., Chen, F.L., 2002. Merging a Pair of Disk Bézier Curves. Proc. 2nd Int. Conf. on Computer Graphics and Interactive Techniques in Australasia and South East Asia, p.65-70.

[10] Zheng, J.M., Wang, G.Z., 2003. Perturbing Bézier coefficients for best constrained degree reduction in the L2-norm. Graphical Model, 65(6):351-368.

Open peer comments: Debate/Discuss/Question/Opinion


Please provide your name, email address and a comment

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2024 Journal of Zhejiang University-SCIENCE