Lu et al. / J Zhejiang Univ Sci A 2007 8(10):1657-1662
1657
A quadratic programming method for optimal degree reduction of
Bézier curves with G
1
-continuity
*
LU Li-zheng
†
, WANG Guo-zhao
(Institute of Computer Graphics and Image Processing, Department of Mathematics, Zhejiang University, Hangzhou 310027, China)
†
E-mail: lulz99@yahoo.com.cn
Received Jan. 22, 2007; revision accepted Apr. 5, 2007
Abstract: This paper presents a quadratic programming method for optimal multi-degree reduction of Bézier curves with
G
1
-continuity. The L
2
and l
2
measures of distances between the two curves are used as the objective functions. The two additional
parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability
condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applica-
tions of degree reduction of Bézier curves with their parameterizations close to arc-length parameterizations are also discussed.
Key words: Degree reduction, Bézier curves, Optimal approximation, G
1
-continuity, Quadratic programming
doi:10.1631/jzus.2007.A1657 Document code: A CLC number: TP391.72
INTRODUCTION
Optimal degree reduction of Bézier curves is an
important task in Computer Aided Geometric Design
(CAGD). Such a process is often required in geomet-
ric modeling, due to its capability to exchange, con-
vert or reduce data, and to compare geometric entities.
It consists of approximating a given curve by another
one of lower degree, and it is frequently required to
preserve some continuity conditions (called con-
straints) at the endpoints.
Many methods have been proposed for degree
reduction of Bézier curves. Watkins and Worsey
(1988) used Chebyshev economization to produce the
best L
∞
-approximation of degree n−1 to a given de-
gree n curve, but without considering endpoints in-
terpolation. Then Ahn (2003) presented a good degree
reduction in L
∞
-norm with constraints of endpoints
continuity by using Jacobi polynomials. Eck (1995)
used constrained Legendre polynomials to minimize
the L
2
-norm between the two curves. Multi-degree
reduction at one time avoiding stepwise computing
was investigated in (Zheng and Wang, 2003; Ahn et
al., 2004), which showed that the optimal approxi-
mation in L
2
-norm can be obtained by different ap-
proaches and that the results are equivalent. Fur-
thermore, one can also find some other methods based
on basis transformations (Lee et al., 2002; Rababah et
al., 2006) and on the active contour model (Pottmann
et al., 2002).
Inspired by geometric Hermite interpolation
[see e.g. (Degen, 2005)], we showed that the problem
of degree reduction can be solved with constraints of
G
1
- and G
2
-continuity (Lu and Wang, 2006a; 2006b).
The main advantage is that the approximation can be
further optimized by the additional parameters pro-
vided by geometric continuity. So we can obtain the
approximating curve with a smaller approximation
error.
In this paper, we propose to use the quadratic
programming method to solve degree reduction of
Bézier curves with G
1
-continuity. We express the
L
2
-distance and the l
2
-distance between two curves as
a quadratic polynomial of two parameters. To meet
Journal of Zhejiang University SCIENCE A
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*
Project supported by the National Natural Science Foundation of
China (No. 60473130) and the National Basic Research Program
(973) of China (No. G2004CB318000)