Computer Aided Geometric Design 22 (2005) 815–817
www.elsevier.com/locate/cagd
A note on the paper in CAGD (2004, 21 (2), 181–191) ✩
Ren-Jiang Zhang a,b , Guo-Jin Wang a,∗
a State Key Lab of CAD&CG and Department of Mathematics, Zhejiang University, Hangzhou, 310027, PR China
b China Jiliang University, Zhejiang, Hangzhou, 310018, PR China
Received 15 February 2004; revised 4 July 2004; accepted 29 March 2005
Available online 14 April 2005
Abstract
Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in L2 -norm equals
the best weighted Euclidean approximation of the Bernstein–Bézier coefficients of f in a paper, published in the
journal, Computer Aided Geometric Design 21 (2) (2004) 181–191. In this note, we point out an error in their
paper and give the correct result.
 2005 Elsevier B.V. All rights reserved.
Keywords: Constrained degree reduction; Bézier curve; Weights
In the paper (Ahn et al., 2004) published in this journal, Ahn, Lee, Park and Yoo showed that the
best constrained degree reduction of a given Bézier curve Pn (t) of degree from n to m with C α−1 continuity at the boundary in L2 -norm is equivalent to the best weighted Euclidean approximation of the
vector of Bernstein–Bézier (BB) coefficients of Pn (t) from the vector of degree raised BB coefficients
of polynomials of degree m with C α−1 -continuity at the boundary, and gave some algorithms of degree
reduction. Unhappily, there exists an error in their theory and hence it has brought a corresponding error
in their algorithm, we begin with their Example 2 to explain. For a degree 5 Bézier polynomial
P5 (t) = 5(1 − t)4 t + 4 × 10(1 − t)3 t 2 + 2 × 10(1 − t)2 t 3 + 5 × 5(1 − t)t 4 ,
✩
t ∈ [0, 1]
The result of this paper was independently (if later) found by Hongwu Bai, Zhenglin Ye and Shuxun Wang.
* Corresponding author.
E-mail addresses: zrj@cjlu.edu.cn (R.-J. Zhang), wgj@math.zju.edu.cn (G.-J. Wang).
0167-8396/$ – see front matter  2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.cagd.2005.03.005
816
R.-J. Zhang, G.-J. Wang / Computer Aided Geometric Design 22 (2005) 815–817
4 (t).
Fig. 2. The approximated effect for P5 (t) by Q
Fig. 1. The approximated effect for P5 (t) by Q4 (t).
its vector of BB coefficients is (0, 1, 4, 2, 5, 0). According to the theory of Ahn et al. they have asserted
that the polynomial
Q4 (t) = 125/36 × 4(1 − t)3 t + 35/54 × 6(1 − t)2 t 2 + 25/4 × 4(1 − t)t 3
is the polynomial of best approximation for P5 (t). But this is not true, in fact, by using the method of
undetermined coefficients, we can obtain the best approximation polynomial
4 (t) = 31/12 × 4(1 − t)3 t + 25/18 × 6(1 − t)2 t 2 + 25/4 × 4(1 − t)t 3
Q
4 (t) and Q4 (t), respecfor P5 (t). Figs. 1 and 2 show the dissimilar approximated effects for P5 (t) by Q
tively. We can see that the effect using Q4 (t) is better than one using Q4 (t). We need to check Ahn et al.’s
paper again, and then we have found that Corollary 5.2 in their paper is not right. It should be corrected
as follows:
α,β
α,β
Corollary 5.2. The orthogonal complements of Pm in Pn
the weighted Euclidean inner product of BB coefficients
B b, B cw :=
n
n
n−β
with respect to the L2 -inner product and
bi ci w̃i
i=α
are equal, where

(i+α)···(i+β+1)

 wi (n−i+α)···(n−i+β+1) ; α > β,
α = β,
w̃i = wi ;

 w (n−i+α+1)···(n−i+β) ; α < β.
i
(i+α+1)···(i+β)
wi = n2
i
.
n
i−α
n
i+β
Proof. We just prove for the case α > β. It is easy to compute
1
pi (k) :=
w̃k
1
0
1
1
n
Bkn (t)t α+i (1 − t)β dt =
t α+i+k (1 − t)n+β−k dt
w̃k k
0
R.-J. Zhang, G.-J. Wang / Computer Aided Geometric Design 22 (2005) 815–817
=
n
k−α
n
k+β
n
k
817
(n − k + α) · · · (n − k + β + 1) (α + i + k)!(n + β − k)!
(k + α) · · · (k + β + 1)
(n + α + β + i + 1)!
β
α
i
n!
=
(k − α + l1 )
(n − k − β + l2 )
(k + α + l3 ).
(n + α + β + i + 1)! l =1
l =1
l =1
1
2
As the assertion in Ahn et al.’s paper, we complete the proof.
3
2
Now we recomputed the above example by using our new corollary. Note that the matrix W51,2 in Ahn
et al.’s paper has become
5 0 0
0 5 0
0 0 8
4 (t).
after correction. Thus we can easily obtain a new Q4 (t) which is the same as the above Q
Acknowledgements
Supported by the National Grand Fundamental Research 973 Program of China (No. 2004CB719400),
the National Natural Science Foundation of China (No. 60373033, No. 60333010) and the National
Natural Science Foundation for Innovative Research Groups (No. 60021201).
References
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.