Computer Aided Geometric Design 19 (2002) 365–377
www.elsevier.com/locate/comaid
Optimal multi-degree reduction of Bézier curves with constraints
of endpoints continuity
Guo-Dong Chen, Guo-Jin Wang ∗
State Key Laboratory of CAD&CG, Institute of Computer Images and Graphics, Zhejiang University,
Hangzhou 310027, China
Abstract
Given a Bézier curve of degree n, the problem of optimal multi-degree reduction (degree reduction of more than
one degree) by a Bézier curve of degree m (m < n−1) with constraints of endpoints continuity is investigated. With
respect to L2 norm, this paper presents one approximate method (MDR by L2 ) that gives an explicit solution to
deal with it. The method has good properties of endpoints interpolation: continuity of any r, s (r, s 0) orders can
be preserved at two endpoints respectively. The method in the paper performs multi-degree reduction at one time
and does not need the stepwise computing. When applied to the multi-degree reduction with endpoints continuity
of any orders, the MDR by L2 obtains the best least squares approximation. Comparison with another method of
multi-degree reduction (MDR by L∞ ), which achieves the nearly best uniform approximation with respect to L∞
norm, is also given. The approximate effect of the MDR by L2 is better than that of the MDR by L∞ . Explicit
approximate error analysis of the multi-degree reduction methods is presented.  2002 Published by Elsevier
Science B.V.
Keywords: Degree reduction; Bézier curve; Optimal approximation; Endpoint continuity
1. Introduction
The exchanging of product model data between various CAD/CAM systems is often needed. However
the representation schemes of parametric curves and surfaces are varied in different geometric modeling
systems. Such as, the maximum degree, which different computer systems can deal with, varies quite
dramatically. Therefore for the data communication between diverse CAD/CAM systems, curves of high
degree must be approximated by curves of lower degree due to variation in the maximum degree allowed.
* Corresponding author.
E-mail address: wgj@math.zju.edu.cn (G.-J. Wang).
0167-8396/02/$ – see front matter  2002 Published by Elsevier Science B.V.
PII: S 0 1 6 7 - 8 3 9 6 ( 0 2 ) 0 0 0 9 3 - 6
366
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
Thus the problem of how to optimally approximate a given parametric curve by a lower degree curve
within a certain error bound has arisen in CAGD.
In recent years, many methods have been used to reduce the degree of Bézier curves. The problem
of degree reduction is viewed as the inverse process of degree elevation (Forrest, 1972; Farin, 1983;
Piegl and Tiller, 1995). In general, degree reduction is not exactly possible in contract to the reverse
process of degree elevation. Thus degree reduction approximation of parametric curves and surfaces has
been widely studied. Discrete points and derivative information of original curve are used in degree
reduction approximation (Danneberg and Nowacki, 1985; Hoschek, 1987). The degree reduction of
Bézier curves can also been done by using Chebyshev polynomials approximation (Watkins and Worsey,
1988; Lachance, 1988). A simple geometric constructive method of degree reduction with constrained
Chebyshev polynomials is presented in (Eck, 1993), while a least squares method of degree reduction
with constrained Legendre polynomials is presented in (Eck, 1995). Using conversion of bases between
Chebyshev and Bernstein bases, a method of degree reduction with the reduction matrix is developed
(Bogacki et al., 1995).
From the practical point of view, when transmitting geometric information from one system to another,
it is our general aim to ensure a high degree of accuracy and the least possible loss of geometric
information. Moreover degree reduction schemes often need to be combined with the subdivision
algorithm, i.e., a high degree curve is approximated by a number of lower degree curve segments and
continuity between adjacent lower degree curve segments should be maintained.
Unfortunately, all methods known up-to-now have some disadvantages. First, they have no explicit
solutions for optimal multi-degree reduction with constraints of endpoints continuity of high order and
have to be determined by numeric algorithms such as Remes-type algorithm. Secondly, for the multidegree reduction, most methods need stepwise approximation and hence a lot of time for computing is
spent. Thirdly, most methods in general cannot achieve the optimal approximation any more.
The aim of this paper is to find out the method of optimal multi-degree reduction with endpoints
continuity of high order. Based on the inverse of degree elevation and orthogonal polynomial
approximation theory, a method called MDR by L2 is presented in this paper, which gives an explicit
solution and has the optimal precision for multi-degree reduction with constrains of endpoint continuity
with respect to L2 norm. The method can perform the degree reduction of more than one degree at a
time and avoid the stepwise computing. The geometric interpolation information of endpoints between
the original curve and the degree-reduced curve can be preserved, i.e., continuity of any r, s (r, s 0)
orders can be preserved at two endpoints respectively. In this paper another method called MDR by L∞
is introduced, which also gives an explicit solution for multi-degree reduction with respect to L∞ norm
and which is compared with the MDR by L2 . For the constraints of endpoint continuity of any orders, the
MDR by L2 can obtain the best least squares approximation of multi-degree reduction and the MDR by
L∞ achieves nearly best uniform approximation. The approximate effect of the MDR by L2 is obviously
better than that of the MDR by L∞ and the computational examples also display it.
The organization of the paper is as follows. The second section is preliminaries. In the third section the
best least square multi-degree reduction of Bézier curves with constrains of endpoints continuity (MDR
by L2 ) and the approximate error are presented. The fourth section compares the MDR by L2 with the
MDR by L∞ . Section five presents the conclusion.
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
367
2. Preliminaries
In this paper, Πn denotes all√real polynomials of degree at most n. The denotation · denotes the
Euclidean vector norm v = v, v, and d∗ (·, ·) denotes as the distance function with respect to L∗
norm.
Given a degree n Bézier curve
P (t) =
n
P i Bin (t),
t ∈ [0, 1],
(1)
i=0
where Bin (t) = ni t i (1 − t)n−i is the Bernstein polynomial of degree n and {P i }ni=0 are the control points.
The problem of degree reduction is to find out a Bézier curve Q(t), t ∈ [0, 1], of degree m (m < n)
such that a suitable distance function d(P , Q) is minimized. Obviously the approximate result of degree
reduction will vary much according to the chosen distance function. In this paper, we use the least squares
(L2 ) norm and the uniform (L∞ ) norm to measure the approximate error. The distance functions between
P (t) and Q(t) with respect to L2 and L∞ norms on the interval [a, b] are defined as follows respectively:
b
(2)
d2 (P , Q) = P (t) − Q(t)2 dt,
a
d∞ (P , Q) = max P (t) − Q(t).
t ∈[a,b]
(3)
Now the optimal multi-degree reduction with constrains of endpoint continuity can be described as
follows.
Definition 1. Given a degree n Bézier curve P n (t), approximate it optimally by a degree m (m < n − 1)
Bézier curve Qm (t) with respect to different distance function and avoid the stepwise computing,
whereby continuity of any r, s (r, s 0) orders should be preserved at two endpoints respectively. This
problem is called optimal multi-degree reduction with constrains of endpoint continuity.
The key of the optimal multi-degree reduction in the paper is the constraints of endpoint continuity
and the minimization of the approximate error distance function, i.e., find the best least squares and best
uniform approximation with respect to the distance functions d2 and d∞ , respectively.
We first present the following property of Bernstein polynomials:
Lemma 1. Bernstein polynomial Bim (t) of degree m, 0 t 1, can be represented by
Bim (t) =
n+i−m
(m,n) n
bi,j
Bj (t),
n > m, i = 0, 1, . . . , m,
(4)
j =i
where
(m,n)
=
bi,j
m n−m
n
.
i
j −i
j
(5)
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G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
Proof. From the definition of Bernstein polynomials, we have
Bim (t)
m
m
m−i i
=
(1 − t) t =
(1 − t)m−i t i (1 − t + t)n−m
i
i
=
n−m
j =0
=
m
m−i i n − m
(1 − t) t
(1 − t)n−m−j t j
i
j
n−m
n+i−m
(m,n)
n−m
n
n
n−(i+j ) i+j
t
=
bi,j Bjn (t).
(1 − t)
j
i +j
i+j
j =i
m
i
j =0
✷
In the following theorem, a part of the control points of the approximate degree reduced Bézier curve
are firstly derived to satisfy the geometric interpolation constraints of two endpoints.
Theorem 1. Given a degree n Bézier curve P n (t) =
then the curve can be expressed as
P n (t) ≡ Q(t) =
r
Qi Bim (t) +
i=0
n−s−1
i=r+1
n
n
i=0 P i Bi (t),
P Ii Bin (t) +
m
t ∈ [0, 1], and let r + s < m < n − 1,
Qi Bim (t),
(6)
i=m−s
if and only if Eqs. (7)–(9) is satisfied:

j −1

1
1

(m,n)

bi,j Qi , j = 1, 2, . . . , r,
Q0 = (m,n) P 0 , Qj = (m,n) P j −



b
b

j,j
0,0
i=max(0,j −(n−m))



1
Qm = (m,n) P n ,

bm,m



j −1


1

(m,n)

bm−i,n−j Qm−i , j = 1, 2, . . . , s.
P n−j −

 Qm−j = b(m,n)
m−j,n−j
i=max(0,j −(n−m))
(7)
When m − s > n + r − m,

r

(m,n)
I


bi,j
Qi , j = r + 1, r + 2, . . . , n + r − m,
Pj = Pj −



i=0
P Ij = P j , m − s − 1 > n + r − m; j = n + r − m + 1, n + r − m + 2, . . . , m − s − 1,

s



(m,n)
I


=
P
−
bm−i,j
Qm−i , j = m − s, m − s + 1, . . . , n − s − 1.
P
j
 j
i=0
(8)
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
369
When m − s n + r − m,

r

(m,n)

I

=
P
−
bi,j
Qi , m − s − 1 > r; j = r + 1, r + 2, . . . , m − s − 1,
P

j
j



i=0


r
s


 I
(m,n)
(m,n)
bi,j
Qi −
bm−i,j
Qm−i , j = m − s, m − s + 1, . . . , n + r − m,
Pj = Pj −
i=0
i=0



s


(m,n)


bm−i,j
Qm−i ,
P Ij = P j −




i=0

m − s − 1 > r; j = n + r − m + 1, n + r − m + 2, . . . , n − s − 1.
(9)
Where {Qi }ri=0 , {Qi }m
i=m−s are the part control points of degree reduced curve Qm (t) of degree m,
I n−s−1
{P i }i=r+1 are the unknown accessorial control points, and the following two equations must be satisfied.
dλ Qm (0) dλ P n (0)
=
,
dt λ
dt λ
dµ Qm (1) dµ P n (1)
=
,
dt µ
dt µ
λ = 0, 1, . . . , r,
µ = 0, 1, . . . , s.
(10)
(m,n) n
bi,j
Bj (t). Substitute it into the right side of Eq. (6)
Proof. By Lemma 1, we have Bim (t) = n+i−m
j =i
(m,n)
and express it in matrix form. Let bi,j = 0 (j < i or j − i > n − m), then


(Q0 , Q1 , . . . , Qr ) 

(m,n)
b0,0
(m,n)
b0,1
(m,n)
b1,1

...
...
...
(m,n)
b0,n−m
(m,n)
b1,n−m
(m,n)
b1,n−m+1
(m,n)
br,r
(m,n)
br,r+1
...



(m,n)
br,n−m+r
B0n
B1n
..
.




n
Bn+r−m


n
Br+1
n

 Br+2

+ P Ir+1 , P Ir+2 , . . . , P In−s−1 
 ..  + (Qm−s , . . . , Qm−1 , Qm )
.


×

n
Bn−s−1
(m,n)
bm−s,m−s

(m,n)
bm−s,n−s
...
...
(m,n)
bm−1,m−1
(m,n)
bm,m
...
...
(m,n)
bm−1,n−1
(m,n)
bm,n−1


n
Bm−s
  .. 
 . 
 n 
Bn−1
(m,n)
bm,n
Bnn
n 
B0
 .. 
. 
= (P 0 , P 1 , . . . , P r , P r+1 , . . . , P n−s−1 , P n−s , . . . , P n ) 
 Bn  ,
n−1
Bnn
with the linear independent property of Bernstein bases {Bin }ni=0 , the necessary and sufficient conditions
of Eqs. (7)–(9) can be derived. In addition, from Lemma 1 and Eq. (7), we have
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G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
m
Qi Bim (t)
=
i=0
m n+i−m
i=0
=
r
(m,n)
bi,j
Qi Bjn (t)
j =i
P j Bjn (t) +
j =0
n−s−1
j =r+1
min(j,m)
(m,n)
bi,j
Qi
i=max(0,j −(n−m))
Bjn (t) +
n
P j Bjn (t).
j =n−s
Then Eqs. (10) can be derived from the above equation and the symmetrical property of the Bézier
curves. That is the end of the proof. ✷
Remark 1. The process in theorem 1 in essence is the part inverse process of degree elevation (Farin,
1991).
Remark 2. Let m = r + s + 1 and {Qi }m
i=0 is shown as (7). Then the approximate degree reduced curve
Qm (t) of degree m with control points {Qi }m
i=0 is the simplest Hermite interpolant, which preserves the
interpolation of r, s orders at two endpoints repectively. In fact all control points {Qi }m
i=0 can be derived
from interpolation conditions.
Remark 3. Let m = n − 1, r = s = [(m − 1)/2]. When m is odd. {Qi }m
i=0 is shown as (7) and when m is
m/2−1
are
shown
as
(7)
and
even, {Qi }i=0 and {Qi }m
i=m/2+1
L
Qm/2 = Qs + QRr /2,
where
nP m−s − (n − m + s)Qm−s
nP r+1 − (r + 1)Qr
,
.
QRr =
n − (r + 1)
m−s
Then the approximate curve Qm (t) with control points {Qi }m
i=0 is the degree reduced Bézier curve in the
paper of (Piegl and Tiller, 1995).
QLs =
3. Best least squares multi-degree reduction (MDR by L2 )
In the approximate theory, the orthogonal polynomial basis functions are always used to solve least
squares approximate problems. We will use constrained Jacobi polynomials to minimize the least squares
distance function. The Jacobi polynomials Jn(r,s)(x) (Szego, 1975) are orthogonal on the domain and can
be explicitly represented in Bernstein forms as
n+r n+s n
i
n−i
(r,s)
n+i
n x+1
, n = 0, 1, . . . ,
(11)
(−1)
Bi
Jn (x) =
n
2
i
i=0
where x ∈ [−1, 1] and r, s > −1.
These Jacobi polynomials are orthogonal on [−1, 1] with respect to the weight function
w (r,s)(x) = (1 + x)r (1 − x)s .
Define the constrained Jacobi polynomials Jn,r,s as
(2r+2,2s+2)
(x),
Jn,r,s (x) = (1 + x)r+1 (1 − x)s+1 Jn−r−s−2
n = r + s + 2, r + s + 3, . . . .
(12)
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
371
It forms the orthogonal basis on [−1, 1] with respect to the weight w(x) = 1, and has roots of
multiplicity r + 1, s + 1 at x = −1, +1, respectively.
If the function f (x) (x ∈ [−1, 1]) can be represented as f (x) = (1 + x)r+1 (1 − x)s+1 f¯(x), then in
Πn , the polynomial
Jn(r,s)(x) =
n
ai Ji,r,s (x)
(13)
i=r+s+2
is the best least squares approximation of degree n to f (x) on x ∈ [−1, 1], where
n > r + s + 1,
1
ai =
δi
1
Ji,r,s (x)f (x) dx,
−1
1
δi =
2
Ji,r,s (x) dx.
−1
The least squares approximate error is
1
1
n
d2 (f, Jn ) = |f (x) − Jn (x)|2 dt = f 2 (t) dt −
δi ai2 .
−1
−1
(14)
i=r+s+2
If f (x) is defined on the interval [a, b], we can transform it to the [−1, 1] interval by a linear transform
as t = (2x − b − a)/(b − a).
From the properties of Bernstein polynomials and Jacobi polynomials, we have
Lemma 2. For k = 0, 1, . . . , n let Jk(r,s)(2t − 1) (0 t 1) be the Jacobi polynomial of degree k. Then
the linear relation between Jacobi polynomials and Bernstein polynomials {Bin (t)}ni=0 (0 t 1) can be
expressed in the matrix form as follows:
−1 (r,s)
(r,s)
= L(r,s)
B n = L(r,s)
J n = E(r,s)
(15)
J (r,s)
n×n B n ,
n×n
n×n J n ,
n
where
T
= J0(r,s)(2t − 1), J1(r,s)(2t − 1), . . . , Jn(r,s)(2t − 1) ,
J (r,s)
n
T
B n = B0n (t), B1n (t), . . . , Bnn (t) ,
(r,s)
L(r,s)
n×n = Lk,j (n+1)×(n+1) ,
(16)
(17)
k+r k+s
k
(k,n)
L(r,s)
(−1)k+i bi,j
, (18)
k,j =
i
k−i
i
i=max(0,j +k−n)
min(j,k)
(k,n)
is shown as Eq. (5).
and bi,j
Proof. By (11), the Jacobi polynomials can be represented in Bernstein form as
n
n+r n+s
n n
(r,s)
n+i
(−1)
Bi (t), n = 0, 1, . . . ,
Jn (2t − 1) =
i
n−i
i
i=0
on the interval [−1, 1]. Then from Lemma 1 and the linear independent property of Jacobi bases and
Bernstein bases, the conclusion can be easily derived. ✷
Now we present the best least squares multi-degree reduction (MDR by L2 ) as follows.
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G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
I n
Denote P In (t) = n−s−1
i=r+1 P i Bi (t). Then the problem of optimal approximation of P n (t) with
constraints of endpoints continuity of any r, s (r, s 0, r + s < n − 1) orders is equal to finding out
the optimal approximation of P In (t) without constraints of endpoints continuity. By the properties of
Bernstein polynomials, we have
n−s−1
P Ii Bin (t)
= (1 − t)
s+1 r+1
t
i=r+1
N
P IIi BiN (t) = (1 − t)s+1 t r+1 P IIN (t),
(19)
i=0
where
N = n − (r + s + 2),
P IIi
=
P Ir+1+i
n
·
r +1+i
N
,
i
i = 0, 1, . . . , N.
Denote
II II
0,P
1 ,...,P
IIN ,
P IIN = P
IIi = P IIi ,
P
i = 0, 1, . . . , N,
IIN (t) =
P
N
IIi BiN (t).
P
i=0
Then from (19), there is
IIN (t) = (1 − t)s+1 t r+1
P IIN B N .
P In (t) = (1 − t)s+1 t r+1 P
On the other hand, by Lemma 2, there is
(2r+2,2s+2)
P IIN E(2r+2,2s+2)J (2r+2,2s+2) = P III
.
P IIN B N = NJ
N×N
N
N
Suppose r + s < m − 1, denote
III III
II (2r+2,2s+2),
III
M = m − (r + s + 2),
P III
N = P 0 , P 1 , . . . , P N = P N EN×N
then it is easy to know that
(r,s)
(2r+2,2s+2)
P III
J m (t) = (1 − t)s+1 t r+1
MJM
is the best least squares approximation of degree n to
(r,s)
(2r+2,2s+2)
P III
J n (t) = P In (t) = (1 − t)s+1 t r+1
N JN
on the interval [−1, 1].
(2r+2,2s+2)
. By Lemma 2,
P III
Now we try to derive the corresponding Bernstein form of (1 − t)s+1 t r+1
MJM
there is
(2r+2,2s+2)
(2r+2,2s+2)
=
P III
BM = P IV
P III
MJ
ML
M BM ,
M
M×M
where
IV IV
III (2r+2,2s+2).
IV
P IV
M = P 0 , P 1 , . . ., P M = PML
M×M
Then
(2r+2,2s+2)
= (1 − t)s+1 t r+1
P III
P IV
(1 − t)s+1 t r+1
MJ M
M BM =
m−s−1
Qi Bim (t),
i=r+1
where
IV
Qi = P
i−r−1
M
m
,
i−r −1
i
i = r + 1, r + 2, . . . , m − s − 1.
(20)
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
The best least squares approximate error can be derived as follows:
n
1/2
2
(r,s)
(r,s)
III
P J (t) =
δi .
J (t), ε2 = d2 P n (t), Qm (t) = d2 n
m
i
373
(21)
i=m+1
Legendre polynomials are special cases of Jocobi polynomials. Therefore, converting a polynomial
in Bézier basis to Jacobi basis is very similar to that from Bézier to Legendre. And as for the detailed
converting process, one can refer to (Li and Zhang, 1998). This converting process is simple and stable.
To sum up, we can obtain the following theorem:
Theorem 2. Given a degree n Bézier curve P n (t),when {Qi }m
i=0 is shown as (7) and (20), m < n − 1,
m
Q
B
(t)
of degree m is its best least squares
the multi-degree reduced Bézier curve Qm (t) = m
i i
i=0
approximation, whereby the continuity of r, s (r, s 0, r + s < m − 1) orders can be preserved at two
endpoints, respectively. The best least squares error of approximate multi-degree reduction is ε2 .
As compared with most degree reduction methods, such as the methods by Eck (1993, 1995), the
MDR by L2 possesses a series of advantages. First, It avoids stepwise approximation for the multi-degree
reduction so that the computing time can correspondingly be decreased and there are no accumulative
calculative errors. Secondly, it can satisfy constraint conditions of endpoint continuity of any orders at
the same time of multi-degree reduction, thus the degree reduction computation can combine with the
subdivision algorithm to increase the precision. Furthermore, it achieves the optimal approximation using
the orthogonal polynomials and acquires the explicit solution of degree reduction curves.
Example 1. Let P 12 (t) be a Bézier curve of degree 12 with control points (−14, 8), (−10, −5), (−7, 5),
(−5, −7), (1, −3), (−3, 5), (1, 11), (4, 9), (7, −7), (10, −4), (12, 9), (14, 11), (19, 0). The best least
squares 3-degree reductions with different endpoint constraints are shown in Fig. 1. Fig. 2 shows the
6-degree reduction with C 1 continuity after one subdivision. As can be seen from the example, the MDR
by L2 in the paper obtains the good approximation of multi-degree reduction and can be combined with
the subdivision algorithm effectively.
4. Comparison between the MDR by L2 and the MDR by L∞
In the paper (Chen and Wang, 2000), we present one simple method (MDR by L∞ ) to the multi-degree
reduction with respect to L∞ norm and the MDR by L∞ obtains the nearly best uniform approximation.
In the following we introduce the MDR by L∞ briefly and compare it with the MDR by L2 in this paper
by some numerical examples.
Chebyshev polynomials are one of the classical orthogonal polynomials that have been studied
extensively. Let Tn (x) denote the Chebyshev polynomial of degree n, which is defined by Tn (x) =
cos(n arccos x) (−1 x 1). Then we can obtain the following explicit Bernstein representation (Eck,
1993)
n
2n
n
n+i
n x +1
, n = 0, 1, . . . .
(22)
(−1)
Bi
Tn (x) =
2
2i
i
i=0
374
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
(a)
(b)
Fig. 1. Reduction from degree 12 (solid) to degree 9 (dash). (a) (r, s) = (2, 2). (b) (r, s) = (3, 3).
(a)
(b)
Fig. 2. Reduction from degree 12 (solid) to degree 6 (dash) ((r, s) = (1, 1)). (a) Without subdivision. (b) With one subdivision.
One of the important properties of Chebyshev polynomials Tn (x) is the so-called equioscillating
property, i.e., that the Chebyshev polynomials have n + 1 extremal values (−1)i at x = cos(iπ/n),
i = 0, 1, . . . , n.
Chebyshev polynomials have been used widely in degree reduction in the past. The constrained
Chebyshev polynomial is firstly introduced to deal with degree reduction with constraints of endpoints
continuity (Lachance, 1988). Unfortunately, these constrained Chebyshev polynomials have no explicit
representations except in certain special cases and have to be determined numerically by a modified
Remez algorithm (Davis, 1963).
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
375
From the properties of Bernstein polynomials and Chebyshev polynomials, we have
Lemma 3. For k = 0, 1, . . . , n, the linear relation between Chebyshev polynomials {Ti (2t − 1)}ni=0 (0 t 1) and Bernstein polynomials {Bin (t)}ni=0 (0 t 1) can be expressed in matrix form as follows:
T n = C n×n B n ,
B n = C −1
n×n T n = An×n T n ,
(23)
where
T
T n = T0 (2t − 1), T1 (2t − 1), . . . , Tn (2t − 1) ,
(24)
min(j,k)
C n×n = (Ck,j )(n+1)×(n+1) ,
Ck,j =
(k,n)
(−1)k+i bi,j
i=max(0,j +k−n)
2k
k
,
2i
i
(25)
(k,n)
is shown as Eq. (5).
and bi,j
The proof of Lemma 3 is similar to the proof of Lemma 2. Thus the MDR by L∞ can be described as
follows.
Denote
P IIN (t) = (1 − t)s+1 t r+1
P In (t) = (1 − t)s+1 t r+1
N
IIi BiN (t) = (1 − t)s+1 t r+1
P
P IIN B N ,
i=0
where
II II
0,P
1 ,...,P
IIN ,
P IIN = P
IIi = P IIi ,
P
i = 0, 1, . . . , N.
Then by (19) and Lemma 3, there is
P IIN AN×N T N = P III
P IIN B N = N T N.
Let r + s < m − 1, denote
M = m − (r + s + 2),
III III
II
III
P III
N = P 0 , P 1 , . . . , P N = P N AN×N .
By Chebyshev polynomial approximation theory (Fox and Parker, 1968), P III
M T M is the nearly best
II
uniform approximation of P N B N among all polynomials of degree M on the interval [−1, 1].
P III
Now we derive the corresponding Bernstein form of (1 − t)s+1 t r+1
M T M . By Lemma 3, there is
P III
(1 − t)s+1 t r+1
MT M
=
(1 − t)s+1 t r+1
P IV
M BM
=
m−s−1
Qi Bim (t),
i=r+1
where
IV IV
III
IV
P IV
M = P 0 , P 1 , . . . , P M = P M C M×M ,
M
m
IV
, i = r + 1, r + 2, . . . , m − s − 1.
Qi = P i−r−1
i−r −1
i
(26)
376
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
(a)
(b)
Fig. 3. The comparison between the MDR by L2 (dash) and the MDR by L∞ (dot) (the solid curve is the original curve).
(a) Reduction from degree 12 to 6 ((r, s) = (0, 0)). (b) Reduction from degree 12 to 8 ((r, s) = (2, 2)).
The error bound of nearly best uniform approximation is presented as follows:
N
III Pi d∞ P n (t), Qm (t) ε∞ = max (1 − t)s+1 t r+1
0t 1
i=M+1
N
(r + 1)r+1 (s + 1)s+1 III P
.
=
i
(r + s + 2)r+s+2 i=M+1
(27)
Therefore we can obtain the following theorem:
m
m
Theorem 3. Let Qm (t) = m
i=0 Qi Bi (t), where {Qi }i=0 is shown as (7) and (26), then the multi-degree
reduced Bézier curve Qm (t) of degree m (m < n − 1) can approximate the given degree n Bézier curve
P n (t) with that the continuity of r, s (r, s 0, r + s < m − 1) orders can be preserved at two endpoints,
respectively. It is the nearly best uniform approximation under the conditions of endpoint interpolaion
and the corresponding error bound of degree reduction approximation is ε∞ .
Fig. 3 presents the comparison of the MDR by L∞ with the MDR by L2 using the input curve in the
Example 1 of Section 3.
Obviously, the MDR by L2 is better than the nearly best uniform approximate method (MDR by L∞ )
under the conditions of endpoint continuity.
5. Conclusions
By using the constrained Jacobi orthogonal polynomial basis functions, this paper derives one best
least squares approximation method for multi-degree reduction of Bézier curves with constraints of
endpoints continuity. The approximate degree reduced Bézier curve is shown as an explicit solution
G.-D. Chen, G.-J. Wang / Computer Aided Geometric Design 19 (2002) 365–377
377
form and can preserve continuity of any r, s (r, s 0) orders at two endpoints, respectively. The error
bound is given and the degree of accuracy for the approximation is optimal with respect to L2 norm
according to the traditional approximate theory. The method in the paper avoids stepwise computing for
the multi-degree reduction so that the computing time can obviously be reduced. The method in this
paper can be effectively combined with the subdivision algorithm for the approximation of multi-degree
reduction within a prescribed error tolerance.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 60173034) and the
Foundation of State Key Basic Research 973 Item (No. G1998030600).
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