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Computer Aided Geometric Design 21 (2004) 151–164 www.elsevier.com/locate/cagd

A unified matrix representation for degree reduction of Bézier curves Hasik Sunwoo a,∗,1 , Namyong Lee b a Department of Mathematics, Konkuk University, Chungju, 380-701, South Korea b Department of Mathematics, Minnesota State University, Mankato, MN 56001, USA

Received 25 October 2002; received in revised form 28 July 2003; accepted 29 July 2003

Abstract In this paper, we show the degree reduction of Bézier curves in a matrix representation. Most degree reduction algorithms have been described as a set of recursive equations which are based on the inverse problem of degree elevation. However, degree elevation can be easily expressed in terms of matrices. Motivated by this observation, we represent most well known degree reduction algorithms in a unified matrix form. In this way, we can simply express the process of degree reduction and achieve greater insight for many known results.  2003 Elsevier B.V. All rights reserved. Keywords: Bézier curve; Degree reduction; Generalized inverse; Matrix algebra

1. Introduction In general, degree reduction of Bézier curves addresses the following problem: Problem 1. Let {bi }ni=0 ⊂ R s be a given set of (control) points which define the Bézier curve xn (t) =

n

bi Bin (t),

t ∈ [0, 1],

i=0

* Corresponding author.

E-mail addresses: [email protected] (H. Sunwoo), [email protected] (N. Lee). 1 Present address: Department of Mathematics, Konkuk University, Chungju, 380-701, South Korea.

0167-8396/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cagd.2003.07.007

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H. Sunwoo, N. Lee / Computer Aided Geometric Design 21 (2004) 151–164

in terms of Bernstein polynomials Bin (t) = ni t i (1 − t)n−i of degree n. Then find another point set s {b¯i }m i=0 ⊂ R defining the (approximative) Bézier curve x¯ m (t) =

m

b¯i Bim (t),

t ∈ [0, 1],

i=0

of lower degree m < n so that a suitable distance function d(xn , x¯ m ) between xn and x¯ m is minimized on the interval [0, 1]. The problem of degree reduction of Bézier curves was initially proposed as an inverse process of degree elevation in the works of Forrest (1972) and Farin (1983) where most of the algorithms were described in terms of two different recursive extrapolation formulas of an inverse of degree elevation. Accordingly, we can classify these methods by the nature of the recursions used: (1) Combining both formulas by taking the left half and the right half: Forrest (1972), Piegl and Tiller (1995). (2) Taking a weighted average: Farin (1983). (3) L∞ -norm approximation using (constrained) Chebyshev polynomials: Watkins and Worsey (1988), Lachance (1988), Eck (1993). (4) Least squares approximation using constrained Legendre polynomials: Eck (1995), Lee and Park (1997). The purpose of this paper is to describe the process of degree reduction of Bézier curves by means of matrix representation. By introducing the Moore–Penrose generalized inverse matrix, we can describe most degree reduction algorithms by a unified matrix representation. In this way, we simplify the degree reduction of Bézier curves and clarify the relationship among other degree reduction algorithms. For simplicity of notation, we restrict ourselves to the case of m = n − 1 and only consider a singlevalued function case. Now, we restate the degree reduction problem as follows: Problem 2. Let {bi }ni=0 be a given set of real numbers which defines the Bézier function xn (t) =

n

bi Bin (t),

t ∈ [0, 1].

i=0

Then, find another set of real coefficients, {b¯i }n−1 i=0 , such that x¯ n−1 (t) =

n−1

b¯i Bin−1 (t),

t ∈ [0, 1],

i=0

defines an approximation to xn (t) for which a suitable distance function, d(xn , x¯ n−1 ), is minimized on the interval [0, 1]. The organization of this paper is as follows: We introduce some useful results from matrix algebra in Section 2, in particular, the concept of generalized inverse. In Section 3, a matrix representation is given for the general construction of degree reduction for Bézier curves. Matrix representations for other well known degree reduction methods are given in Section 4.


153

2. Some basic properties of matrices The degree elevation matrix is well known and is of a very simple form. However, since the exact degree reduction is not possible, the exact degree reduction matrix does not exist. But, by introducing a generalized inverse matrix, we can find a degree reduction process in a matrix form for each method described in Section 1. A discussion of generalized inverses for matrices can be found in (Rao and Mitra (1971) and Searle (1982)). Now, we recall their definitions and some of their useful properties. Consider a linear system Ax = b. If A is a square matrix and is nonsingular, then the linear system has a unique solution as x = A−1 b. But, if A is singular or if A is not a square matrix, then the linear system has no solution or has infinitely many solutions. In either case, there is no inverse matrix of A. But, when the linear system has infinitely many solutions, we can extend the concept of inverse matrix to a generalized inverse. Definition 1. Let A be an m × n rectangular matrix. Then an n × m matrix G is called a generalized inverse of A if AGA = A and is denoted by G = A− . Clearly, if A−1 exists, then A− = A−1 is the unique generalized inverse of A. But if not, there exist infinitely many generalized inverses. Lemma 1. For a given linear system Ax = b, if the system is consistent, then x = A− b is a solution. Furthermore, the general solutions are given by x = A− b + (I − A− A)z,

for any vector z.

If we impose some additional conditions, we can get a unique generalized inverse, namely the Moore– Penrose inverse. The definition of the Moore–Penrose inverse is as follows: Definition 2. For any matrix A there exists a unique matrix G satisfying the following four conditions: (i) (ii) (iii) (iv)

AGA = A, GAG = G, AG is symmetric, GA is symmetric,

and is denoted by G = A† . The Moore–Penrose inverse has a beautiful application in the study of an inconsistent linear system Ax = b. Since there is no exact solution to the inconsistent linear system, we consider least squares solutions which minimize Ax − b2 . A vector x∗ is called a least squares solution to Ax = b if Ax∗ − b2 Ax − b2 for all x. A vector x∗ is called a minimal least squares solution to Ax = b if x∗ is a least squares solution to Ax = b and x∗ 2 is the smallest among all least squares solutions.

154


With the help of matrix algebra it is easy to see that any solution to the normal equations At Ax = At b would be a least squares solution. As seen in the next lemma, the minimal least squares solution can be obtained by using the Moore–Penrose inverse. Lemma 2. x = A† b is the minimal least squares solution to Ax = b.

3. The general construction In this section, we describe, first, the general process of degree reduction of Bézier curves in terms of recursive formula. Then we will represent the procedure in a unified matrix representation. Also some useful results about matrix representation will be given. The well known procedure of elevating the degree of a Bézier curve from n − 1 to n can be written in terms of the control points as bi =

1 ¯ i · bi−1 + (n − i) · b¯i for i = 0, . . . , n. n

(1)

We solve this overdetermined system for the unknowns {b¯i }n−1 i=0 twice: first by neglecting the last equation (i = n), and second by ignoring the first equation (i = 0) and obtaining the recursively defined points b¯iI =

1 I n · bi − i · b¯i−1 n−i

for i = 0, . . . , n − 1

(2)

and 1 II = n · bi − (n − i) · b¯iII for i = n, . . . , 1 (3) b¯i−1 i which are distinguished by upper indices I and II. Based on this construction, the key idea of the degree reduction process is to write the unknown points b¯i of the final degree reduced curve x¯ n−1 as a linear combination of b¯iI and b¯iII , b¯i = (1 − λi )b¯iI + λi b¯iII

for i = 0, . . . , n − 1.

(4)

Now, in view of (4), the problem of computing the points {b¯i } is shifted to the problem of computing factors {λi }n−1 i=0 . These unknown factors were introduced by Eck (1993). However the earlier known degree reduction procedures, for example, Forrest (1972) and Farin (1983), can be expressed of the form in (4). This observation enables us to find a unified approach of degree reduction of Bézier curves. Now we express the procedure of degree reduction in matrix-vector form. First, we rewrite the degree elevation equation (1) in matrix-vector notation as follows ¯ b = Ab,

(5)

where b = [b0 . . . bn ]t and b¯ = [b¯0 . . . b¯n−1 ]t with the degree elevation matrix A = (aij ) being an (n + 1) × n matrix given by   (n − i)/n, i = j, (6) aij = i/n, i = j + 1,  0, otherwise,


for i = 0, . . . , n and j = 0, . . . , n − 1. Note that, A has the form   n 1 n−1   1 . .  . .. .. A=   n n−1 1 n

155

(7)

Note also that Eck (1993) and Brunnet et al. (1996), show that the two coefficients of b¯iI and b¯iII , in Eqs. (2) and (3), can be rewritten as,

i (−1)i n (−1)j (8) bj b¯iI = n−1 j i

and

j =0

n i+1 j n ¯biII = (−1) (−1) bj n−1 j i j =i+1

(9)

for i = 0, . . . , n − 1. Using these two formulas, we can easily find two n × (n + 1) matrices L = (lij ) and U = (uij ) as follows (nj) , for i = 0, . . . , n − 1; j = 0, . . . , i, (−1)i+j n−1 (10) lij = (i ) 0, elsewhere, and

uij =

(−1)i+j +1 0,

(nj) , (n−1 i )

for i = 0, . . . , n − 1; j = i + 1, . . . , n,

(11)

elsewhere.

Note that L is a lower triangular matrix and U is an upper triangular matrix such that b¯ I = Lb and

b¯ II = U b

(12)

I II ]t and b¯ II = [b¯0II . . . b¯n−1 ]t . where b¯ I = [b¯0I . . . b¯n−1 Now, as noted earlier, the key idea of the degree reduction process is to write the unknown points ¯b of the final degree reduced curve x¯ n−1 as a linear combination of b¯ I and b¯ II . Furthermore, b¯ can be represented in matrix form: (13) b¯ = (I − D)b¯ I + D b¯ II = (I − D)L + DU b = Gb

where D = diag(λ0 , . . . , λn−1 ) is a diagonal matrix and G = (I − D)L + DU , which is actually a generalized inverse of the degree elevation matrix. Lemma 3. Both L and U defined in (10) and (11) are generalized inverses of the degree elevation matrix A. Furthermore LA = U A = In where In denotes an n × n identity matrix. Proof. We will prove only that LA = In . It follows that ALA = A, which implies that L is a generalized inverse of A.

156


Since there are only two nonzero elements in each column of A, the ij th element of LA is n lik akj = lij ajj + li,j +1 aj +1,j . (LA)ij = k=0

Now if j > i, then all lij = 0 and this implies (LA)ij = 0. If j < i, then n n n−j j +1 j i+j i+j +1 j +1 + (−1) (LA)ij = (−1) n−1 · n−1 · n n i

i 1 n n i+j = (−1) · (n − j ) − · (j + 1) = 0. j j +1 n n−1 i Finally if i = j , then li,i+1 = 0 and therefore n n−i i = 1. (LA)ii = lii aii = (−1)i+i n−1 · n i The result now follows with the case of U A = In being proved similarly.

✷

Lemma 3 gives us the following result. Corollary of Lemma 3. For any matrix D, G = L − D(L − U ) is a generalized inverse of A. Though G = L − D(L − U ) is a generalized inverse of A for any matrix D, we are interested in D being a diagonal matrix. For any diagonal matrix Dλ = diag(λ0 , λ1 , . . . , λn−1 ), let us define − A− λ = A (λ0 , λ1 , . . . , λn−1 ) = L − Dλ (L − U ).

A− λ

(14) A− λ

is also a generalized inverse of A. We have written = L − Dλ (L − U ) instead of Then = (I − D )L + D U because Lb and (L − U )b are more meaningful when we compute A− A− λ λ λ λ b. In fact, (L − U )b = (−1)n n b0 · h

i n−1 −1

where h = (hi ) with hi = (−1) i defined by

n n n n+j (−1) bj . b0 = j

(15) for i = 0, . . . , n − 1, and the nth forward difference n b0 is

j =0

From Eq. (15), one can easily see that b¯ I = b¯ II if and only if n b0 = 0. For this reason, we assume that n b0 = 0 for the rest of this paper. Note that, b¯ I and b¯ II represent the degree reduced Bézier curves. If we multiply A to both vectors, then we obtain artificially degree elevated Bézier curves whose control points are given by the following lemma. Lemma 4. Let bI = ALb and bII = AU b. Then bI = b − n b0 en

and

bII = b + (−1)n+1 n b0 e1

where ei = (0, . . . , 0, 1, 0, . . . , 0)t .


157

Proof. From the definitions of matrices A, L, and U in Eqs. (6), (10), and (11), respectively, we can easily see that     0 d1 . . . dn         0    In−1 AU =  AL =  ,  0 I    n−1      c0 . . . cn−1 0 where cj = −(−1)n+j nj for j = 0, . . . , n − 1 and dj = (−1)j +1 nj for j = 1, . . . , n. Hence, we have     b0 b0 .. ..     . . =  ALb =     bn−1 bn−1  n+j n − n−1 bn − n b0 b j =0 (−1) j j and  n (−1)j +1 nb   b + (−1)n+1 n b  0 0 j =1 j j     b b 1 1 = . AU b =  .. ..     . . bn bn This completes the proof. ✷ This lemma is useful in describing the relationship between xn (t) and x¯ In−1 (t), or between xn (t) and x¯ IIn−1 (t). We will discuss this at the end of this section. − ¯ Similarly, if b¯ = A− λ b and c = AAλ b, then two Bézier curves, b and c, represent the same curves geometrically and parametrically. That is, x¯ n−1 (t) =

n−1

b¯i Bin−1 (t) =

i=0

n

ci Bin (t).

(16)

i=0

In the following theorem we can obtain the explicit form of c and thus the error between b and AAλ b. Theorem 1.



(−1)n λ0 .. .



    −λ λ i n+i i−1   n (−1) n b − AA−  b0 . ( ) λb= i   ..   . 1 − λn−1 Proof. Using the fact ALb = b − n b0 en , we have AA− λ b = A L − Dλ (L − U ) b = ALb − ADλ (L − U )b = b − n b0 en − ADλ (L − U )b.

158


Hence n b − AA− λ b = b0 en + ADλ (L − U )b.

On the other hand,



 (−1)n λ0 ..   .   n−1 n+i (−1)    .. .. λ   (n−1) i  n b0 . .  i   .. n − 1 1   . n −λn−1  (−1)n λ  0 ..   .   λ −λ n+i i i−1   n n =  (−1) b0 . (i )    .   .. −λn−1 This completes the theorem. ✷ 



n 1 1 ADλ (L − U )b =  n 

While Theorem 1 gives us the error I − AA− λ , the next theorem gives a pointwise representation of the error between the original curve and the degree-reduced curve. Theorem 2. By letting λ−1 = 0 and λn = 1, we have n λi − λi−1 (−1)n+i n Bin (t). xn (t) − x¯ n−1 (t) = n b0 i

i=0

As a consequence of Theorem 2, we can easily see the relationship between xn (t) and x¯ n−1 (t) by investigating the diagonal elements. Now, we describe more properties of b¯ I and b¯ II . Since bI = Ab¯ I , both b¯ I and bI represent the same Bézier curves as do b¯ II and bII . That is, x¯ In−1 (t)

=

n−1

b¯iI Bin−1 (t) =

n

i=0

x¯ IIn−1 (t) =

n−1 i=0

biI Bin (t),

(17)

i=0

b¯iII Bin−1 (t) =

n

biII Bin (t).

(18)

i=0

Hence with the help of Lemma 4, we have xn (t) − x¯ In−1 (t) = n b0 Bnn (t), xn (t) − x¯ IIn−1 (t) = (−1)n n b0 B0n (t). Since Bnn (0) = 0 and B0n (1) = 0, xn (t) and

(19) (20) x¯ In−1 (t)

Also the errors are given by d∞ xn , x¯ In−1 = d∞ xn , x¯ IIn−1 = n b0

meet at t = 0, and xn (t) and

x¯ IIn−1 (t)

meet at t = 1. (21)


159

and |n b0 | , d2 xn , x¯ In−1 = d2 xn , x¯ IIn−1 = √ 2n + 1 where d∞ (f, g) = max f (t) − g(t) t ∈[0,1]

and

(22)

1 2 d2 (f, g) = f (t) − g(t) dt. 0

4. Matrix representations of degree reduction of Bézier curves We may describe the problem of degree reduction as that of finding a suitable vector b¯ from b such that Ab¯ = b where A is the degree elevation matrix defined in (5). As we described in the previous section, the ¯ the control points of degree reduced Bézier curve, can be represented in the form vector b, b¯ = A− λb

(23)

A− λ

= L − Dλ (L − U ) for a diagonal matrix Dλ = diag(λ0 , . . . , λn−1 ). Using the results in where Section 3, we now illustrate the use of matrix representations in each of several well known methods. 4.1. Forrest’s work Forrest (1972) proposed to combine both formulas by taking the left half of the coefficients from (2) and the right half of the coefficients from (3) as b¯i = b¯iI (i = 0, 1, . . .), b¯i = b¯iII (i = . . . , n − 2, n − 1). If n is odd, the midpoint is defined by b¯i = 12 (b¯iI + b¯iII ) with i = (n − 1)/2. Hence Forrest’s method is given by − A (0, . . . , 0, 1, . . . , 1), for even n, (24) = A− Forrest 1 − A (0, . . . , 0, 2 , 1, . . . , 1), for odd n. For even n,

λi − λi−1 = and for odd n, λi − λi−1 =

1 0 1/2 0

if i = n/2, otherwise, if i = (n − 1)/2, (n + 1)/2, otherwise.

160


Thus, by Theorem 2, xn (t) − x¯ n−1 (t) can be represented as follows:  n b n (t), for even n,  (−1)n/2 ( n )0 Bn/2 n/2 xn (t) − x¯ n−1 (t) = n  (−1)(n−1)/2 nnb0 B n (n−1)/2 (t) − B(n+1)/2 (t) , for odd n. 2((n−1)/2) From the definition of the Bernstein polynomial, Bin (t) = ni t i (1 − t)n−i , it is easily seen that for even n, xn (t) − xn−1 (t) = n b0 · t n/2 (1 − t)n/2 and for odd n, n xn (t) − xn−1 (t) = | b0 | t (n−1)/2 (1 − t)(n−1)/2 (1 − 2t) 2 because

n n n (t) − B(n+1)/2 (t) = t (n−1)/2 (1 − t)(n−1)/2 (1 − 2t). B(n−1)/2 (n − 1)/2 Hence, the error is given by max xn (t) − x¯ n−1 (t)

( 12 )n |n b0 |

0t 1

1 √ (1 2 n 4

−

1 (n−1)/2 ) |n b0 | 4n

if n is even, if n is odd,

(25)

guiding the same the result of Park and Choi (1994). 4.2. Weighted average: Farin (1983) Farin (1983) proposed the method by taking a weighted average of the form

i ¯ II ¯bi = 1 − i b , for i = 0, . . . , n − 1. b¯iI + n−1 n−1 i Hence Farin’s method is given by

2 1 − − , ,...,1 . AFarin = A 0, n−1 n−1 Since λi − λi−1 =

for i = 1, . . . , n − 2 and λ0 = 1 − λn−1 = 0. Thus, by Theorem 1, we have   0 . ..    1 (−1) n+i   n  n b − AA− λ b =  n−1 ( i )  b0 .   ..   . 0

(26)

1 n−1

(27)

Also, by Theorem 2, we have n b0 (−1)n+i n n Bi (t). n − 1 i=1 i n−1

xn (t) − x¯ n−1 (t) =

(28)


161

Note that n−1 (−1)n+i n n Bi (t) = − t n (1 − t) + (−1)n t (1 − t)n . i=1

(29)

i

Therefore, the error bound is given by

n−1 |n b0 | max (−1)n+i t i (1 − t)n−i max xn (t) − x¯ n−1 (t) = 0t 1 n − 1 0t 1 i=1

|n b0 | max t n (1 − t) + (−1)n t (1 − t)n n − 1 0t 1 |n b0 | max t n (1 − t) 2 n − 1 0t 1

n n n 2 b0 . = 2 n −1 n+1 =

This result is different from the result of Park and Choi (1994). We obtain better error bound for the degree reduction of Farin’s method because the terms, when i = 0 and i = n, do not appear in the summation of (28). 4.3. L∞ -norm approximation: Eck (1993) Many authors have used the Chebyshev polynomials to find the degree reduction of Bézier curves, such as the earlier works by Watkins and Worsey (1988), Lachance (1988), and Eck (1993). The Chebyshev polynomial of degree m is defined by Tm (t) = cos(m · arccos t),

t ∈ [−1, 1],

and can be represented using a Bernstein basis by

m t +1 2m m where ci,m = (−1)m+i ci,m Bim . Tm (t) = 2 2i i i=0

(30)

(31)

The main idea in the works cited above is to express a Bézier curve xn (t) in terms of Chebyshev polynomials (shifted to the interval [0, 1]) as xn (t) =

n

ai Ti (2t − 1),

t ∈ [0, 1],

(32)

i=0

and truncate the highest coefficient in (32) to obtain the best L∞ -norm approximation as x¯ n−1 (t) =

n−1

ai Ti (2t − 1),

t ∈ [0, 1].

(33)

i=0

Immediately, from Eqs. (32) and (33), we have xn (t) − x¯ n−1 (t) = an Tn (2t − 1)

(34)

162


or equivalently, in matrix-vector notation, we have (I − AA− λ )b = an · h

(35)

A− λ

= L − Dλ (L − U ) for some diagonal matrix Dλ and h = [c0,n . . . cn,n ] . Since LA = I , by where multiplying L to both sides, we obtain t

Dλ (L − U )b = an Lh. Since



.. . n+i



 .. .   (−1)i i (−1)j nj cj,n  , n−1 j =0 Lh =   ( i ) .. . 

 n  (−1) n−1 λi  b0 and Dλ (L − U )b =   (i )  .. . we have

i i an an 2n n n+j (−1) cj,n = n λi = n j b0 j =0 b0 j =0 2j

for i = 0, . . . , n − 1.

(36)

By Theorem 1, the last row of Eq. (35) can be written as (1 − λn−1 )n b0 = an cn,n = an . By substituting λn−1 in (36) and (37), and by using the identity

n

2n

j =0 2j

(37) = 22n−1 , we have

n b0 . 22n−1 Therefore, the diagonal elements are given by i 1 2n for i = 0, . . . , n − 1, λi = 2n−1 2j 2 an =

(38)

(39)

j =0

which are deduced for best L∞ -approximation using Chebyshev polynomials, and thus recover Eck’s results of (1993). 4.4. The least squares approximation: Eck (1995) Eck (1995) found the best L2 norm degree reduction of a Bézier curve by using so-called constrained Legendre polynomials which are explicitly known. We now derive explicit expressions for the diagonal elements {λi }n−1 i=0 . Theorem 3 (Eck, 1995). Consider a curve xn with n b0 = 0. Iff the factors {λi }n−1 i=0 are given by

i 1 n n λi = 2n j −α j +α n+2α

(40)

j =0

¯ n−1 that minimizes d2 (xn , x¯ n−1 ) with the additional then the control points {b¯i }n−1 i=0 determine the curve x constraints for t0 = 0 and t0 = 1 and 2α n dr dr , 0 r α − 1. ¯ x (t) = (t) (41) x n n−1 dt r dt r t =t0 t =t0


163

Eck (1995) showed that the diagonal elements are the solution of the following equation: b − AA− λb=a·h where h = [h0 . . . hn ]t and

−1 n n n n+i . hi = (−1) i i−α i+α

(42)

(43)

We can solve Eq. (42) by using almost the same procedure as described in the previous section. By introducing α, we obtain the C (α−1) -continuous degree reduced Bézier curve at the two endpoints (t = 0 and t = 1). Note that we have b − AA− λ b in Theorem 1. The following result shows the relationship between continuity at the endpoints and the vector b − AA− λ b. Theorem 4. Let e = (ei ) where e = b − AA− λ b. If ei = 0,

i = 0, . . . , α − 1,

then xn (t) and x¯ n−1 (t) meet C (α−1) -continuously at t = 0. If en−i = 0,

i = 0, . . . , α − 1,

then xn (t) and x¯ n−1 (t) meet C (α−1) -continuously at t = 1. Since hi = hn−i = 0, for i = 0, . . . , α − 1, in (43), we see that ei = en−i = 0,

for i = 0, . . . , α − 1.

Hence we have C (α−1) -continuous degree reduced Bézier curve at both of two endpoints. Finally, the best constrained Legendre polynomials appear for the best constrained L2 -approximation from Eck’s results of (1995). And as a special case, the best L2 -approximation and thus the choice of the λi , when α = 0, which generate the matrix A− λ that is in fact the Moore–Penrose inverse according to (Lee and Park (1997)).

5. Conclusion In this paper, we have described the degree reduction of Bézier curves in terms of matrix-vector representation. With the help of matrix algebra, we can easily express the procedures in matrix-vector form. We have also presented a unified matrix representation for these methods using two different recursive extrapolation formulas for an inverse of degree elevation. As an application, we described its uses in many well known degree reduction results in Section 4. In some case, we obtain better error bounds for the degree reduction.

Acknowledgements The authors would like to thank the editor and the referee for their valuable comments.

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