Optimal properties of the uniform algebraic ... - Semantic Scholar
Computer Aided Geometric Design 23 (2006) 226–238 www.elsevier.com/locate/cagd
Optimal properties of the uniform algebraic trigonometric B-splines Guozhao Wang, Yajuan Li ∗ Institute of Computer Graphics and Image Processing, Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China Received 24 March 2005; received in revised form 31 March 2005; accepted 23 September 2005
Abstract In this paper, we construct a matrix, which transforms a generalized C-Bézier basis into a generalized uniform algebraic-trigonometric B-spline (C-B-spline or UAT B-spline) basis. We also show that it is a totally positive matrix and give a normalized B-basis of the generalized UAT B-splines. 2005 Elsevier B.V. All rights reserved. Keywords: C-Bézier basis; C-B-spline basis; C-Bézier curve; UAT B-spline; NUAT B-spline; B-basis; Totally positive
1. Introduction The normalized B-basis is the unique normalized basis of a space with optimal shape preserving properties. It is well known that both the Bézier basis and the B-spline basis are normalized B-bases of the polynomial space. However, they have many shortcomings. For instance, they cannot represent exactly transcendental curves such as the helix and the cycloid. Furthermore, repeated differentiation of the rational form produces curves of very high degree. So new bases in new spaces have been extensively studied in the literature (Zhang, 1996, 1997; Mainar and Peña, 2004; Lü et al., 2002; Chen and Wang, 2003; Wang and Chen, 2004). * Corresponding author.
E-mail addresses: [email protected] (G. Wang), [email protected], [email protected] (Y. Li). 0167-8396/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cagd.2005.09.002
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Zhang proposed the cubic C-Bézier basis and the cubic UAT B-spline basis (Zhang, 1996, 1997) in the space Γ3 = span{cos t, sin t, 1, t}, and Mainar and Peña obtained the matrix for transforming the cubic C-Bézier basis into the UAT B-spline basis and proved that the cubic UAT B-spline basis can be extended to a B-basis (Mainar and Peña, 2002). The cubic C-Bézier curves described in (Zhang, 1996) are special cases of generalized C-Bézier curves described in (Chen and Wang, 2003); The cubic UAT B-spline curves described in (Zhang, 1997) and the quadratic-cycloidal curves described in (Mainar and Peña, 2004) are special cases of the UAT B-spline curves (Lü et al., 2002). In fact, not only the cubic UAT B-spline basis has such properties, but the general UAT B-spline basis of arbitrary order n can also be extended to a B-basis of the space Γn = span{cos t, sin t, 1, t, . . . , t n−2 }. In this paper, we compute the matrix which transforms the C-Bézier basis into the UAT B-spline basis and decompose this matrix into a product of bidiagonal and stochastic factors. Thus we deduce that the UAT B-spline basis is normalized totally positive (NTP). Furthermore, we give a normalized B-basis of the UAT B-splines. This paper is organized as follows. In Section 2, the basic definitions and notations will be presented. Then in Section 3, we will present the matrix transforming the general C-Bézier basis into the general UAT B-spline basis. In Sections 4 and 5, the total positivity of the general UAT B-spline basis will be proven. As a result, the general UAT B-spline basis can be extended to a normalized B-basis. The convergence of the UAT B-spline basis will be proved in the last section.
2. Preliminaries 2.1. The UAT B-spline basis The main result of this paper is to show some optimal properties of the UAT B-spline basis. Let ti = iα (i = 0, ±1, ±2, . . .) be a set of knots which partition the parameter axis t uniformly. Then the UAT Bspline basis functions of order n + 1 {Mi,n+1 (t)}+∞ i=−∞ of the space Γn on t ∈ [−∞, +∞], 0 < α < π were given in (Lü et al., 2002): −α sin t/[2(cos α − 1)], 0 t α, M0,2 (t) = −α sin(2α − t)/[2(cos α − 1)], α t 2α, 0, otherwise. t 1 Mi,n (x) dx, n 2, i = ±1, ±2, . . . . Mi,n+1 (t) = α t−α
Mi,n+1 (t) = M0,n+1 (t − ti ),
i = ±1, ±2, . . . .
It is easy to get the following equation: Mi,n+1 (0) = Mi+1,n+1 (α),
i = 0, 1, . . . , n.
(1)
The UAT B-spline basis has many optimal properties the same as those of the B-spline basis, such as the subdivision property, the variation diminishing property, the convexity preserving property, etc. By the properties of the UAT B-spline, it is not difficult to show that it can be used as a quasi-interpolant to given data points in the same way as the B-spline. Lü et al. (2002) gave the derivative of Mi,n+1 : dMi,n+1 (t) 1 = Mi,n (t) − Mi+1,n (t) , i = −n, −n + 1, . . . , 0. dt α
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They also showed that {M−n,n+1 (t), M−n+1,n+1 (t), . . . , M0,n+1 (t)} is a basis of the space Γn on t ∈ [0, α]. By the definition of the UAT B-spline basis, we know M−n,n (t) = M1,n (t) = 0, t ∈ [0, α]. Let mn denote the UAT B-spline basis of order n + 1, that is, mn = (M−n,n+1 (t), M−n+1,n+1 (t), . . . , M0,n+1 (t)), we have dmn = mn−1 C, (2) dt where −1 1 1 −1 1 . (3) C= ··· ··· α −1 1 n×(n+1) 2.2. The C-Bézier basis Mainar and Peña obtained the matrix for transforming the cubic C-Bézier basis into the UAT B-spline basis and extended the cubic UAT B-spline basis to a B-basis (Mainar and Peña, 2002). In this paper, we will present the same result for the general UAT B-spline basis by transforming the general C-Bézier basis. We review the C-Bézier curves presented in (Chen and Wang, 2003). Two initial functions were given as: sin(α − t) sin t , u1,1 (t) = , t ∈ [0, α], α ∈ (0, π ) sin α sin α for n > 1. The C-Bézier basis functions {u0,n , u1,n , . . . , un,n } of the space Γn = span{cos t, sin t, 1, t, . . . , t n−2 } were defined recursively by u0,1 (t) =
t u0,n (t) = 1 −
δ0,n−1 u0,n−1 (s) ds, 0
t
δi−1,n−1 ui−1,n−1 (s) − δi,n−1 ui,n−1 (s) ds
ui,n (t) =
for 1 i n − 1,
0
t un,n (t) =
δi−1,n−1 ui−1,n−1 (s) ds, 0
where δi,n = 1/
α 0
ui,n (t) dt. By the definition of the C-Bézier basis functions, we obtain:
u0,n (0) = 1,
u1,n (0) = u2,n (0) = · · · = un,n (0) = 0,
un,n (α) = 1,
u0,n (α) = u1,n (α) = · · · = un−1,n (α) = 0.
The derivatives of the functions were also presented in (Chen and Wang, 2003): dui,n (t) = δi−1,n−1 ui−1,n−1 (t) − δi,n−1 ui,n−1 (t). dt
(4)
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Denoting un = (u0,n (t), u1,n (t), . . . , un,n (t)), we can get: dun = un−1 D, dt where u−1,n−1 (t) = un,n−1 (t) = 0 and −δ0,n−1 δ0,n−1 −δ1,n−1 δ1,n−1 D= ···
(5) .
···
−δn−1,n−1
(6)
δn−1,n−1
2.3. The NUAT B-spline basis To get the transforming matrix between the UAT B-spline basis and the C-Bézier basis, we must use the definition and the properties of the non-uniform algebraic trigonometric (NUAT) B-spline basis. Given a knot sequence T = (ti )+∞ −∞ , some initial functions were given in (Wang and Chen, 2004): ti < t ti+1 , sin(t − ti )/ sin(ti+1 − ti ), Ni,2 (t) = sin(ti+2 − t)/ sin(ti+2 − ti+1 ), ti+1 < t ti+2 , 0, otherwise. Then the NUAT B-spline basis functions of order n + 1 can be defined as: t Ni,n (t) =
δi,n−1 Ni,n−1 (s) − δi+1,n−1 Ni+1,n−1 (s) ds,
n 3.
−∞
+∞ Where 0 ti+1 − ti < π , δi,n = 1/ −∞ Ni,n (t) dt. Here 0/0 = 0. The derivatives of the basis functions are: dNi,n (t) = δi−1,n−1 ui−1,n−1 (t) − δi,n−1 Ni,n−1 (t). dt Considering NUAT B-spline basis functions {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} of order n + 1 defined on knot sequence t−n+1 = · · · = t0 < t1 < · · · < th = · · · = th+n−1 . By the properties of the NUAT basis functions (Wang and Chen, 2004), we know that the NUAT B-spline curve: s(t) =
h−1
Ni,n+1 (t)pi ,
t ∈ [t0 , th ]
i=−n
must have the endpoint properties, V.D. property and boundary tangent properties (Fig. 1). By (Carnicer and Peña, 1994; Mainar and Peña, 1999), {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} is NTP. Here pi , i = −n, −n + 1, . . . , h − 1 are control points. The knot_insertion theorem is as follows (Theorem 4.2 of (Wang and Chen, 2004)): [1] = (ti[1] )+∞ Lemma 1. Let T = (ti )+∞ −∞ be a given knot sequence with 0 ti+1 − ti < π , and let T −∞ be a new knot sequence obtained by inserting a new knot u into T with ti u < ti+1 , where Nj,n (t) and [1] (t) are defined on the knot sequence T and T [1] respectively. Then we have for all j, n 2: Nj,n
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Fig. 1. The NUAT B-spline curve with endpoint properties and boundary tangent properties. [1] Ni,n (t) = αj,n Nj,n (t) + βj +1,n Nj[1] +1,n (t).
Here αj,n , βj +1,n are real numbers. In (Wang and Chen, 2004) it was also shown that αj,n 0 and βj +1,n 0 for all j and αj,n + βj,n = 1 for all j, n 3. In fact, the UAT B-spline basis is just the NUAT B-spline basis with uniform knots ti+1 − ti = α, 0 < α < π . Wang and Chen (2004) also showed that, in the case ti−n+1 = ti−n+2 = · · · = ti < ti+1 = · · · = ti+n , {Ni−n+1,n (t), . . . , Ni,n (t)} is just the C-Bézier basis of order n on [ti , ti+1 ]. 2.4. The B-basis Carnicer and Peña studied the problem of finding an optimal shape preserving basis functions in a given space, and they showed that it can be solved by the concept of B-basis (Carnicer and Peña, 1994). Mainar, Peña and Sánchez-Reyes applied it to spaces mixing trigonometric and algebraic functions (Mainar et al., 2001). Definition 1. Let {u0 , u1 , . . . , un } be a totally positive (TP) basis (Carnicer and Peña, 1994) of a space Γn of the functions defined on I ∈ R. Then {u0 , u1 , . . . , un } is a B-basis if and only if the following conditions hold for all i = j .
ui (t) inf t ∈ I, uj (t) = 0 = 0. u (t) j
If {u0 , u1 , . . . , un } is also a normalized basis, then it is a normalized B-basis. Carnicer and Peña proved the existence of the B-basis and the normalized B-basis (Carnicer and Peña, 1994) that is, if there is a TP (or NTP) basis in a space, then there is a B-basis (or a normalized B-basis). They also showed that the Bernstein basis and the B-spline basis are both B-bases. In (Mainar et al., 2001), other examples were
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presented. In the same way, it is easy to prove that C-Bézier basis is normalized B-basis in the space Γn . To prove that the UAT B-spline basis can be extended to a B-basis, we will compute the transformation between the UAT B-spline and the C-Bézier basis in the next section.
3. The exchange between UAT B-spline basis and C-Bézier basis Let {M−n,n+1 (t), M−n+1,n+1 (t), . . . , M0,n+1 (t)} be the UAT B-spline basis of order n + 1 with uniform knots (ti )+∞ −∞ , ti+1 − ti = α, 0 < α < π , and {u0,n (t), u1,n (t), . . . , un,n (t)} be the C-Bézier basis of order n + 1. Then both of them are bases of the space Γn on [0, α]. Thus each of them can be expressed in [n+1] terms of the other. That is, there exists a nonsingular matrix An+1 = (ai,j )(n+1)×(n+1) such that mn = un An+1 .
(7)
By definition, we immediately get A3 : 1/2 1/2 0 1 0 , A3 = 0 0 1/2 1/2 and (Mainar and Peña, 2002) obtained the matrix A4 : P Q P 0 0 Q 2P 0 A4 = 0 2P Q 0 , 0 P Q P where P=
α − sin α , 2α(1 − cos α)
Q=
sin α − α cos α . α(1 − cos α)
In the following, we shall construct the matrix An+1 recursively, which can be represented by An . We use the following notations: −1 D . F= 0 ... 0 1 C , [n] [n] G= ai,1 ai,n 0 α1 ni=1 δi−1,n−1 . . . α1 ni=1 δi−1,n−1 where C, D are defined in (3) and (6). Then we get the following theorem: Theorem 1. The matrix An+1 transforming the UAT B-spline basis of order n into the C-Bézier basis of order n can be computed recursively: [n+1] An 0 =F G. An+1 = ai,j 0 1
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Proof. By Eq. (7), we have dmn dun = An+1 . dt dt Substituting Eqs. (2) and (5) in (8), we get mn−1 C = un−1 DAn+1 . By (7), we obtain: An C = DAn+1 .
(8)
(9)
By (1) and (4), we have [n+1] M−n,n+1 (0) = a1,1 ,
[n+1] M−n+1,n+1 (α) = an+1,2
[n+1] [n+1] and a1,1 = an+1,2 . Similarly, we have [n+1] [n+1] = an+1,i+n+2 = Mi+1,n+1 (α), Mi,n+1 (0) = a1,i+n+1
M−n,n+1 (α) = 0 =
Eq. (9) can be rewritten as: An 0 [n+1] 0 1 0 a1,1 That is
An+1 =
−1
D
i = −1, −2, . . . , −n,
[n+1] an+1,1 .
0 1
C
[n+1] a1,2
0 1
An 0
=
[n+1] . . . a1,n
D
0 1
An+1 .
C [n+1] 0 a1,1
[n+1] a1,2
[n+1] . . . a1,n
.
(10)
[n+1] [n+1] [n+1] , a1,2 , . . . , a1,n are obtained, then we obtain Theorem 1. In the folNow once the values of a1,1 [n+1] [n+1] [n+1] by recursion. Letting the first row vector lowing we will compute the values of a1,1 , a1,2 , . . . , a1,n of −1 An 0 D 0 1 0 1
be l 1 , then l1 = −
n−1 [n] ai+1,1 i=0
δi,n−1
,
−
n−1 [n] ai+1,2 i=0
δi,n−1
Letting the column vectors of C [n+1] 0 a1,1
[n+1] a1,2
...
,
...,
−
n−1 [n] ai+1,n i=0
δi,n−1
,1 .
[n+1] a1,n
be c1 , c2 , . . . , cn+1 respectively, we immediately get c1 = (−1/α, 0, . . . , 0). Then [n]
[n+1] a1,1
1 ai+1,1 = l 1 c1 = , α i=0 δi,n−1
[n+1] a1,2
1 ai+1,2 = l 1 c2 = , α i=0 δi,n−1
n−1
n−1
thus we get c2 ;
[n]
thus we get c3 ;
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··· [n+1] = l 1 cn+1 = a1,n+1
233
[n]
1 ai+1,n+1 . α i=0 δi,n−1 n−1
[n+1] [n+1] [n+1] , a1,2 , . . . , a1,n have been explicitly obtained, that is, the transforming matrix An+1 has been So a1,1 obtained by (10) and Theorem 1 has been proved: −1 C D An 0 [n] [n] [n] An+1 = . 2 ai+1,1 ai+1,n 1 n−1 ai+1,2 0 1 0 1 · · · α1 n−1 0 α1 n−1 i=0 δi,n−1 i=0 δi,n−1 i=0 δi,n−1 α
As an example, we can compute A4 from A3 by the Theorem 1: A3 0 G, A4 = F 0 1 where
−2αP 0 F= 0 0
2αP − αQ −2αP 2αP − αQ −2αP 0 −2αP 0 0
1 1 , 1 1
−1/α 0 G= 0 0
1/α −1/α 0 P
0 1/α −1/α Q
0 0 . 1/α P
4. The total positivity (TP) of the UAT B-spline basis Obviously, A3 and A4 are TP matrices (Zhang, 1996). We want to prove the matrix An+1 is TP by decomposing it into a product of bidiagonal and stochastic factors. Here a stochastic matrix is a nonnegative matrix such that the sum of the entries of each row is 1. We will prove the following theorem: Theorem 2. The matrix An+1 transforming the UAT B-spline basis of order n into the C-Bézier basis of order n is totally positive and stochastic. Proof. For the UAT B-spline basis functions of order n + 1, 0 t α, inserting a knot at t = 0 and t = α repeatedly until it get to the multiplicities of knots t = 0 and t = α are n respectively, and the old basis can be expressed by the new basis. From Section 2.3 we know that the UAT B-spline basis is the NUAT B-spline basis with uniform knots, so we denote the basis functions after the l’th inserting knot as [l] [0] , here Ni,n = Mi,n . According to Lemma 1, which is illustrated in Fig. 2, it’s easy to get that Ni,n [1] (t) = 0, N−n,n+1
[0] [1] N0,n+1 (t) = N1,n+1 (t),
when t ∈ [0, α], α ∈ (0, π ).
Then β1[1] = 1, α0[1] = 0, so β0[1] = 1. Thus [0] [1] [1] (t) = β−n+1 N−n+1,n+1 (t), N−n,n+1 [0] [1] [1] [1] (t) = αi[1] Ni,n+1 (t) + βi+1 Ni+1,n+1 (t) Ni,n+1 [0] (t) = N0,n+1
[1] N1,n+1 (t).
for i = −1, −2, . . . , −n + 1,
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Fig. 2. Inserting a knot at t = 0.
[1] [l] [l] Here αi[1] > 0, βi+1 > 0, αi[1] + βi[1] = 1, i = −n + 1, −n + 2, . . . , 0. Let m[l] n = (N−n,n+1 (t), N−n+1,n+1 (t), [l] [0] [1] . . . , N0,n+1 (t)), l = 0, 1, . . . , n − 1, m[0] n = mn , we have mn = mn Q0 . Continuing to insert new knots [l] [l+1] until the multiplicity of knot t = 0 is n, we have mn = mn Ql for l = 0, 1, . . . , n − 2: [2] [n−1] Qn−2 . . . Q1 Q0 . mn = m[1] n Q0 = mn Q1 Q0 = · · · = mn
Where Ql are all one banded nonnegative and stochastic matrices: l+1 l+1 β−n+1 α−n+1 l+1 l+1 β−n+2 α−n+2 , l = 0, 1, . . . , n − 2. Ql = ··· ··· 1 0 1 Analogously, continuing by inserting new knots at t = α until obtaining multiplicity n, we get mn = mn[n−1] Qn−2 . . . Q1 Q0 = · · · = mn[2n−2] Q2n−3 . . . Q1 Q0 . So we can deduce from (7) that the matrix An+1 can be expressed as product of one banded nonnegative and stochastic matrices. Since each of such matrices is TP and the product of TP matrices is also TP (Boor and Pinkus, 1982), we conclude that An+1 is TP and stochastic. 2 Since {u0,n (t), u1,n (t), . . . , un,n (t)} is the normalized B-basis and An+1 is a nonsingular and TP matrix, by Property 1.5 of (Lü et al., 2002) and Theorem 4.2 of (Carnicer and Peña, 1994), we get the following theorem. Theorem 3. The UAT B-spline basis {N−n,n+1 (t), N−n+1,n+1 (t), . . . , N0,n+1 (t)}) is a NTP basis of the space Γn = span{cos t, sin t, 1, t, . . . , t n−2 }, n 2, for 0 < α < π .
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5. A normalized B-basis The UAT B-spline functions {M0,n+1 (t), M1,n+1 (t), . . . , Mh−n−1,n+1 (t)} form a set of functions whose support is inside of [0, hα], n + 1 h ∈ z+ . We can add 2n NUAT basis functions into the UAT function set to form a normalized B-basis (Figs. 3, 4, 5). This is the following theorem: Theorem 4. The NUAT basis {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} defined on the knot sequence t−n+1 = · · · = t0 = 0, ti = iα, i = 1, . . . , h, th = · · · = th+n−1 = hα is a normalized B-basis for t ∈ [0, hα], 0 < α < π , h > 0. See Fig. 5. Proof. By the Section 2.3, we know that the basis {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} is NTP. By Definition 1, the only thing we need to do is proving the following equation: inf Ni,n+1 (t)/Nj,n+1 (t) | Nj,n+1 (t) = 0 = 0. The local support of Ni,n+1 (t) and Nj,n+1 (t) (i = j ) are I = (ti , ti+n+1 ) and J = (tj , tj +n+1 ). Let’s give the proof respectively: (1) When i < j , (a) If ti+n+1 < tj +n+1 , let t ∗ = (ti+n+1 + tj +n+1 )/2, we have Ni,n+1 (t ∗ ) = 0, Nj,n+1 (t ∗ ) = 0, So:
Ni,n+1 (t) Ni,n+1 (t ∗ ) N = 0. (t) = 0 = inf j,n+1 N (t) N (t ∗ ) j,n+1
j,n+1
− ti+n+1 ,
(b) If ti+n+1 = tj +n+1 , let t → higher than that of Mj,n+1 (t), so
then the infinite minimum order of Ni,n+1 (t) is j − i order
Ni,n+1 (t) = 0. t→ti+n+1 Nj,n+1 (t) lim −
Fig. 3. The UAT B-spline basis {N0,n+1 (t), N1,n+1 (t), . . . , Nh−n−1,n+1 (t)}.
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Fig. 4. The adding 2n NUAT basis functions.
Fig. 5. The normalized B-basis.
(2) When j < i, (a) If ti > tj , let t ∗ = (ti + tj )/2, we have Ni,n+1 (t ∗ ) = 0, Nj,n+1 (t ∗ ) = 0, So:
Ni,n+1 (t) inf Nj,n+1 (t)
∗ Nj,n+1 (t) = 0 = Ni,n+1 (t ) = 0. Nj,n+1 (t ∗ )
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(a)
(b)
(c)
(d)
237
Fig. 6. The B-spline (real line) and the UAT B-spline (dashed line). (a) The B-spline basis and the UAT B-spline basis when α = 7π/8. (b) The B-spline basis and the UAT B-spline basis when α = 1. (c) The B-spline curve and the UAT B-spline curve when α = 7π/8. (d) The B-spline curve and the UAT B-spline curve when α = 1.
(b) If ti = tj , let t → ti+ , then the infinite minimum order of Ni,n+1 (t) is i − j order higher than that of Nj,n+1 (t), so lim+
t→ti
Ni,n+1 (t) = 0. Nj,n+1 (t)
That completes the proof.
2
In fact, the normalized B-basis functions defined in Theorem 4 are composed of two parts. One is the set of UAT B-spline basis functions, i.e., Mi,n+1 = Ni,n+1 , i = 0, 1, . . . , h − n − 1, and the other is the set of NUAT B-spline basis functions, i.e., Nj,n+1 , j = −n, . . . , −1 or j = h − n, . . . , h − 1 (Figs. 3, 4, 5). Furthermore, the normalized B-basis of the cubic C-B-splines presented in (Mainar and Peña, 2002) is the special case of the normalized B-basis with n = 3.
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6. The convergence of the UAT B-spline basis Finally, we present an interesting result about the UAT B-spline curve. When α changes, the UAT B-spline curve also changes. That is the following proposition: Theorem 5. The UAT B-spline basis Mi,n (t), i = 0, 1, . . . , n converge to B-spline basis function bi,n (t) when α → 0. Therefore, the UAT B-spline curve converges to the B-spline curve (Fig. 6). Proof. Zhang (1997) proved this proposition on the space Γ3 . Suppose now that it holds in the space Γn−1 : lim Mi,n−1 (t) = bi,n−1 (t).
α→0
So we have: Mi,n−1 (t) − Mi+1,n−1 (t) 1 = bi,n−1 (t) − bi+1,n−1 (t) = bi,n−1 (t). α→0 α→0 α α By Mi,n (0) = bi,n (0), the following limit is obtained: (t) = lim lim Mi,n
lim Mi,n (t) = bi,n (t).
α→0
From the definitions of the UAT B-spline and the B-spline curve, we have this proposition.
2
Acknowledgements We are very grateful to the referees for their helpful suggestions and comments. This work was partially supported by the Natural Science Foundation of China (No. 60473130) and the Foundation of State Key Basic Research 973 Development Programming Item of China (No. G2004CB318000).
References Boor, C., Pinkus, A., 1982. The approximation of a totally positive band matrix by a strictly banded totally positive one. Lin. Alg. Appl. 42, 81–98. Carnicer, J., Peña, J., 1994. Totally positive bases for shape preserving curve design and optimality of B-splines. Computer Aided Geometric Design 11, 635–656. Chen, Q., Wang, G., 2003. A class of Bézier-like curves. Computer Aided Geometric Design 20 (1), 29–39. Lü, Y., Wang, G., Yang, X., 2002. Uniform trigonometric polynomial B-spline curves. Science in China (Series F) 45, 335–343. Mainar, E., Peña, J., 1999. Corner cutting algorithms associated with optimal shape preserving representations. Computer Aided Geometric Design 16, 883–906. Mainar, E., Peña, J., 2002. A basis of C-Bézier splines with optimal properties. Computer Aided Geometric Design 19, 291–295. Mainar, E., Peña, J., 2004. Quadratic-cycloidal curves. Adv. Comput. Math. 20, 161–175. Mainar, E., Peña, J., Sánchez-Reyes, J., 2001. Shape preserving alternatives to the rational Bézier model. Computer Aided Geometric Design 18 (1), 37–60. Wang, G., Chen, Q., 2004. NUAT B-spline curves. Computer Aided Geometric Design 21, 193–205. Zhang, J., 1996. An extension of cubic curves. Computer Aided Geometric Design 13, 199–217. Zhang, J., 1997. Two different forms of C-B-splines. Computer Aided Geometric Design 14 (1), 31–41.
Optimal properties of the uniform algebraic trigonometric B-splines Guozhao Wang, Yajuan Li ∗ Institute of Computer Graphics and Image Processing, Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China Received 24 March 2005; received in revised form 31 March 2005; accepted 23 September 2005
Abstract In this paper, we construct a matrix, which transforms a generalized C-Bézier basis into a generalized uniform algebraic-trigonometric B-spline (C-B-spline or UAT B-spline) basis. We also show that it is a totally positive matrix and give a normalized B-basis of the generalized UAT B-splines. 2005 Elsevier B.V. All rights reserved. Keywords: C-Bézier basis; C-B-spline basis; C-Bézier curve; UAT B-spline; NUAT B-spline; B-basis; Totally positive
1. Introduction The normalized B-basis is the unique normalized basis of a space with optimal shape preserving properties. It is well known that both the Bézier basis and the B-spline basis are normalized B-bases of the polynomial space. However, they have many shortcomings. For instance, they cannot represent exactly transcendental curves such as the helix and the cycloid. Furthermore, repeated differentiation of the rational form produces curves of very high degree. So new bases in new spaces have been extensively studied in the literature (Zhang, 1996, 1997; Mainar and Peña, 2004; Lü et al., 2002; Chen and Wang, 2003; Wang and Chen, 2004). * Corresponding author.
E-mail addresses: [email protected] (G. Wang), [email protected], [email protected] (Y. Li). 0167-8396/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cagd.2005.09.002
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Zhang proposed the cubic C-Bézier basis and the cubic UAT B-spline basis (Zhang, 1996, 1997) in the space Γ3 = span{cos t, sin t, 1, t}, and Mainar and Peña obtained the matrix for transforming the cubic C-Bézier basis into the UAT B-spline basis and proved that the cubic UAT B-spline basis can be extended to a B-basis (Mainar and Peña, 2002). The cubic C-Bézier curves described in (Zhang, 1996) are special cases of generalized C-Bézier curves described in (Chen and Wang, 2003); The cubic UAT B-spline curves described in (Zhang, 1997) and the quadratic-cycloidal curves described in (Mainar and Peña, 2004) are special cases of the UAT B-spline curves (Lü et al., 2002). In fact, not only the cubic UAT B-spline basis has such properties, but the general UAT B-spline basis of arbitrary order n can also be extended to a B-basis of the space Γn = span{cos t, sin t, 1, t, . . . , t n−2 }. In this paper, we compute the matrix which transforms the C-Bézier basis into the UAT B-spline basis and decompose this matrix into a product of bidiagonal and stochastic factors. Thus we deduce that the UAT B-spline basis is normalized totally positive (NTP). Furthermore, we give a normalized B-basis of the UAT B-splines. This paper is organized as follows. In Section 2, the basic definitions and notations will be presented. Then in Section 3, we will present the matrix transforming the general C-Bézier basis into the general UAT B-spline basis. In Sections 4 and 5, the total positivity of the general UAT B-spline basis will be proven. As a result, the general UAT B-spline basis can be extended to a normalized B-basis. The convergence of the UAT B-spline basis will be proved in the last section.
2. Preliminaries 2.1. The UAT B-spline basis The main result of this paper is to show some optimal properties of the UAT B-spline basis. Let ti = iα (i = 0, ±1, ±2, . . .) be a set of knots which partition the parameter axis t uniformly. Then the UAT Bspline basis functions of order n + 1 {Mi,n+1 (t)}+∞ i=−∞ of the space Γn on t ∈ [−∞, +∞], 0 < α < π were given in (Lü et al., 2002): −α sin t/[2(cos α − 1)], 0 t α, M0,2 (t) = −α sin(2α − t)/[2(cos α − 1)], α t 2α, 0, otherwise. t 1 Mi,n (x) dx, n 2, i = ±1, ±2, . . . . Mi,n+1 (t) = α t−α
Mi,n+1 (t) = M0,n+1 (t − ti ),
i = ±1, ±2, . . . .
It is easy to get the following equation: Mi,n+1 (0) = Mi+1,n+1 (α),
i = 0, 1, . . . , n.
(1)
The UAT B-spline basis has many optimal properties the same as those of the B-spline basis, such as the subdivision property, the variation diminishing property, the convexity preserving property, etc. By the properties of the UAT B-spline, it is not difficult to show that it can be used as a quasi-interpolant to given data points in the same way as the B-spline. Lü et al. (2002) gave the derivative of Mi,n+1 : dMi,n+1 (t) 1 = Mi,n (t) − Mi+1,n (t) , i = −n, −n + 1, . . . , 0. dt α
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They also showed that {M−n,n+1 (t), M−n+1,n+1 (t), . . . , M0,n+1 (t)} is a basis of the space Γn on t ∈ [0, α]. By the definition of the UAT B-spline basis, we know M−n,n (t) = M1,n (t) = 0, t ∈ [0, α]. Let mn denote the UAT B-spline basis of order n + 1, that is, mn = (M−n,n+1 (t), M−n+1,n+1 (t), . . . , M0,n+1 (t)), we have dmn = mn−1 C, (2) dt where −1 1 1 −1 1 . (3) C= ··· ··· α −1 1 n×(n+1) 2.2. The C-Bézier basis Mainar and Peña obtained the matrix for transforming the cubic C-Bézier basis into the UAT B-spline basis and extended the cubic UAT B-spline basis to a B-basis (Mainar and Peña, 2002). In this paper, we will present the same result for the general UAT B-spline basis by transforming the general C-Bézier basis. We review the C-Bézier curves presented in (Chen and Wang, 2003). Two initial functions were given as: sin(α − t) sin t , u1,1 (t) = , t ∈ [0, α], α ∈ (0, π ) sin α sin α for n > 1. The C-Bézier basis functions {u0,n , u1,n , . . . , un,n } of the space Γn = span{cos t, sin t, 1, t, . . . , t n−2 } were defined recursively by u0,1 (t) =
t u0,n (t) = 1 −
δ0,n−1 u0,n−1 (s) ds, 0
t
δi−1,n−1 ui−1,n−1 (s) − δi,n−1 ui,n−1 (s) ds
ui,n (t) =
for 1 i n − 1,
0
t un,n (t) =
δi−1,n−1 ui−1,n−1 (s) ds, 0
where δi,n = 1/
α 0
ui,n (t) dt. By the definition of the C-Bézier basis functions, we obtain:
u0,n (0) = 1,
u1,n (0) = u2,n (0) = · · · = un,n (0) = 0,
un,n (α) = 1,
u0,n (α) = u1,n (α) = · · · = un−1,n (α) = 0.
The derivatives of the functions were also presented in (Chen and Wang, 2003): dui,n (t) = δi−1,n−1 ui−1,n−1 (t) − δi,n−1 ui,n−1 (t). dt
(4)
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Denoting un = (u0,n (t), u1,n (t), . . . , un,n (t)), we can get: dun = un−1 D, dt where u−1,n−1 (t) = un,n−1 (t) = 0 and −δ0,n−1 δ0,n−1 −δ1,n−1 δ1,n−1 D= ···
(5) .
···
−δn−1,n−1
(6)
δn−1,n−1
2.3. The NUAT B-spline basis To get the transforming matrix between the UAT B-spline basis and the C-Bézier basis, we must use the definition and the properties of the non-uniform algebraic trigonometric (NUAT) B-spline basis. Given a knot sequence T = (ti )+∞ −∞ , some initial functions were given in (Wang and Chen, 2004): ti < t ti+1 , sin(t − ti )/ sin(ti+1 − ti ), Ni,2 (t) = sin(ti+2 − t)/ sin(ti+2 − ti+1 ), ti+1 < t ti+2 , 0, otherwise. Then the NUAT B-spline basis functions of order n + 1 can be defined as: t Ni,n (t) =
δi,n−1 Ni,n−1 (s) − δi+1,n−1 Ni+1,n−1 (s) ds,
n 3.
−∞
+∞ Where 0 ti+1 − ti < π , δi,n = 1/ −∞ Ni,n (t) dt. Here 0/0 = 0. The derivatives of the basis functions are: dNi,n (t) = δi−1,n−1 ui−1,n−1 (t) − δi,n−1 Ni,n−1 (t). dt Considering NUAT B-spline basis functions {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} of order n + 1 defined on knot sequence t−n+1 = · · · = t0 < t1 < · · · < th = · · · = th+n−1 . By the properties of the NUAT basis functions (Wang and Chen, 2004), we know that the NUAT B-spline curve: s(t) =
h−1
Ni,n+1 (t)pi ,
t ∈ [t0 , th ]
i=−n
must have the endpoint properties, V.D. property and boundary tangent properties (Fig. 1). By (Carnicer and Peña, 1994; Mainar and Peña, 1999), {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} is NTP. Here pi , i = −n, −n + 1, . . . , h − 1 are control points. The knot_insertion theorem is as follows (Theorem 4.2 of (Wang and Chen, 2004)): [1] = (ti[1] )+∞ Lemma 1. Let T = (ti )+∞ −∞ be a given knot sequence with 0 ti+1 − ti < π , and let T −∞ be a new knot sequence obtained by inserting a new knot u into T with ti u < ti+1 , where Nj,n (t) and [1] (t) are defined on the knot sequence T and T [1] respectively. Then we have for all j, n 2: Nj,n
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Fig. 1. The NUAT B-spline curve with endpoint properties and boundary tangent properties. [1] Ni,n (t) = αj,n Nj,n (t) + βj +1,n Nj[1] +1,n (t).
Here αj,n , βj +1,n are real numbers. In (Wang and Chen, 2004) it was also shown that αj,n 0 and βj +1,n 0 for all j and αj,n + βj,n = 1 for all j, n 3. In fact, the UAT B-spline basis is just the NUAT B-spline basis with uniform knots ti+1 − ti = α, 0 < α < π . Wang and Chen (2004) also showed that, in the case ti−n+1 = ti−n+2 = · · · = ti < ti+1 = · · · = ti+n , {Ni−n+1,n (t), . . . , Ni,n (t)} is just the C-Bézier basis of order n on [ti , ti+1 ]. 2.4. The B-basis Carnicer and Peña studied the problem of finding an optimal shape preserving basis functions in a given space, and they showed that it can be solved by the concept of B-basis (Carnicer and Peña, 1994). Mainar, Peña and Sánchez-Reyes applied it to spaces mixing trigonometric and algebraic functions (Mainar et al., 2001). Definition 1. Let {u0 , u1 , . . . , un } be a totally positive (TP) basis (Carnicer and Peña, 1994) of a space Γn of the functions defined on I ∈ R. Then {u0 , u1 , . . . , un } is a B-basis if and only if the following conditions hold for all i = j .
ui (t) inf t ∈ I, uj (t) = 0 = 0. u (t) j
If {u0 , u1 , . . . , un } is also a normalized basis, then it is a normalized B-basis. Carnicer and Peña proved the existence of the B-basis and the normalized B-basis (Carnicer and Peña, 1994) that is, if there is a TP (or NTP) basis in a space, then there is a B-basis (or a normalized B-basis). They also showed that the Bernstein basis and the B-spline basis are both B-bases. In (Mainar et al., 2001), other examples were
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presented. In the same way, it is easy to prove that C-Bézier basis is normalized B-basis in the space Γn . To prove that the UAT B-spline basis can be extended to a B-basis, we will compute the transformation between the UAT B-spline and the C-Bézier basis in the next section.
3. The exchange between UAT B-spline basis and C-Bézier basis Let {M−n,n+1 (t), M−n+1,n+1 (t), . . . , M0,n+1 (t)} be the UAT B-spline basis of order n + 1 with uniform knots (ti )+∞ −∞ , ti+1 − ti = α, 0 < α < π , and {u0,n (t), u1,n (t), . . . , un,n (t)} be the C-Bézier basis of order n + 1. Then both of them are bases of the space Γn on [0, α]. Thus each of them can be expressed in [n+1] terms of the other. That is, there exists a nonsingular matrix An+1 = (ai,j )(n+1)×(n+1) such that mn = un An+1 .
(7)
By definition, we immediately get A3 : 1/2 1/2 0 1 0 , A3 = 0 0 1/2 1/2 and (Mainar and Peña, 2002) obtained the matrix A4 : P Q P 0 0 Q 2P 0 A4 = 0 2P Q 0 , 0 P Q P where P=
α − sin α , 2α(1 − cos α)
Q=
sin α − α cos α . α(1 − cos α)
In the following, we shall construct the matrix An+1 recursively, which can be represented by An . We use the following notations: −1 D . F= 0 ... 0 1 C , [n] [n] G= ai,1 ai,n 0 α1 ni=1 δi−1,n−1 . . . α1 ni=1 δi−1,n−1 where C, D are defined in (3) and (6). Then we get the following theorem: Theorem 1. The matrix An+1 transforming the UAT B-spline basis of order n into the C-Bézier basis of order n can be computed recursively: [n+1] An 0 =F G. An+1 = ai,j 0 1
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Proof. By Eq. (7), we have dmn dun = An+1 . dt dt Substituting Eqs. (2) and (5) in (8), we get mn−1 C = un−1 DAn+1 . By (7), we obtain: An C = DAn+1 .
(8)
(9)
By (1) and (4), we have [n+1] M−n,n+1 (0) = a1,1 ,
[n+1] M−n+1,n+1 (α) = an+1,2
[n+1] [n+1] and a1,1 = an+1,2 . Similarly, we have [n+1] [n+1] = an+1,i+n+2 = Mi+1,n+1 (α), Mi,n+1 (0) = a1,i+n+1
M−n,n+1 (α) = 0 =
Eq. (9) can be rewritten as: An 0 [n+1] 0 1 0 a1,1 That is
An+1 =
−1
D
i = −1, −2, . . . , −n,
[n+1] an+1,1 .
0 1
C
[n+1] a1,2
0 1
An 0
=
[n+1] . . . a1,n
D
0 1
An+1 .
C [n+1] 0 a1,1
[n+1] a1,2
[n+1] . . . a1,n
.
(10)
[n+1] [n+1] [n+1] , a1,2 , . . . , a1,n are obtained, then we obtain Theorem 1. In the folNow once the values of a1,1 [n+1] [n+1] [n+1] by recursion. Letting the first row vector lowing we will compute the values of a1,1 , a1,2 , . . . , a1,n of −1 An 0 D 0 1 0 1
be l 1 , then l1 = −
n−1 [n] ai+1,1 i=0
δi,n−1
,
−
n−1 [n] ai+1,2 i=0
δi,n−1
Letting the column vectors of C [n+1] 0 a1,1
[n+1] a1,2
...
,
...,
−
n−1 [n] ai+1,n i=0
δi,n−1
,1 .
[n+1] a1,n
be c1 , c2 , . . . , cn+1 respectively, we immediately get c1 = (−1/α, 0, . . . , 0). Then [n]
[n+1] a1,1
1 ai+1,1 = l 1 c1 = , α i=0 δi,n−1
[n+1] a1,2
1 ai+1,2 = l 1 c2 = , α i=0 δi,n−1
n−1
n−1
thus we get c2 ;
[n]
thus we get c3 ;
G. Wang, Y. Li / Computer Aided Geometric Design 23 (2006) 226–238
··· [n+1] = l 1 cn+1 = a1,n+1
233
[n]
1 ai+1,n+1 . α i=0 δi,n−1 n−1
[n+1] [n+1] [n+1] , a1,2 , . . . , a1,n have been explicitly obtained, that is, the transforming matrix An+1 has been So a1,1 obtained by (10) and Theorem 1 has been proved: −1 C D An 0 [n] [n] [n] An+1 = . 2 ai+1,1 ai+1,n 1 n−1 ai+1,2 0 1 0 1 · · · α1 n−1 0 α1 n−1 i=0 δi,n−1 i=0 δi,n−1 i=0 δi,n−1 α
As an example, we can compute A4 from A3 by the Theorem 1: A3 0 G, A4 = F 0 1 where
−2αP 0 F= 0 0
2αP − αQ −2αP 2αP − αQ −2αP 0 −2αP 0 0
1 1 , 1 1
−1/α 0 G= 0 0
1/α −1/α 0 P
0 1/α −1/α Q
0 0 . 1/α P
4. The total positivity (TP) of the UAT B-spline basis Obviously, A3 and A4 are TP matrices (Zhang, 1996). We want to prove the matrix An+1 is TP by decomposing it into a product of bidiagonal and stochastic factors. Here a stochastic matrix is a nonnegative matrix such that the sum of the entries of each row is 1. We will prove the following theorem: Theorem 2. The matrix An+1 transforming the UAT B-spline basis of order n into the C-Bézier basis of order n is totally positive and stochastic. Proof. For the UAT B-spline basis functions of order n + 1, 0 t α, inserting a knot at t = 0 and t = α repeatedly until it get to the multiplicities of knots t = 0 and t = α are n respectively, and the old basis can be expressed by the new basis. From Section 2.3 we know that the UAT B-spline basis is the NUAT B-spline basis with uniform knots, so we denote the basis functions after the l’th inserting knot as [l] [0] , here Ni,n = Mi,n . According to Lemma 1, which is illustrated in Fig. 2, it’s easy to get that Ni,n [1] (t) = 0, N−n,n+1
[0] [1] N0,n+1 (t) = N1,n+1 (t),
when t ∈ [0, α], α ∈ (0, π ).
Then β1[1] = 1, α0[1] = 0, so β0[1] = 1. Thus [0] [1] [1] (t) = β−n+1 N−n+1,n+1 (t), N−n,n+1 [0] [1] [1] [1] (t) = αi[1] Ni,n+1 (t) + βi+1 Ni+1,n+1 (t) Ni,n+1 [0] (t) = N0,n+1
[1] N1,n+1 (t).
for i = −1, −2, . . . , −n + 1,
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Fig. 2. Inserting a knot at t = 0.
[1] [l] [l] Here αi[1] > 0, βi+1 > 0, αi[1] + βi[1] = 1, i = −n + 1, −n + 2, . . . , 0. Let m[l] n = (N−n,n+1 (t), N−n+1,n+1 (t), [l] [0] [1] . . . , N0,n+1 (t)), l = 0, 1, . . . , n − 1, m[0] n = mn , we have mn = mn Q0 . Continuing to insert new knots [l] [l+1] until the multiplicity of knot t = 0 is n, we have mn = mn Ql for l = 0, 1, . . . , n − 2: [2] [n−1] Qn−2 . . . Q1 Q0 . mn = m[1] n Q0 = mn Q1 Q0 = · · · = mn
Where Ql are all one banded nonnegative and stochastic matrices: l+1 l+1 β−n+1 α−n+1 l+1 l+1 β−n+2 α−n+2 , l = 0, 1, . . . , n − 2. Ql = ··· ··· 1 0 1 Analogously, continuing by inserting new knots at t = α until obtaining multiplicity n, we get mn = mn[n−1] Qn−2 . . . Q1 Q0 = · · · = mn[2n−2] Q2n−3 . . . Q1 Q0 . So we can deduce from (7) that the matrix An+1 can be expressed as product of one banded nonnegative and stochastic matrices. Since each of such matrices is TP and the product of TP matrices is also TP (Boor and Pinkus, 1982), we conclude that An+1 is TP and stochastic. 2 Since {u0,n (t), u1,n (t), . . . , un,n (t)} is the normalized B-basis and An+1 is a nonsingular and TP matrix, by Property 1.5 of (Lü et al., 2002) and Theorem 4.2 of (Carnicer and Peña, 1994), we get the following theorem. Theorem 3. The UAT B-spline basis {N−n,n+1 (t), N−n+1,n+1 (t), . . . , N0,n+1 (t)}) is a NTP basis of the space Γn = span{cos t, sin t, 1, t, . . . , t n−2 }, n 2, for 0 < α < π .
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5. A normalized B-basis The UAT B-spline functions {M0,n+1 (t), M1,n+1 (t), . . . , Mh−n−1,n+1 (t)} form a set of functions whose support is inside of [0, hα], n + 1 h ∈ z+ . We can add 2n NUAT basis functions into the UAT function set to form a normalized B-basis (Figs. 3, 4, 5). This is the following theorem: Theorem 4. The NUAT basis {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} defined on the knot sequence t−n+1 = · · · = t0 = 0, ti = iα, i = 1, . . . , h, th = · · · = th+n−1 = hα is a normalized B-basis for t ∈ [0, hα], 0 < α < π , h > 0. See Fig. 5. Proof. By the Section 2.3, we know that the basis {N−n,n+1 (t), N−n+1,n+1 (t), . . . , Nh−1,n+1 (t)} is NTP. By Definition 1, the only thing we need to do is proving the following equation: inf Ni,n+1 (t)/Nj,n+1 (t) | Nj,n+1 (t) = 0 = 0. The local support of Ni,n+1 (t) and Nj,n+1 (t) (i = j ) are I = (ti , ti+n+1 ) and J = (tj , tj +n+1 ). Let’s give the proof respectively: (1) When i < j , (a) If ti+n+1 < tj +n+1 , let t ∗ = (ti+n+1 + tj +n+1 )/2, we have Ni,n+1 (t ∗ ) = 0, Nj,n+1 (t ∗ ) = 0, So:
Ni,n+1 (t) Ni,n+1 (t ∗ ) N = 0. (t) = 0 = inf j,n+1 N (t) N (t ∗ ) j,n+1
j,n+1
− ti+n+1 ,
(b) If ti+n+1 = tj +n+1 , let t → higher than that of Mj,n+1 (t), so
then the infinite minimum order of Ni,n+1 (t) is j − i order
Ni,n+1 (t) = 0. t→ti+n+1 Nj,n+1 (t) lim −
Fig. 3. The UAT B-spline basis {N0,n+1 (t), N1,n+1 (t), . . . , Nh−n−1,n+1 (t)}.
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Fig. 4. The adding 2n NUAT basis functions.
Fig. 5. The normalized B-basis.
(2) When j < i, (a) If ti > tj , let t ∗ = (ti + tj )/2, we have Ni,n+1 (t ∗ ) = 0, Nj,n+1 (t ∗ ) = 0, So:
Ni,n+1 (t) inf Nj,n+1 (t)
∗ Nj,n+1 (t) = 0 = Ni,n+1 (t ) = 0. Nj,n+1 (t ∗ )
G. Wang, Y. Li / Computer Aided Geometric Design 23 (2006) 226–238
(a)
(b)
(c)
(d)
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Fig. 6. The B-spline (real line) and the UAT B-spline (dashed line). (a) The B-spline basis and the UAT B-spline basis when α = 7π/8. (b) The B-spline basis and the UAT B-spline basis when α = 1. (c) The B-spline curve and the UAT B-spline curve when α = 7π/8. (d) The B-spline curve and the UAT B-spline curve when α = 1.
(b) If ti = tj , let t → ti+ , then the infinite minimum order of Ni,n+1 (t) is i − j order higher than that of Nj,n+1 (t), so lim+
t→ti
Ni,n+1 (t) = 0. Nj,n+1 (t)
That completes the proof.
2
In fact, the normalized B-basis functions defined in Theorem 4 are composed of two parts. One is the set of UAT B-spline basis functions, i.e., Mi,n+1 = Ni,n+1 , i = 0, 1, . . . , h − n − 1, and the other is the set of NUAT B-spline basis functions, i.e., Nj,n+1 , j = −n, . . . , −1 or j = h − n, . . . , h − 1 (Figs. 3, 4, 5). Furthermore, the normalized B-basis of the cubic C-B-splines presented in (Mainar and Peña, 2002) is the special case of the normalized B-basis with n = 3.
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6. The convergence of the UAT B-spline basis Finally, we present an interesting result about the UAT B-spline curve. When α changes, the UAT B-spline curve also changes. That is the following proposition: Theorem 5. The UAT B-spline basis Mi,n (t), i = 0, 1, . . . , n converge to B-spline basis function bi,n (t) when α → 0. Therefore, the UAT B-spline curve converges to the B-spline curve (Fig. 6). Proof. Zhang (1997) proved this proposition on the space Γ3 . Suppose now that it holds in the space Γn−1 : lim Mi,n−1 (t) = bi,n−1 (t).
α→0
So we have: Mi,n−1 (t) − Mi+1,n−1 (t) 1 = bi,n−1 (t) − bi+1,n−1 (t) = bi,n−1 (t). α→0 α→0 α α By Mi,n (0) = bi,n (0), the following limit is obtained: (t) = lim lim Mi,n
lim Mi,n (t) = bi,n (t).
α→0
From the definitions of the UAT B-spline and the B-spline curve, we have this proposition.
2
Acknowledgements We are very grateful to the referees for their helpful suggestions and comments. This work was partially supported by the Natural Science Foundation of China (No. 60473130) and the Foundation of State Key Basic Research 973 Development Programming Item of China (No. G2004CB318000).
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