Brief Paper ACTA AUTOMATICA SINICA Vol. 34, No. 8 August, 2008

Extended Cubic UniformB-spline and ααα-B-spline

XU Gang1 WANG Guo-Zhao1

Abstract Spline curve and surface play an important role inCAD and computer graphics. In this paper, we propose severalextensions of cubic uniform B-spline. Then, we present the ex-tensions of interpolating α-B-spline based on the new B-splinesand the singular blending technique. The advantage of the ex-tensions is that they have global and local shape parameters.Furthermore, we also investigate their applications in data in-terpolation and polygonal shape deformation.

Key words B-spline, α-B-spline, singular blending, interpo-lation, curve/surface modeling

Curve and surface modeling is an important subject inCAD and computer graphics[1−3]. Spline curve/surfacemodeling is the most traditional modeling method based onthe theory of computer aided geometric design (CAGD)[4].Several kinds of splines have been proposed in the fieldof CAGD, such as B-splines and T-splines[5]. In prac-tice, we often use cubic uniform B-spline for curve/surfacemodeling. However, once the control points of the cu-bic uniform B-spline curve are determined, the shape ofthe curve is determined. In order to overcome this dis-advantage, several extensions of cubic uniform B-splinehave been proposed[6−8]. On the other hand, cubic B-spline interpolation is the traditional global interpolationmethod[9−10]. Unfortunately, it also has some disadvan-tages, making them less desirable for certain applications.First, it cannot provide parameter for curve local modifi-cation; second, it may exhibit undesirable oscillations; andfinally, solving the linear system is very expensive compu-tationally. Moreover, the interpolation method is global;thus, changes to any data point will require solving againall the linear systems.

In order to avoid the above limitations, Tai et al. pro-posed a new method to solve the interpolation problem[11]

based on the α-B-spline presented in [12]. The general-izations and applications of α-B-spline were investigated in[13−15]. However, the above interpolating splines based onsingular blending cannot conveniently modify the curves. Ifthe users want to globally modify the curves, they must setall the local parameters to be equal. In this paper, wefirst propose two kinds of new extensions of cubic uniformB-spline. Then we propose the extensions of interpolatingα-B-spline by the singular blending technique, and namethem α-EB-spline. They not only have local shape parame-ters, but also have global parameters. We also present theirapplications in data interpolation and shape deformation.

1 Extended cubic uniform B-spline blend-ing functions

The blending function of degree 4 was proposed in [6],which is an extension of cubic uniform B-spline basis func-tion.

Received June 6, 2007; in revised form September 7, 2007Supported by National Natural Science Foundation of China

(60773179) and the National Basic Research Program of China (973Program) (2004CB318000)1. Institute of Computer Graphics and Image Processing, De-

partment of Mathematics, Zhejiang University, Hangzhou 310027P.R.ChinaDOI: 10.3724/SP.J.1004.2008.00980

Definition 1. For t ∈ [0, 1], the blending function ofdegree 4 is defined as follows.

b40(t) =

1

24(4− λ− 3λt)(1− t)3

b41(t) =

1

24[16 + 2λ− 12(2 + λ)t2 + 12(1 + λ)t3 − 3λt4]

b42(t) =

1

24[4− λ + 12t + 6(2 + λ)t2 − 12t3 − 3λt4]

b43(t) =

1

24[4(1− λ) + 3λt]t3

In particular, in case of λ = 0, the blending func-tion of degree 4 will degenerate to the cubic uniform B-spline blending function. In the following, we will list twonew kinds of extensions of blending functions of degree 4.Hence, they can be considered as further extensions of cubicuniform B-spline blending function.

Definition 2. For t ∈ [0, 1], the blending function ofdegree 5 is defined as follows.

b50(t) =

1

40(5− λ− 4λt)(1− t)4

b51(t) =

1

40[30 + 2λ− 20(3 + λ)t2 + 40(1 + λ)t3 − 5(1+

7λ)t4 + 12λt5]

b52(t) =

1

40[5− λ + 20t + 10(3 + λ)t2 − 20(1 + λ)t3−

5(1− 5λ)t4 − 12λt5]

b53(t) =

1

40[5(1− λ) + 4λt]t4

Definition 3. For t ∈ [0, 1], the blending function ofdegree 6 is defined as follows.

b60(t) =

1

60(6− λ− 5λt)(1− t)5

b61(t) =

1

60[48 + 2λ− 30(4 + λ)t2 + 40(3 + 2λ)t3 − 30(2+

3λ)t4 + 6(3 + 7λ)t5 − 5λt6]

b62(t) =

1

60[6− λ + 30t + 15(4 + λ)t2 − 20(3 + 2λ)t3+

15(2 + 3λ)t4 − 6(3 + 2λ)t5 − 5λt6]

b63(t) =

1

60[6(1− λ) + 5λt]t5

When λ = 0, bki (t) will be bk−1

i (t) with λ = 1, k =5, 6. The above three kinds of blending functions have thefollowing theorem.

Theorem 1. The blending functions bki (t), where k =

4, 5, 6, and i = 0, 1, 2, 3, satisfy1)

∑3i=0 bk

i (t) = 1;

2) bki (t) = bk

3−i(1− t);

3) When −k(k − 2) ≤ λ ≤ 1, bki (t) ≥ 0, t ∈ [0, 1].

2 Extended cubic uniform B-spline curvesThe properties mentioned in Theorem 1 enable the

blending functions to be used for curve design.Definition 4. Given control points PPP i ∈ Rd (d =

2, 3, i = 0, 1, · · · , n), and the knots u1 < u2 < · · · < un+1,for u ∈ [ui, ui+1], i = 3, 4, · · · , n, the polynomial curve seg-ments are defined as follows.

CCCj,k(λ; t) =

3∑i=0

bki (t)PPP j+i−3, k = 4, 5, 6

No. 8 XU Gang and WANG Guo-Zhao: Extended Cubic Uniform B-spline and α-B-spline 981

where t = u− uihi

, hi = ui+1 − ui. The polynomial curve is

defined as follows.

CCCk(λ; u) = CCCi;k(λ;u− ui

hi), u ∈ [ui, ui+1]

They can be considered as extensions of cubic uniformB-spline curve. They have many nice properties such asconvex full property, symmetry, and geometric invariability.

Theorem 2. For the case of uniform knot, CCCk(λ; u) isC2 continuous; for the case of nonuniform knot, CCCk(λ; u)is G2 continuous.

Proof. After direct computation for k = 5 and k = 6,we have

CCCi,k(λ; 0) =1

2k(k−1){(k − λ)PPP i−3+2[k(k − 2) + λ]PPP i−2+

(k − λ)PPP i−1}CCCi,k(λ; 1) =

1

2k(k−1){(k − λ)PPP i−2+2[k(k − 2) + λ]PPP i−1+

(k − λ)PPP i}CCC′i,k(λ; 0) =

1

2(PPP i−1 −PPP i−3), CCC′i,k(λ; 1) =

1

2(PPP i −PPP i−2)

CCC′′i,k(λ; 0) =1

2(k + λ− 2)(PPP i−3 − 2PPP i−2 + PPP i−1)

CCC′′i,k(λ; 1) =1

2(k + λ− 2)(PPP i−2 − 2PPP i−1 + PPP i)

Hence, for i = 4, 5, · · · , n, we can obtain

CCC(l)k (λ; u−i ) = (

hi

hi−1)lCCC

(l)k (λ; u+

i ), l = 0, 1, 2, k = 5, 6

Thus, the conclusion is proved. ¤Fig. 1 presents the curves constructed by various blend-

ing functions with different shape parameters. We can findthat the approaching degrees of the curves of degree 5 anddegree 6 to their control polygon are higher than that ofthe curves of degree 4 with the same shape parameters.

(a) (b)

(c)

Fig. 1 Curve examples constructed by blending functions withλ = −4,−3,−2,−1, 0, 1 ((a) Curves constructed by blendingfunction of degree 4; (b) Curves constructed by blending func-tion of degree 5; (c) Curves constructed by blending function ofdegree 6)

3 Singularly reparameterized line seg-ment

A singularly reparameterized (SR) line segment is a linesegment that possesses parametric derivatives equal to zeroat each end. It is obtained by blending two endpointswith a singular blending function[11, 15]. An m-level singu-lar blending function S(t), t ∈ [0, 1], satisfies the followingconditions

S(0) = 0, S(1) = 1

S(k)(0) = S(k)(1) = 0, k = 1, 2, · · · , m(1)

Loe used a 2-level singular blending function and suggestedS(t) = 1 − (1 − t3)3 in [12]. Another choice is the quinticHermite polynomial S(t) = 10t3 − 15t4 + 6t5. For the re-mainder of this paper, we will adopt the singular blendingfunction presented in [14]. It is a piecewise cubic polyno-mial defined on three subintervals.

S(t) =

9

2t3, 0 ≤ t ≤ 1

3

9

2[t3 − 3(t− 1

3)3],

1

3≤ t ≤ 2

3

1− 9

2(1− t)3,

2

3≤ t ≤ 1

(2)

Using the singular blending function S(t), we can blendtwo adjacent vertices VVV j and VVV j+1 to produce a SR linesegment LLLj(t), that is,

LLLj(t) = (1− S(t))VVV j + S(t)VVV j+1, t ∈ [0, 1]

From (1), we have

LLLj(0) = VVV j , LLLj(1) = VVV j+1

LLL′j(0) = LLL′j(1) = LLL′′j (0) = LLL′′j (1) = 0

4 ααα-EB-Spline curvesFor the interpolating α-B-spline presented in [11] and

[13], a local tension parameter αj will be assigned to eachvertex PPP j . And we interpolate these tension parametersusing the singular blending function.

αj(t) = (1− S(t))αj + S(t)αj+1

By using the singular blending technique, we construct theinterpolating α-EB-spline as follows.

QQQkj (λ; t) = (1− αj(t))CCCj,k(λ; t) + αj(t)LLLj(t) =

(1− αj(t))CCCj,k(λ; t) + αj(t)[(1− S(t))VVV j + S(t)VVV j+1]

where 0 ≤ αj ≤ 1, S(t) is defined as (2), and

VVV j = PPP j +1− αj

αj(PPP j −CCCj,k(λ; 0))

We easily obtain that

QQQj(0) = PPP j , QQQj(1) = PPP j+1

For open control polygon, we introduce two additionalpoints PPP−1 and PPP n+2, where

982 ACTA AUTOMATICA SINICA Vol. 34

PPP−1 = 2PPP 0 −PPP 1,PPP n+2 = 2PPP n+1 −PPP n

The plantom points are chosen so that the extended B-spline curve interpolates the endpoints PPP 0 and PPP n+1 (SeeFig. 2).

To produce the closed curve for closed polygon, threeadditional points PPP−1, PPP n+2, and PPP n+3 are introduced asfollows (See Fig. 3).

PPP−1 = PPP n,PPP n+2 = PPP 0,PPP n+3 = PPP 1

The interpolating α-EB-spline interpolates the polygonvertices as shown in Figs. 2 and 3. As LLLj(t) and CCCj,k(λ; t)are both C2 continuous, the α-EB-spline is also C2 con-tinuous. We can also change the shape of the curves byadjusting the shape parameters αj and λ (See Figs. 2 and3).

(a) (b)

(c) (d)

Fig. 2 The butterfly-like curves constructed by the blendingfunction of degree 5

(a) (b)

(c) (d)

Fig. 3 The closed curve constructed by the blending functionof degree 4

5 ααα-EB-spline surface

For the blending of extended B-spline curve with SRline segments, we can blend extended B-spline surfacewith network of singularly reparametrized bilinear patches.An SR bilinear patch is determined by four vertices{VVV i,j ,VVV i,j+1,VVV i+1,j ,VVV i+1,j+1}:

LLLi,j(u, v) = (1− S(u))(1− S(v))VVV i,j + (1− S(u))S(v)×VVV i,j+1 + S(u)(1− S(v))VVV i+1,j + S(u)S(v)VVV i+1,j+1

where S(u) and S(v) are defined as (2).Obviously, its first and second order partial derivatives

vanish at the boundaries.

∂uLLLi,j(0, v) = ∂uLLLi,j(1, v) = ∂vLLLi,j(u, 0) = ∂vLLLi,j(u, 1) = 0

∂2uLLLi,j(0, v) = ∂2

uLLLi,j(1, v) = ∂2vLLLi,j(u, 0) = ∂2

vLLLi,j(u, 1) = 0

∂2uvLLLi,j(0, v) = ∂2

uvLLLi,j(1, v) = ∂2uvLLLi,j(u, 0) =

∂2uvLLLi,j(u, 1) = 0

The extended B-spline surface is defined by

CCCki,j(u, v) =

3∑n=0

3∑m=0

bn,k(λ; u)bm,k(λ; v)PPP i+n−1,j+m−1

where bn,k(λ; u) and bm,k(λ; v) are the extended blendingfunctions, and PPP i+n−1,j+m−1 is the control point. The in-terpolating α-EB-spline surface is given by

QQQki,j(u, v) = (1− αij(u, v))CCCk

i,j(u, v) + αij(u, v)LLLij(u, v)

where

αi,j(u, v) = (1− S(u))(1− S(v))αi,j + (1− S(u))S(v)×αi,j+1 + S(u)(1− S(v))αi+1,j + S(u)S(v)αi+1,j+1

VVV i,j in LLLij(u, v) are defined as follows.

VVV i,j = PPP i,j +1− αij

αij(PPP ij −CCCk

i,j(0, 0))

Obviously, QQQki,j(0, 0) = PPP i,j . That is, interpolating α-

EB-spline surface interpolates the vertices of the network.It is also C2 continuous. We can change the shape of thesurface by adjusting the shape parameters λ and αij .

6 Applications

As the α-B-spline[11−14], the interpolating α-EB-splinecan also be used for curve/surface interpolation. It avoidsthe necessity of solving linear systems for interpolation ofB-spline curve/surface (See Fig. 4). As the interpolating

α-B-spline[15], we can also use α-EB-spline for deformingpolygonal shapes into smooth surfaces. The user only needsto input a polygonal shape, then the modeling system willgenerate an original smooth surface interpolating all thepolygonal vertices. The user can modify the smooth sur-face, both globally and locally, by changing the global shapeparameter λ and the local tension parameters αij , respec-tively. Fig. 5 shows the effects of the global shape parame-ter in a interpolating surfaces constructed by the blendingfunctions of degree 6.

No. 8 XU Gang and WANG Guo-Zhao: Extended Cubic Uniform B-spline and α-B-spline 983

(a)

(b)

Fig. 4 Data points interpolation with the same λ but differentαj

(a) (b)

(c)

Fig. 5 The effects of global shape parameter λ ((a) λ = −2;(b) λ = 0; (c) λ = 1)

7 ConclusionIn this paper, we first proposed two kinds of new ex-

tensions of cubic uniform B-spline. The extensions haveglobal shape parameters, so the users can modify the curveglobally by adjusting the shape parameters. Then, we pro-posed the extensions of interpolating α-B-spline by the sin-gular blending technique. We also studied their applica-tion in curve interpolation and deformation. The advan-tage of the extensions is that they have global and localparameters. The modeling examples illustrate that thesenew interpolating splines are valuable for curve/surface de-sign in CAD systems. Convexity-preserving interpolation

and monotone-preserving interpolation are very importantin practice. In the future, we will study the convexity-preserving conditions and the monotone-preserving condi-tions of these interpolation splines.

References

1 Farin G. Curves and Surfaces for CAGD (Fifth Edition).USA: Morgan Kaufmann, 2002

2 Zheng J M, Cai Y Y. Interpolation over arbitrary topologymeshes using a two-phase subdivision scheme. IEEE Trans-actions on Visualization and Computer Graphics, 2006,12(3): 301−310

3 Zheng J M, Wang G Z, Liang Y D. GCn continuity con-ditions for adjacent rational parametric surfaces. ComputerAided Geometric Design, 1995, 12(2): 111−129

4 Wang Guo-Jin, Wang Guo-Zhao, Zheng Jian-Min. ComputerAided Geometric Design. Springer Press, 2001 (in Chinese)

5 Sederberg T W, Zheng J M, Bakenov A, Nasri A. T-splinesand T-NURCCs. In: Proceedings of the International Con-ference on Computer Graphics and Interactive Techniques.California, USA: 2003. 477−484

6 Han Xu-Li, Liu Sheng-Jun. An extension of the cubic uni-form B-spline curves. Journal of Computer Aided Design andComputer Graphics, 2003, 15(5): 576−578 (in Chinese)

7 Wang Wen-Tao, Wang Guo-Zhao. Uniform B-Spline withshape parameter. Chinese Journal of Computer Aided De-sign and Computer Graphics, 2004, 16(6): 783−788 (in Chi-nese)

8 Wang Wen-Tao, Wang Guo-Zhao. Bezier curves with shapeparameter. Journal of Zhejiang University Science A, 2005,6(6): 497−501 (in Chinese)

9 Cai Y Y, Ye X Z, Chui C K, Anderson J H. Constructivealgorithm of vascular network modeling for training of mini-mally invasive catheterization procedure. Advances in Engi-neering Software, 2003, 34(7): 439−450

10 Piegl L A, Ma W Y, Tiller W. An alternative method ofcurve interpolation. The Visual Computer, 2005, 21(1-2):104−117

11 Tai C L, Loe K F. Alpha-spline: a C2 continuous spline withweights and tension control. In: Proceedings of the Confer-ence on Shape Modeling and Applications. AizuWaka-matsu,Japan: IEEE, 1999. 138−143

12 Loe K F. Alpha-B-spline: a linear singular blending B-spline.The Visual Computer, 1996, 12(1): 18−25

13 Tai C L, Barsky B A, Loe K F. An interpolation methodwith weights and relaxation parameters. In: Proceedings ofthe 4th International Conference on Curves and Surfaces.Saint-Malo, France: IEEE, 1999. 393−401

14 Tai C L, Wang G J. Interpolation with slackness and continu-ity control and convexity-preservation using singular blend-ing. Journal of Computational and Applied Mathematics,2004, 172(2): 337−361

15 Tai C L, Loe K F, Barsky B A, Chan Y H. A method for de-forming polygonal shapes into smooth spline surface models.In: Proceedings of the International Conference on Informa-tion Visualization, CAGD Symposium. 1999. 302−308

XU Gang Received his bachelor and Ph.D. degrees fromShandong University and Zhejiang University in 2003 and 2008,respectively. He will join INRIA Sophia-Antipolis as a post-doctoral fellow in October, 2008. His research interest coverscomputer aided geometric design, digital geometry processing,and computer graphics. Corresponding author of this paper.E-mail: xugangzju@yahoo.com.cn

WANG Guo-Zhao Received his bachelor and master de-grees from Zhejiang University in 1968 and 1982, respectively.He is currently a professor in the Department of Mathematicsat Zhejiang University. His research interest covers computergraphics, computer aided geometric design, and medical imageprocessing. E-mail: wanggz@zju.edu.cn