# Cubic B-Spline Curves with Shape Parameter and Their Applications

Authors: Houjun Hang; Xing Yao; Qingqing Li; Michel Artiles
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The present studies on the extension of B-spline mainly focus on Bezier methods and uniform B-spline and are confined to the adjustment role of shape parameters to curves. Researchers pay little attention to nonuniform B-spline. This paper discusses deeply the extension of the quasi-uniform B-spline curves. Firstly, by introducing shape parameters in the basis function, the spline curves are defined in matrix form. Secondly, the application of the shape parameter in shape design is analyzed deeply. With shape parameters, we get another means for adjusting the curves. Meanwhile, some examples are given. Thirdly, we discuss the smooth connection between adjacent B-spline segments in detail and present the adjusting methods. Without moving the control points position, through assigning appropriate value to the shape parameter, C1 continuity of combined spline curves can be realized easily. Results show that the methods are simple and suitable for the engineering application.


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## Body

## 1. Introduction

B-spline methods are very popular in computer-aided geometric design and associated fields because of their distinct advantages. In recent years, some other methods have been presented for representing curves and surfaces. Papers [1–7] presented successively C-curves, T-curves, TC-curves, and β-spline in trigonometric functions space. In order to improve the flexibility of product design, researchers give further consideration to introduce shape parameters. Through the parameters, designers can adjust flexibly the shape of curves and surfaces. Wang et al. [8, 9] introduced successively shape parameters to uniform quadratic TC-B-spline curves and quadratic TC-Bezier curves. Xiong et al. [10] discussed extension of uniform C-B-spline curves and surfaces. Bashir et al. [11] researched the G2 and C2 rational quadratic trigonometric Bezier curve with two shape parameters, and Liu et al. [12] discussed further hyperbolic polynomial uniform B-spline curves and surfaces with shape parameter. In recent years, researchers also paid attention to extension of traditional B-spline methods. But they mainly concentrated on Bezier curves [13], quadratic and cubic uniform B-spline curves [14–19]. Uniform B-splines can represent overall continuity closed curves and surfaces. But they use equally spaced knots; the spline does not interpolate the first and last control points. Because a nonuniform B-spline uses repeated knots technology, the curves have clamped property. The designers can locate more easily the two end points of the curve and achieve smooth connection between adjacent B-spline segments. So it has more practical significance for us to study extension of nonuniform B-spline curves and surfaces.This paper discusses mainly cubic B-spline curves with shape parameter and presents the matrix representation and analyzes the influence of shape parameter on the curve shape. The application of the shape parameter in shape design is discussed deeply. With shape parameters, we get another means for adjusting the curves. In the end, we focus on discussions about how to realize C1 continuity between adjacent B-spline segments by only adjusting the value of the shape parameters without changing the position of the control points. Results show that the methods given by this paper are simple and suitable for the engineering application.


## 2. Definition of Basis Functions and Curves

Given control pointsd0,d1,…,dn, let the knot vector be U=[0,0,0,1,2,…,n-2,n-1,n-1,n-1]. Then we have a piecewise representation of the quadratic open uniform B-spline with parameter (1)pu=∑j=i-2idjNj,2λ,u,u∈i,i+1⊂0,n-1,i=2,3,…,n,where -1≤λ≤1 is called shape parameter and Nj,2λ,u(j=0,1,…,n) are basis functions.(1) Whenn=2, the curve only has one segment. (2)pt=∑j=02djNj,20λ,t,t∈0,1, where N0,20(λ,t), N1,20(λ,t), and N2,20(λ,t) are(3)N0,20λ,t=1+λt1-t2,N1,20λ,t=2-λ+2λt1-tt,N2,20λ,t=1-λ+λtt2.Also it can be expressed in matrix form:(4)pt=1tt2t3Mλd0d1d2,t∈0,1,M=1+λ00-21+λ2-λ01+λ3λ-21-λ0-2λλ.Figure1 shows the influence of the parameter λ on N0,2(λ,u), N1,2(λ,u), and N2,2(λ,u), where solid line, dash line, and dotted line correspond to λ=1, λ-1, and λ=0.Figure 1
The influence of the parameter on the spline shape (n=2).(2) Whenn=3, the spline curve is composed of two segments.(5)p1t=∑j=02djNj,21λ,t,p2t=∑j=13djNj,22λ,t,t∈0,1(6)N0,21λ,t=1+λt1-t2,N1,21λ,t=142+λt2-tt+1-λ1-tt1-t,N2,21λ,t=1421-λ+λtt2,(7)N1,22λ,t=142+λ+λt1-t2,N2,22λ,t=142-λ1-t1-t1+t+1+λtt1-t,N3,22λ,t=λt-λ+1t2.Also it can be expressed in the following matrix form:(8)p1t=1tt2t3M1λd0d1d2,t∈0,1,M1=100λ-22-λ01-2λ52λ-32121-λλ-54λ14λ,p2t=1tt2t3M2λd1d2d3,t∈0,1,M2=12+14λ142-λ0-1-14λ1+14λ012-14λ54λ-321-λ14λ-54λλ.Figure2 shows the influence of λ on the shape of N0,2(λ,u), N1,2(λ,u), N2,2(λ,u), and N3,2(λ,u), where solid line and dash line correspond to λ=0.6 and λ=-0.5.Figure 2
The influence of the parameter on the spline shape (n=3).(3) Whenn>3, the spline curve is composed of n-1 segments.(9)p1t=∑j=02djNj,21λ,t,pn-1t=∑j=n-2ndjNj,2n-1λ,t,pi-1t=∑j=i-2idjNj,23λ,t,i=3,…,n-1,t∈0,1,where N0,21, N1,21, and N2,21 are shown in (6).(10)Nn-2,2n-1λ,t=142+λ+λt1-t2,Nn-1,2n-1λ,t=142-λ1-t1-t1+t+1+λtt1-t,Nn,2n-1λ,t=λt-λ+1t2,Ni-2,23λ,t=142+λ+λt1-t2,Ni-1,23λ,t=142-λ1-t1-t1+t+142+λtt2-t,Ni,23λ,t=14λt-2λ+2t2,i=3,4,…,n-1.Also it can be expressed in the following matrix form:(11)pn-1t=1tt2t3Mn-1λdn-2dn-1dn,Mn-1=12+14λ142-λ0-1-14λ1+14λ012-14λ54λ-321-λ14λ-54λλ,pi-1t=1tt2t3Mi-1λdi-2di-1di,Mi-1=142+λ142-λ0-144+λ144+λ0142-λ14-4+3λ121-λ14λ-12λ14λ.Figure3 shows the influence of the parameter λ on the shape of basis functions N0,2(λ,u), N1,2(λ,u), N2,2(λ,u), N3,2(λ,u), N4,2(λ,u), and N5,2(λ,u), where solid line and dash line correspond to λ=0.5 and λ=-0.7.Figure 3
The influence of the parameter on the spline shape (n=5).


## 3. The Properties of Basis Functions and Spline Curves

(1)(12)Nijλ,u≥0,∀u∈0,1,λ∈-1,1.(2)(13)∑j=02Nj,20λ,u=1,∑j=02Nj,21λ,u=1,∑j=13Nj,22λ,u=1,∑j=n-2nNj,2n-1λ,u=1∑j=i-2iNj,23λ,u=1,i=3,4,…,n-1.(3)(14)N0,2jλ,t1t=0>0t∈0,10t=1,j=0,1,N2,20λ,t0t=0>0t∈0,11t=1,N3,22λ,t0t=0>0t∈0,11t=1,Nn,2n-1λ,t0t=0>0t∈0,11t=1,whereλ=0, that is, quadratic open B-spline.(4)(15)p0=d0,pn-1=dn.(5)(16)dpduu=0=dp1dtt=0=2-λd1-d0,dpduu=n-1=dpn-1dtt=1=2+λdn-dn-1.(6)(17)pi-1′λ,1=1-14λdi-di-1,pi′λ,0=1+14λdi-di-1,i=2,3,…,n-2.That is to say, the curve is G1 at inner knots. Especially when λ=0, the curve is C1.


## 4. Application in Shape Design

Nonuniform B-spline methods have important applications in shape design. By modifying the shape parameters, the designers get additional choice in two-dimensional design.Figures4–6 illustrate the influence of the parameter λ on the shape of curve, where Figure 4 shows α=-0.65, 0, and 0.65 corresponding, respectively, to the thick line, the thin line, and the dash line. Figure 5 shows λ=0.5 corresponding to the solid line and λ=-0.4 the dash line. Figure 6 shows α=-0.4, 0, and 0.4 corresponding, respectively, to the solid line, the dotted line, and the dash line. Figures 7–11 show the application of the spline cure in this paper in fractal modeling.Figure 4
α
=
-
0.65, 0, and 0.65 corresponded, respectively, to the thick line, the thin line, and the dash line.Figure 5
λ
=
0.5 corresponds to the solid line and λ=-0.4 the dash line.Figure 6
α
=
-
0.4, 0, and 0.4 corresponded, respectively, to the solid line, the dotted line, and the dash line.Figure 7
Fractal curve generated by the spline curve in this paper (α=-0.6, 4 iterations).Figure 8
Fractal curve generated by the spline curve in this paper (α=-0.6, 5 iterations).Figure 9
Fractal curve generated by the spline curve in this paper (α=-0.55).Figure 10
Fractal curve generated by the spline curve in this paper(α=-0.8).Figure 11
Fractal curve generated by the spline curve in this paper(α=0).


## 5. Composite Spline Curves

In the practical application, we usually construct composite spline curves that satisfy some smooth conditions. By adjusting shape parameters, designers can achieve the goal.Two spline curves are given: The first one is(18)pu=∑j=i-2ipjNj,2λ1,u,u∈i-2,i-1⊂0,n-1,i=2,3,…,n,λ1∈-1,1.The knot vector U=[0,0,0,1,2,…,n-2,n-1,n-1,n-1].The second one is(19)qu=∑j=i-2iqjNj,2λ2,v,v∈i-2,i-1⊂0,m-1,i=2,3,…,m,λ2∈-1,1.The knot vector V=0,0,0,1,2,…,m-2,m-1,m-1,m-1.From the equations above,pn≡q0.If curvesp(t) and q(t) satisfy G1 continuity that ∃δ>0, Δq0=δpn-1, we can modify the parameter to enable the two curves’ C1 continuity without changing the position of the control points. The adjusting methods are shown below.(1) If[2-3/δ,2-1/δ]∩-1,1≠Φ, we choose λ2∈2-3/δ,2-1/δ∩-1,1 and modify λ1=(2-λ2)δ-2.(2) If[δ-2,3δ-2]∩[-1,1]≠Φ, we choose λ1∈[δ-2,3δ-2]∩[-1,1] and adjust λ2=2-2+λ1/δ.We only prove that(1) and (2) can be gotten in the same way.If[2-3/δ,2-1/δ]∩[-1,1]≠Φ, we can choose freely the value of λ2 as long as λ2∈[2-3/δ,2-1/δ]∩[-1,1]. From 2-3/δ≤λ2≤2-1/δ, we find -1≤(2-λ2)δ-2≤1 and adjust λ1=(2-λ2)δ-2.(20)dqdvv=0=2-λ2Q1-Q0=2-λ2δpn-pn-1,dpduu=n-1=2+λ1pn-pn-1.From λ1=(2-λ2)δ-2, we find δ=2+λ1/2-λ2. So dq/dvv=0=dp/duu=n-1.Two spline curves are given.The first one isp(u)=∑j=05pjNj,2(λ1,u)=∑j=i-2ipjNj,2(λ1,u),u∈[i-2,i-1]⊂[0,4],i=2,3,4,5,λ1∈[-1,1]. The knot vector U=0,0,0,1,2,3,4,4,4.The second one isq(u)=∑j=04qjNj,2(λ2,v)=∑j=i-2iqjNj,2(λ2,v),v∈[i-2,i-1]⊂[0,3],i=2,3,4,λ2∈[-1,1]. The knot vector V=0,0,0,1,2,3,3,3, where λ1=0.6 and λ2=0.4 and the control points are(21)p00.5,1,p11,2,p22,2.25,p33,1.5,p44,2.25,p55,1.25q05,1.25,q15.5,0.75,q26.5,2,q37,1,q48,2.5.Looking at the control points, we see that p5≡q0 and Δq0=δp4, δ=1/2.Because2-3/δ,2-1/δ∩-1,1=-4,0∩-1,1≠Φ, we can choose λ2∈-4,0∩-1,1. Here we let λ2=-0.5. From the above method, we adjust λ1=0.6 to λ1=(2-λ2)δ-2=-0.75. Now, the curves p(t) and q(t) are of C1 continuity, as shown in Figure 12, where the solid line is t, the original curve, and the dash line is the adjusted curve.Figure 12
C1 composite spline curve.On one hand, the adjusting methods are very simple to do, as shown in (1) and (2) in Section5. On the other hand, the adjusting methods let the designers achieve easily smooth connection between adjacent B-spline segments without moving the control points. So it is suitable for the engineering application.


## 6. Conclusion

This paper proposes a class of cubic spline curves with parameter. It is actually the extension of quadratic open B-spline. Through the parameter, we can adjust flexibly the shape of spline curves. With different parameter values, the curve is dragged near or pushed away from the curve. The spline curve is global G1 and keeps clamped property. We can realize C1 smooth connection between the spline curves by only modifying the shape parameter value without changing the control points. The adjusting methods proposed in this paper have very important application value.



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