# Two Extensions of the Quadratic NonuniformB-Spline Curve with Local Shape Parameter Series

Authors: Xiang Kong; Jun Chen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Two extensions of the quadratic nonuniformB-spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the paper. The new extensions not only inherit most excellent properties of the quadratic nonuniform B-spline curve but also can move locally toward or against the fixed control polygon by varying the shape parameter series. They are C1 and C2 continuous separately. Furthermore, the W3D3C1P2 spline curve includes the quadratic nonuniform B-spline curve as a special case. Two applications, the interpolation of the position and the corresponding tangent direction and the interpolation of a line segment, are discussed without solving a system of linear functions. Several numerical examples indicated that the new extensions are valid and can easily be applied.


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## Body

## 1. Introduction

The nonuniform polynomialB-spline curve has been popularly applied in computer-aided geometric design (CAGD) [1]. In particular, the quadratic one, which is the simplest nonuniform B-spline curve, has gained widespread application based on its efficiency and convenience.However, with the specified parameterization method of control point, the position of the classicalB-spline curve is determined by the control polygon. The only way to adjust the shape of the curve is the modification of the control points, which can be seen as a shortcoming for shape adjusting in some cases. For example, in the shape designing, the control polygon as the outline curve is given firstly. Then, the unique B-spline curve, which is smooth and approaches to the control polygon, can be obtained correspondingly. In fact, when the designer specifies the control polygon, he or she may already be satisfied with the rough shape of curves they wanted. Hence, the designer prefers to get more candidate curves characterized by the given control polygon which keeps the original rough shape. In other words, the curve should be modified tinily with the same control polygon.To overcome the rigidity of the nonuniform polynomialB-spline curve, the nonuniform rational B-spline (NURBS) curve is developed by assigning a weight for each control point. Therefore, the NURBS curve can be modified by altering the weights with fixed control polygon. However, in the light of the complexity of fractional expressions, it also suffers from several new drawbacks, such as difficulty in choosing the value of the weights, unstability with the increasing order of rational fraction, and trouble in derivatives and integrals computation. Furthermore, changing the weights to adjust the shape of a NURBS curve is hard to the designer [2].In order to avoid the inconveniences of the NURBS curve, some new spline curves, the expressions of which include the shape parameters, have been extensively studied. Thus, these new spline curves can be altered by the shape parameters effectively without changing the control polygon. In [3], the cubic β-spline curve with two global shape parameters was presented, which is C2 continuous for a uniform knot sequence. The CB-spline curve or HB-spline curve constructed in the space spanned by polynomial and trigonometric/hyperbolic functions is another feasible method [4–8]. The definition domain can be seen as a global shape parameter. In [9], the variable degree polynomial spline curve with two degrees which play the role of design parameters was described. In [10], using a simple modification of the well-known geometric construction of C4 quintic splines, a class of C2∩FC3 spline curves possessing tension properties is described. In [11, 12], the shape parameters were given by blending the B-spline curves and a singularly parameterized sequence of connected line segments. In [13–15], with the help of the blending functions, the shape parameter series were incorporated into the usual spline basis functions, and consequently several kinds of the piecewise quartic polynomial spline curves with shape parameters were investigated under the uniform/nonuniform knot sequence.The main purpose of this paper is to increase the flexibility of the widely used quadratic nonuniformB-spline curve by imitating the way of [13–15]. Two kinds of nonuniform piecewise polynomial spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the present paper. They are considered as the extensions of the quadratic nonuniform B-spline curve, because of the following:(i)
They have three consecutive control points for each curve segment, which means that they have the same structure of the expression as the quadraticB-spline curve(ii)
They inherit the most excellent geometric properties of the quadraticB-spline curve, such as the properties of geometric invariance, affine invariance, local support, convex hull, variation diminishing (VD), as well as C1/C2 continuity(iii)
They include the quadratic nonuniformB-spline curve as a special case for the W3D3C1P2 spline curve(iv)
They can be adjusted locally by varying the values of shape parameters with the fixed control polygon, which exceed the quadratic nonuniformB-spline curveWhen the shape parameter is global, the change of a shape parameter value affects the entire curve leading to a global change, which is not so suitable for CAGD applications. In view of this, the shape parameters in the present paper are local. Varying a shape parameter value only allows modifying the shape of the curve locally in two neighboring curve segment. In addition, these two extensions can be closer to or more far away from the given control polygon than the quadraticB-spline curves with appropriate shape parameters.The rest of this paper is organized as follows. In Section2, two kinds of basis functions with local shape parameter series are described and the properties of them are analyzed in detail. The corresponding spline curves are constructed, and the local control of them is shown in Section 3. Open and closed spline curves are illustrated. Section 4 deals with two interpolations without solving a system of linear functions, the interpolation of the position and the corresponding tangent direction, and the interpolation of a line segment. The problems with the clamped knot vector are discussed in Section 5. Conclusion is given in Section 6. Our results are supported by numerical examples in Section 2, Section 3, Section 4, and Section 5.


## 2. Basis Functions

### 2.1. Definition

Given a nonuniform knot sequenceuii=−∞+∞, let ⋯<u0<u1<⋯<un−1<un<⋯, where we refer to U=…,u0,u1,…,un−1,un,… as a knot vector. The piecewise linear transformation function tu can transform the point in the interval ui,ui+1 to the one in the unit interval 0,1 as follows:(1)tu=⋮tiu,u∈ui,ui+1,ti+1u,u∈ui+1,ui+2,ti+2u,u∈ui+2,ui+3,⋮tiu=u−uiui+1−ui.With the help of the linear transformation functiontu, two kinds of piecewise polynomial basis functions with local shape parameter series are defined as follows.Definition 1.
Given a knot vector U, theW3D3C1P2 basis functions NI,iui=−∞+∞ with two local shape parameter series αii=−∞+∞andβii=−∞+∞ and the W3D4C2P1 basis functions NII,iui=−∞+∞ with one local shape parameter series γii=−∞+∞ are defined to be the following functions:(2)Ne,iu=N0e,itu,u∈ui,ui+1,N1e,itu,u∈ui+1,ui+2,N2e,itu,u∈ui+2,ui+3,0,u∉ui,ui+3,e=I,II,where(3)N0I,itN1I,itN2I,it=00αi1+di−βidiαiαiβi1−αi+11+di+1+βi+1di+11−αi+11−αi+11−βi+100B03tB13tB23tB33t,(4)0<αi<1,max0,αi1+di−1di<βi<minαi1+didi,1,(5)N0II,itN1II,itN2II,it=000m14iγiγim22i1m24i1−γi+11−γi+1m32i000B04tB14tB24tB34tB44t,(6)m14i=12γi+2diγi+di2γi−di21+di,m24i=−12−2+γi+1+2di+1γi+1−di+12+di+12γi+1−2di+11+di+1,m22i=12γi+2diγi+di2γi+di2di1+di,m32i=−12γi+1−2di+1+2di+1γi+1−di+12+di+12γi+1di+11+di+1,di21+di2<γi<1−11+di2,di=ui+1−uiui+2−ui+1,Bint=ni1−tn−iti,i=0,1,…,n.
The implication of the acronym WiDjCkPl is shown in Table 1. More details of the properties mentioned in Table 1 can be seen in Section 2.2.Table 1
The implication of the acronyms WiDjCkPl.
AcronymsW3D3C1P2 basis functionsW3D4C2P1 basis functionsW: width of the support intervals33D: degree of polynomial basis functions34C: continuity of the basis functionsC1C2P: number of local shape parameter series2 series:αi, βi1 series:γi


### 2.2. Properties

If the control polygon is maintained, the properties of the spline curve depend on its basis functions. To analyze the spline curve in Section3, we list the properties of the W3D3C1P2 and the W3D4C2P1 basis functions Ne,iui=−∞+∞e=I,II as follows, all of which can be proved from a direct calculation.Property 1.
(local support). Foru∈ui,ui+1, the basis functions Ne,iui=−∞+∞e=I,II are zero, which means that the width of the support intervals is three.Property 2.
(nonnegativity). Foru∈ui,ui+3, the basis functions Ne,iu≥0e=I,II if inequalities (4) or (6) hold.Property 3.
(normalization). The basis functionsNe,iui=−∞+∞e=I,II satisfy(7)∑i=−∞+∞Ne,iu=1,e=I,II.Property 4.
(linear independence). TheW3D3C1P2 basis functions NI,iui=−∞+∞ are linear independent if inequalities (4) and (8) hold; the W3D4C2P1 basis functions NII,iui=−∞+∞ are linear independent if inequality (6) holds.(8)αi−11−αi−1≠βi−11−βi−1orαi1−αi≠αi1+di−βidi1−αi1+di+βidi.Property 5.
(continuity). TheW3D3C1P2 basis functions NI,iui=−∞+∞ are C1 continuous, and the W3D4C2P1 basis functions NII,iui=−∞+∞ are C2 continuous.Property 6.
(degeneracy). Ifαi=di/1+di,βi=di+2/3/1+di, the W3D3C1P2 basis functions NI,iui=−∞+∞ degenerate into the quadratic nonuniform B-spline basis functions.
We illustrate the basis functionsNe,iui=−∞+∞e=I,II in Figure 1.Figure 1
(a) TheW3D3C1P2 basis functions {NI,i (u)} with different α4; (b) The W3D4C2P1 basis functions {NII,i (u)} with different γ4.
(a)(b)


### 2.3. The Case of Multiple Knots

So far, in Definition (1), we have assumed that each point is simple. We discuss the multiple knots with multiplicity k (k = 2, 3) in this section. The construction of basis functions with multiple knots is analogous to Definition (2). The only difference between them is that we delete the corresponding pieces if the knot is multiple. For example, when ui0=ui0+1 is a double knot (here, we set di0−1=0 to make sense of the corresponding definition), we define the basis functions Ne,i0−1ue=I,II as follows:(9)Ne,i0−1u=N0e,i0−1tu,u∈ui0−1,ui0,N2e,i0−1tu,u∈ui0+1,ui0+2,0,u∉ui0−1,ui0+2,e=I,II,where we delete the piece on ui0,ui0+1.According to the definition above, the support intervals width of the basis functions is reduced from 3 segments to3−k.Property 7.
(continuity at double knots). Suppose thatui0=ui0+1 in the knot sequence, the basis functions Ne,juj=i0−2i0e=I,II are C0 continuous at u=ui0when αi0−1=1,αi0=0e=I or γi0−1=1,γi0=0e=II.Property 8.
(discontinuity at triple knots). Suppose thatui0=ui0+1=ui0+2 in the knot sequence, the basis functions Ne,juj=i0−2i0+1e=I,II are discontinuous at u=ui0.Remark 1.
The basis functions with double knots are still nonnegative although the values of the local shape parameters do not satisfy inequalities in (4) and (6). We take NI,i0−1u in (9), for example: (10)N0I,i0−1tuN2I,i0−1tu=00αi0−11+di0−1−βi0−1di0−1αi0−11−αi01−βi000B03tuB13tuB23tuB33tu.
Consideringαi0−1=1,αi0=0 and di0−1=0, we can see that NI,i0−1u with a local shape parameter βi0 is nonnegative.
We display the basis functionsNe,iui=−∞+∞e=I,II with double knots in Figures 2 and 3.Figure 2
TheW3D3C1P2 basis functions {NI,i (u)} with different β3 (a) double knots and (b) consecutive double knots.
(a)(b)Figure 3
TheW3D4C2P1 basis functions {NII,i (u)} with (a) double knots and (b) consecutive double knots.
(a)(b)


## 2.1. Definition

Given a nonuniform knot sequenceuii=−∞+∞, let ⋯<u0<u1<⋯<un−1<un<⋯, where we refer to U=…,u0,u1,…,un−1,un,… as a knot vector. The piecewise linear transformation function tu can transform the point in the interval ui,ui+1 to the one in the unit interval 0,1 as follows:(1)tu=⋮tiu,u∈ui,ui+1,ti+1u,u∈ui+1,ui+2,ti+2u,u∈ui+2,ui+3,⋮tiu=u−uiui+1−ui.With the help of the linear transformation functiontu, two kinds of piecewise polynomial basis functions with local shape parameter series are defined as follows.Definition 1.
Given a knot vector U, theW3D3C1P2 basis functions NI,iui=−∞+∞ with two local shape parameter series αii=−∞+∞andβii=−∞+∞ and the W3D4C2P1 basis functions NII,iui=−∞+∞ with one local shape parameter series γii=−∞+∞ are defined to be the following functions:(2)Ne,iu=N0e,itu,u∈ui,ui+1,N1e,itu,u∈ui+1,ui+2,N2e,itu,u∈ui+2,ui+3,0,u∉ui,ui+3,e=I,II,where(3)N0I,itN1I,itN2I,it=00αi1+di−βidiαiαiβi1−αi+11+di+1+βi+1di+11−αi+11−αi+11−βi+100B03tB13tB23tB33t,(4)0<αi<1,max0,αi1+di−1di<βi<minαi1+didi,1,(5)N0II,itN1II,itN2II,it=000m14iγiγim22i1m24i1−γi+11−γi+1m32i000B04tB14tB24tB34tB44t,(6)m14i=12γi+2diγi+di2γi−di21+di,m24i=−12−2+γi+1+2di+1γi+1−di+12+di+12γi+1−2di+11+di+1,m22i=12γi+2diγi+di2γi+di2di1+di,m32i=−12γi+1−2di+1+2di+1γi+1−di+12+di+12γi+1di+11+di+1,di21+di2<γi<1−11+di2,di=ui+1−uiui+2−ui+1,Bint=ni1−tn−iti,i=0,1,…,n.
The implication of the acronym WiDjCkPl is shown in Table 1. More details of the properties mentioned in Table 1 can be seen in Section 2.2.Table 1
The implication of the acronyms WiDjCkPl.
AcronymsW3D3C1P2 basis functionsW3D4C2P1 basis functionsW: width of the support intervals33D: degree of polynomial basis functions34C: continuity of the basis functionsC1C2P: number of local shape parameter series2 series:αi, βi1 series:γi


## 2.2. Properties

If the control polygon is maintained, the properties of the spline curve depend on its basis functions. To analyze the spline curve in Section3, we list the properties of the W3D3C1P2 and the W3D4C2P1 basis functions Ne,iui=−∞+∞e=I,II as follows, all of which can be proved from a direct calculation.Property 1.
(local support). Foru∈ui,ui+1, the basis functions Ne,iui=−∞+∞e=I,II are zero, which means that the width of the support intervals is three.Property 2.
(nonnegativity). Foru∈ui,ui+3, the basis functions Ne,iu≥0e=I,II if inequalities (4) or (6) hold.Property 3.
(normalization). The basis functionsNe,iui=−∞+∞e=I,II satisfy(7)∑i=−∞+∞Ne,iu=1,e=I,II.Property 4.
(linear independence). TheW3D3C1P2 basis functions NI,iui=−∞+∞ are linear independent if inequalities (4) and (8) hold; the W3D4C2P1 basis functions NII,iui=−∞+∞ are linear independent if inequality (6) holds.(8)αi−11−αi−1≠βi−11−βi−1orαi1−αi≠αi1+di−βidi1−αi1+di+βidi.Property 5.
(continuity). TheW3D3C1P2 basis functions NI,iui=−∞+∞ are C1 continuous, and the W3D4C2P1 basis functions NII,iui=−∞+∞ are C2 continuous.Property 6.
(degeneracy). Ifαi=di/1+di,βi=di+2/3/1+di, the W3D3C1P2 basis functions NI,iui=−∞+∞ degenerate into the quadratic nonuniform B-spline basis functions.
We illustrate the basis functionsNe,iui=−∞+∞e=I,II in Figure 1.Figure 1
(a) TheW3D3C1P2 basis functions {NI,i (u)} with different α4; (b) The W3D4C2P1 basis functions {NII,i (u)} with different γ4.
(a)(b)


## 2.3. The Case of Multiple Knots

So far, in Definition (1), we have assumed that each point is simple. We discuss the multiple knots with multiplicity k (k = 2, 3) in this section. The construction of basis functions with multiple knots is analogous to Definition (2). The only difference between them is that we delete the corresponding pieces if the knot is multiple. For example, when ui0=ui0+1 is a double knot (here, we set di0−1=0 to make sense of the corresponding definition), we define the basis functions Ne,i0−1ue=I,II as follows:(9)Ne,i0−1u=N0e,i0−1tu,u∈ui0−1,ui0,N2e,i0−1tu,u∈ui0+1,ui0+2,0,u∉ui0−1,ui0+2,e=I,II,where we delete the piece on ui0,ui0+1.According to the definition above, the support intervals width of the basis functions is reduced from 3 segments to3−k.Property 7.
(continuity at double knots). Suppose thatui0=ui0+1 in the knot sequence, the basis functions Ne,juj=i0−2i0e=I,II are C0 continuous at u=ui0when αi0−1=1,αi0=0e=I or γi0−1=1,γi0=0e=II.Property 8.
(discontinuity at triple knots). Suppose thatui0=ui0+1=ui0+2 in the knot sequence, the basis functions Ne,juj=i0−2i0+1e=I,II are discontinuous at u=ui0.Remark 1.
The basis functions with double knots are still nonnegative although the values of the local shape parameters do not satisfy inequalities in (4) and (6). We take NI,i0−1u in (9), for example: (10)N0I,i0−1tuN2I,i0−1tu=00αi0−11+di0−1−βi0−1di0−1αi0−11−αi01−βi000B03tuB13tuB23tuB33tu.
Consideringαi0−1=1,αi0=0 and di0−1=0, we can see that NI,i0−1u with a local shape parameter βi0 is nonnegative.
We display the basis functionsNe,iui=−∞+∞e=I,II with double knots in Figures 2 and 3.Figure 2
TheW3D3C1P2 basis functions {NI,i (u)} with different β3 (a) double knots and (b) consecutive double knots.
(a)(b)Figure 3
TheW3D4C2P1 basis functions {NII,i (u)} with (a) double knots and (b) consecutive double knots.
(a)(b)


## 3. Spline Curve

### 3.1. Definition

Given a set of control pointsPii=1n and a nonuniform knot sequence uii=−∞+∞, two kinds of nonuniform piecewise polynomial spline curves with local shape parameter series are defined as follows.Definition 2.
Given a set of control pointsPii=1n and a nonuniform knot sequence uii=1n+3, the W3D3C1P2 spline curve, the W3D4C2P1 spline curve Peue=I,II, and the corresponding ith 3≤i≤n curve segment is defined as follows:(11)Peu=∑i=1nNe,iuPi,u∈u3,un+1,e=I,II,Pe,iu=N2e,i−2tuPi−2+N1e,i−1tuPi−1+N0e,ituPi,u∈ui,ui+1,e=I,II.
We plot the spline curvePeue=I,II in Figures 4 and 5.Figure 4
TheW3D3C1P2 spline curve (P)I (u) with different shape parameter series: (a) open curves; (b) closed curves.
(a)(b)Figure 5
TheW3D4C2P1 spline curve (P)II (u) with different shape parameter series: (a) open curves; (b) closed curves.
(a)(b)


### 3.2. Properties

The properties of theW3D3C1P2 spline curve PIu and the W3D4C2P1 spline curve PIIu are already shown in this section.Property 9.
(local shape adjustability). When theith shape parameter is varying, such as αi, βi, or γi, only two curve segments Pe,iu and Pe,i+1u in Peue=I,II will be changed.Proof.
Here, we takeαi in PI,iu as example. Based on the definition of PI,iu in (9), the matrix form of the curve segment PI,iu is written as follows:(12)PI,iu=Pi−2,Pi−1,PiN2I,i−2tuN1I,i−1tuN0I,itu,=Pi−2,Pi−1,Pi1−αi−11−βi−100αi−1βi−11−αi1+di+βidi1−αi00αi1+di−βidiαiB03tuB13tuB23tuB33tu.
Hence, the curve segmentPI,iu can be adjusted by varying the value of αi−1 and αi. And, we also can obtain that PI,i+1u can be adjusted by varying the value of αi and αi+1. That is to say, only PI,iu and PI,i+1u will modify when αi changes.
The rest of Property9 can be proved similarly.Figure 6
The curvature distributions of theW3D4C2P1 spline curve with different shape parameter series.
(a)(b)(c)Property 10.
(geometric invariance). The spline curvesPeue=I,II only rely on the control points, whereas they are unrelated to the coordinate system.Property 11.
(affine invariance). Barycentric combinations (7) are invariant under affine maps.Property 12.
(convex hull). The curve segmentPeue=I,II is located in the convex hull of the control points Pjj=i−2i.Property 13.
(continuity). TheW3D3C1P2 spline curve PIu is C1 continuous, and the W3D4C2P1 spline curve PIIu is C2 continuous.Remark 2.
The curvature ofW3D4C2P1 spline curve is continuous based on its C2 continuity. We take the W3D4C2P1 spline curves in Figure 5(b), for example.
It is obvious that different shape parameter series bring different curvature distributions in Figure6. How to get the most stable curvature? That is a challenge in our future work.Property 14.
(degeneracy). Ifαi=di/1+di,βi=di+2/3/1+di, the W3D3C1P2 spline curve PIu degenerates into the quadratic nonuniform B-spline curve.Property 15.
(shape-preserving property). The W3D3C1P2 spline curve PIu is shape-preserving if (13) holds; the W3D4C2P1 spline curve PIIu is shape-preserving if (14) holds.(13)αi<βi,(14)1−di2γi+di2>0.Proof.
According to the matrix form of the curve segment (12), PI,iu is the cubic Bézier curve with control points Qjij=03 as follows:(15)Q0i,Q1i,Q2i,Q3i=Pi−2,Pi−1,Pi1−αi−11−βi−100αi−1βi−11−αi1+di+βidi1−αi00αi1+di−βidiαi.
We takeQ0i, for example:(16)Q0i=1−αi−1Pi−2+αi−1Pi−1.
Thus,Q0i is the linear combination of Pi−2andPi−1 and lies on the line segment Pi−2Pi−1. Q1i,Q2i,andQ3i are similar to Q0i. When inequality (13) holds, the order of these points is Pi−2⟶Q0i⟶Q1i⟶Pi−1⟶Q2i⟶Q3i⟶Pi (Figure 7). Meanwhile, if the control polygon Pi−2Pi−1Pi is monotone or convex, the control polygon Q0iQ1iQ2iQ3i is also monotone or convex. So, the curve segment PI,iu considered as a cubic Bézier curve is monotone or convex, too. The shape-preserving property of the W3D3C1P2 spline curve PIu can be obtained thus. Similarly, the rest of Property 15 about the W3D4C2P1 spline curve PIIu also can be proved.Figure 7
The Bézier representation of the curve segment PI,i(u).Property 16.
(variation diminishing property). TheW3D3C1P2 spline curve PIu is variation diminishing if (13) holds; the W3D4C2P1 spline curve PIIu is variation diminishing if (14) holds.Proof.
It is a direct application of the Bézier representation ofPe,iue=I,II.Remark 3.
The properties of shape-preserving and variation diminishing are very important for the spline curve. Therefore, when we construct theW3D3C1P2 spline curve and the W3D4C2P1 spline curve, we choose the shape parameter series based on not only (4) and (6) but also (13) and (14).


### 3.3. Geometric Effect of the Shape Parameters

From Property9, we can adjust the W3D3C1P2 spline curve segment PI,iu by altering the value of αi−1,αi,βi−1,andβi and adjust the W3D4C2P1 spline curve segment PII,iu by altering the value of γi−1andγi. We present the effect of the shape parameters in the shape modification by the derivatives as follows:(17)∂∂αiPI,iu=1+diB23tu+B33tuPi−Pi−1,∂∂αi−1PI,iu=B03tuPi−1−Pi−2,(18)∂∂βiPI,iu=diB23tuPi−1−Pi,∂∂βi−1PI,iu=B13tuPi−1−Pi−2,(19)∂∂γiPII,iu=121+diB34tu+B44tuPi−Pi−1,∂∂γi−1PII,iu=B04tu+121+di−1di−1B14tuPi−1−Pi−2.These 6 derivatives are similar; hence, we take (17) as example to explain the geometric effect of the shape parameters αi with respect to the curve segment PI,iu. When we increase αi only while fixing αi−1,βi−1,βi, the curve segment PI,iu will be away from Pi−1 and closed to Pi. Figure 8 shows the geometric effect of αi,βi,andγi from (17)–(19).Figure 8
The curve segmentsPe,iue=I,IIon the interval ui,ui+1 with different shape parameters: (a) (P)I,i (u) is away from the (P)i-1 with the increase in αi; (b) (P)I,i (u) is closed to the (P)i-1with the increase in βi; (c) (P)II,i (u) is away from the (P)i-1 with the increase in γi.
(a)(b)(c)Based on the shape parameters, the two extensions are more flexible than the quadraticB-spline curve. We show that the W3D3C1P2 spline curve and the W3D4C2P1 spline curve can be closer to or more far away from the given control polygon than the quadratic B-spline curves with appropriate shape parameters in Figure 9.Figure 9
The two extensions can be closer to or more far away from the control polygon than theB-spline curves: (a) the W3D3C1P2 spline curve segment PI,i (u); (b) the W3D4C2P1 spline curve segment PII,i (u).
(a)(b)


## 3.1. Definition

Given a set of control pointsPii=1n and a nonuniform knot sequence uii=−∞+∞, two kinds of nonuniform piecewise polynomial spline curves with local shape parameter series are defined as follows.Definition 2.
Given a set of control pointsPii=1n and a nonuniform knot sequence uii=1n+3, the W3D3C1P2 spline curve, the W3D4C2P1 spline curve Peue=I,II, and the corresponding ith 3≤i≤n curve segment is defined as follows:(11)Peu=∑i=1nNe,iuPi,u∈u3,un+1,e=I,II,Pe,iu=N2e,i−2tuPi−2+N1e,i−1tuPi−1+N0e,ituPi,u∈ui,ui+1,e=I,II.
We plot the spline curvePeue=I,II in Figures 4 and 5.Figure 4
TheW3D3C1P2 spline curve (P)I (u) with different shape parameter series: (a) open curves; (b) closed curves.
(a)(b)Figure 5
TheW3D4C2P1 spline curve (P)II (u) with different shape parameter series: (a) open curves; (b) closed curves.
(a)(b)


## 3.2. Properties

The properties of theW3D3C1P2 spline curve PIu and the W3D4C2P1 spline curve PIIu are already shown in this section.Property 9.
(local shape adjustability). When theith shape parameter is varying, such as αi, βi, or γi, only two curve segments Pe,iu and Pe,i+1u in Peue=I,II will be changed.Proof.
Here, we takeαi in PI,iu as example. Based on the definition of PI,iu in (9), the matrix form of the curve segment PI,iu is written as follows:(12)PI,iu=Pi−2,Pi−1,PiN2I,i−2tuN1I,i−1tuN0I,itu,=Pi−2,Pi−1,Pi1−αi−11−βi−100αi−1βi−11−αi1+di+βidi1−αi00αi1+di−βidiαiB03tuB13tuB23tuB33tu.
Hence, the curve segmentPI,iu can be adjusted by varying the value of αi−1 and αi. And, we also can obtain that PI,i+1u can be adjusted by varying the value of αi and αi+1. That is to say, only PI,iu and PI,i+1u will modify when αi changes.
The rest of Property9 can be proved similarly.Figure 6
The curvature distributions of theW3D4C2P1 spline curve with different shape parameter series.
(a)(b)(c)Property 10.
(geometric invariance). The spline curvesPeue=I,II only rely on the control points, whereas they are unrelated to the coordinate system.Property 11.
(affine invariance). Barycentric combinations (7) are invariant under affine maps.Property 12.
(convex hull). The curve segmentPeue=I,II is located in the convex hull of the control points Pjj=i−2i.Property 13.
(continuity). TheW3D3C1P2 spline curve PIu is C1 continuous, and the W3D4C2P1 spline curve PIIu is C2 continuous.Remark 2.
The curvature ofW3D4C2P1 spline curve is continuous based on its C2 continuity. We take the W3D4C2P1 spline curves in Figure 5(b), for example.
It is obvious that different shape parameter series bring different curvature distributions in Figure6. How to get the most stable curvature? That is a challenge in our future work.Property 14.
(degeneracy). Ifαi=di/1+di,βi=di+2/3/1+di, the W3D3C1P2 spline curve PIu degenerates into the quadratic nonuniform B-spline curve.Property 15.
(shape-preserving property). The W3D3C1P2 spline curve PIu is shape-preserving if (13) holds; the W3D4C2P1 spline curve PIIu is shape-preserving if (14) holds.(13)αi<βi,(14)1−di2γi+di2>0.Proof.
According to the matrix form of the curve segment (12), PI,iu is the cubic Bézier curve with control points Qjij=03 as follows:(15)Q0i,Q1i,Q2i,Q3i=Pi−2,Pi−1,Pi1−αi−11−βi−100αi−1βi−11−αi1+di+βidi1−αi00αi1+di−βidiαi.
We takeQ0i, for example:(16)Q0i=1−αi−1Pi−2+αi−1Pi−1.
Thus,Q0i is the linear combination of Pi−2andPi−1 and lies on the line segment Pi−2Pi−1. Q1i,Q2i,andQ3i are similar to Q0i. When inequality (13) holds, the order of these points is Pi−2⟶Q0i⟶Q1i⟶Pi−1⟶Q2i⟶Q3i⟶Pi (Figure 7). Meanwhile, if the control polygon Pi−2Pi−1Pi is monotone or convex, the control polygon Q0iQ1iQ2iQ3i is also monotone or convex. So, the curve segment PI,iu considered as a cubic Bézier curve is monotone or convex, too. The shape-preserving property of the W3D3C1P2 spline curve PIu can be obtained thus. Similarly, the rest of Property 15 about the W3D4C2P1 spline curve PIIu also can be proved.Figure 7
The Bézier representation of the curve segment PI,i(u).Property 16.
(variation diminishing property). TheW3D3C1P2 spline curve PIu is variation diminishing if (13) holds; the W3D4C2P1 spline curve PIIu is variation diminishing if (14) holds.Proof.
It is a direct application of the Bézier representation ofPe,iue=I,II.Remark 3.
The properties of shape-preserving and variation diminishing are very important for the spline curve. Therefore, when we construct theW3D3C1P2 spline curve and the W3D4C2P1 spline curve, we choose the shape parameter series based on not only (4) and (6) but also (13) and (14).


## 3.3. Geometric Effect of the Shape Parameters

From Property9, we can adjust the W3D3C1P2 spline curve segment PI,iu by altering the value of αi−1,αi,βi−1,andβi and adjust the W3D4C2P1 spline curve segment PII,iu by altering the value of γi−1andγi. We present the effect of the shape parameters in the shape modification by the derivatives as follows:(17)∂∂αiPI,iu=1+diB23tu+B33tuPi−Pi−1,∂∂αi−1PI,iu=B03tuPi−1−Pi−2,(18)∂∂βiPI,iu=diB23tuPi−1−Pi,∂∂βi−1PI,iu=B13tuPi−1−Pi−2,(19)∂∂γiPII,iu=121+diB34tu+B44tuPi−Pi−1,∂∂γi−1PII,iu=B04tu+121+di−1di−1B14tuPi−1−Pi−2.These 6 derivatives are similar; hence, we take (17) as example to explain the geometric effect of the shape parameters αi with respect to the curve segment PI,iu. When we increase αi only while fixing αi−1,βi−1,βi, the curve segment PI,iu will be away from Pi−1 and closed to Pi. Figure 8 shows the geometric effect of αi,βi,andγi from (17)–(19).Figure 8
The curve segmentsPe,iue=I,IIon the interval ui,ui+1 with different shape parameters: (a) (P)I,i (u) is away from the (P)i-1 with the increase in αi; (b) (P)I,i (u) is closed to the (P)i-1with the increase in βi; (c) (P)II,i (u) is away from the (P)i-1 with the increase in γi.
(a)(b)(c)Based on the shape parameters, the two extensions are more flexible than the quadraticB-spline curve. We show that the W3D3C1P2 spline curve and the W3D4C2P1 spline curve can be closer to or more far away from the given control polygon than the quadratic B-spline curves with appropriate shape parameters in Figure 9.Figure 9
The two extensions can be closer to or more far away from the control polygon than theB-spline curves: (a) the W3D3C1P2 spline curve segment PI,i (u); (b) the W3D4C2P1 spline curve segment PII,i (u).
(a)(b)


## 4. Interpolation

### 4.1. The Geometric Significance of the Shape Parameter

Before we apply the two extensions of the quadratic nonuniformB-spline curve to interpolation, we present the position and the corresponding tangent direction of them at u=ui+1. It follows from a routine computation that(20)PIui+1=1−αiPi−1+αiPi,PI′ui+1=3hi+1βi−αiPi−Pi−1,(21)PIIui+1=1−γiPi−1+γiPi,PII′ui+1=2hi+1γi+di21−γidi1+diPi−Pi−1.Consequently, the pointPeui+1e=I,II lies on the line segment Pi−1Pi, and the tangent direction at Peui+1e=I,II is parallel to Pi−1Pi from (20) and (21). Furthermore, we can obtain the geometric significance of the shape parameter as follows:(22)αi=Pi−1PIui+1Pi−1Pi,γi=Pi−1PIIui+1Pi−1Pi.Figures8(a), 8(c), and Figure 9 show it intuitively.Based on the geometric significance of the shape parameters (20) and (21), two interpolations are presented as follows.


### 4.2. Interpolation of the Position and Tangent Direction

Given a set of pointsQii=1n and the unit direct vectors Tii=1n on them as the tangent directions, we can construct a W3D3C1P2 spline curve PIuu∈u3,un+1 from “IPT (Interpolation of the Position and Tangent direction) algorithm,” without solving a system of linear functions such that(23)PIui+2=Qi,PI′ui+2Ti.


#### 4.2.1. IPT Algorithm

Step 1. We construct the control polygon P1P2,…,PnPn+1 first, such that Qi lies on the line segment PiPi+1 and Ti‖PiPi+1.Step 2. We set the local shape parameter series αii=2n+1 from (22) as(24)αi=Pi−1Qi−1Pi−1Pi,i=2,3,…,n+1.Step 3. We select the arbitrary local shape parameter series βii=2n+1 from the scopes in (4) and (13) based on αii=2n+1 above.Remark 3.
We can set the first control pointP1 and the last control point Pn+1 as follows:(25)P1Q1=Q1P2,PnQn=QnPn+1.
The corresponding shape parametersα2=αn+1=0.5.
Comparing (24) with (22), we can see that the W3D3C1P2 spline curve PIu from the “IPT algorithm” interpolates both the position Qii=1n and the corresponding tangent direction Tii=1n with local shape parameter series βii=2n+1 (Figure 10).
Unfortunately, theW3D3C1P2 spline curve PIIu is not suitable to interpolate the position and tangent direction. When we set the local shape parameter series γii=−∞+∞ based on (22) like (24), the value of shape parameter γi may be against the scope in (6), which keeps the nonnegativity of the basis function NII,iui=−∞+∞.Figure 10
TheW3D3C1P2 spline curve PI (u) interpolating the position Qii=14 and tangent directionTii=14 with different shape parameter series βii=25


### 4.3. Interpolation of a Line Segment

Given a line segmentAB, we can construct a W3D3C1P2 spline curve PIu as follows, such that the curve segment PI,iuu∈ui,ui+1 is the line segment AB.We setPi−2,A,Pi−1,B,andPi as collinear. A is between Pi−2 and Pi−1, and B is between Pi−1 and Pi (Figure 11). Then, we set the corresponding shape parameter from (18) as follows:(26)αi−1=Pi−2APi−2Pi−1,αi=Pi−1BPi−1Pi.Figure 11
Interpolation of a line segment: (a) theW3D3C1P2 spline curve PI (u); (b) the W3D4C2P1 spline curve PII (u).
(a)(b)Hence, (26) gives(27)PI,iui=A,PI,iui+1=B.In view of the convex hull property, the curve segmentPI,iu is in the convex hull of Pjj=i−2i, which is a line segment itself. The local shape parameter series βii=2n+1 can be selected freely from the scopes in (4) and (13). Hence, the curve segment PI,iu is the line segment AB. Note that the whole spline curve PIu is still C1 continuous.Similarly, we can get a W3D4C2P1 spline curve PIIu in which the curve segment PII,iuu∈ui,ui+1 is the line segment AB, too. Note that the whole spline curve PIIu is still C2 continuous. From (22), the corresponding shape parameters are set as follows:(28)γi−1=Pi−2APi−2Pi−1,γi=Pi−1BPi−1Pi.Compared withPIu, the corresponding shape parameters in PIIu are fixed.


## 4.1. The Geometric Significance of the Shape Parameter

Before we apply the two extensions of the quadratic nonuniformB-spline curve to interpolation, we present the position and the corresponding tangent direction of them at u=ui+1. It follows from a routine computation that(20)PIui+1=1−αiPi−1+αiPi,PI′ui+1=3hi+1βi−αiPi−Pi−1,(21)PIIui+1=1−γiPi−1+γiPi,PII′ui+1=2hi+1γi+di21−γidi1+diPi−Pi−1.Consequently, the pointPeui+1e=I,II lies on the line segment Pi−1Pi, and the tangent direction at Peui+1e=I,II is parallel to Pi−1Pi from (20) and (21). Furthermore, we can obtain the geometric significance of the shape parameter as follows:(22)αi=Pi−1PIui+1Pi−1Pi,γi=Pi−1PIIui+1Pi−1Pi.Figures8(a), 8(c), and Figure 9 show it intuitively.Based on the geometric significance of the shape parameters (20) and (21), two interpolations are presented as follows.


## 4.2. Interpolation of the Position and Tangent Direction

Given a set of pointsQii=1n and the unit direct vectors Tii=1n on them as the tangent directions, we can construct a W3D3C1P2 spline curve PIuu∈u3,un+1 from “IPT (Interpolation of the Position and Tangent direction) algorithm,” without solving a system of linear functions such that(23)PIui+2=Qi,PI′ui+2Ti.


### 4.2.1. IPT Algorithm

Step 1. We construct the control polygon P1P2,…,PnPn+1 first, such that Qi lies on the line segment PiPi+1 and Ti‖PiPi+1.Step 2. We set the local shape parameter series αii=2n+1 from (22) as(24)αi=Pi−1Qi−1Pi−1Pi,i=2,3,…,n+1.Step 3. We select the arbitrary local shape parameter series βii=2n+1 from the scopes in (4) and (13) based on αii=2n+1 above.Remark 3.
We can set the first control pointP1 and the last control point Pn+1 as follows:(25)P1Q1=Q1P2,PnQn=QnPn+1.
The corresponding shape parametersα2=αn+1=0.5.
Comparing (24) with (22), we can see that the W3D3C1P2 spline curve PIu from the “IPT algorithm” interpolates both the position Qii=1n and the corresponding tangent direction Tii=1n with local shape parameter series βii=2n+1 (Figure 10).
Unfortunately, theW3D3C1P2 spline curve PIIu is not suitable to interpolate the position and tangent direction. When we set the local shape parameter series γii=−∞+∞ based on (22) like (24), the value of shape parameter γi may be against the scope in (6), which keeps the nonnegativity of the basis function NII,iui=−∞+∞.Figure 10
TheW3D3C1P2 spline curve PI (u) interpolating the position Qii=14 and tangent directionTii=14 with different shape parameter series βii=25


## 4.2.1. IPT Algorithm

Step 1. We construct the control polygon P1P2,…,PnPn+1 first, such that Qi lies on the line segment PiPi+1 and Ti‖PiPi+1.Step 2. We set the local shape parameter series αii=2n+1 from (22) as(24)αi=Pi−1Qi−1Pi−1Pi,i=2,3,…,n+1.Step 3. We select the arbitrary local shape parameter series βii=2n+1 from the scopes in (4) and (13) based on αii=2n+1 above.Remark 3.
We can set the first control pointP1 and the last control point Pn+1 as follows:(25)P1Q1=Q1P2,PnQn=QnPn+1.
The corresponding shape parametersα2=αn+1=0.5.
Comparing (24) with (22), we can see that the W3D3C1P2 spline curve PIu from the “IPT algorithm” interpolates both the position Qii=1n and the corresponding tangent direction Tii=1n with local shape parameter series βii=2n+1 (Figure 10).
Unfortunately, theW3D3C1P2 spline curve PIIu is not suitable to interpolate the position and tangent direction. When we set the local shape parameter series γii=−∞+∞ based on (22) like (24), the value of shape parameter γi may be against the scope in (6), which keeps the nonnegativity of the basis function NII,iui=−∞+∞.Figure 10
TheW3D3C1P2 spline curve PI (u) interpolating the position Qii=14 and tangent directionTii=14 with different shape parameter series βii=25


## 4.3. Interpolation of a Line Segment

Given a line segmentAB, we can construct a W3D3C1P2 spline curve PIu as follows, such that the curve segment PI,iuu∈ui,ui+1 is the line segment AB.We setPi−2,A,Pi−1,B,andPi as collinear. A is between Pi−2 and Pi−1, and B is between Pi−1 and Pi (Figure 11). Then, we set the corresponding shape parameter from (18) as follows:(26)αi−1=Pi−2APi−2Pi−1,αi=Pi−1BPi−1Pi.Figure 11
Interpolation of a line segment: (a) theW3D3C1P2 spline curve PI (u); (b) the W3D4C2P1 spline curve PII (u).
(a)(b)Hence, (26) gives(27)PI,iui=A,PI,iui+1=B.In view of the convex hull property, the curve segmentPI,iu is in the convex hull of Pjj=i−2i, which is a line segment itself. The local shape parameter series βii=2n+1 can be selected freely from the scopes in (4) and (13). Hence, the curve segment PI,iu is the line segment AB. Note that the whole spline curve PIu is still C1 continuous.Similarly, we can get a W3D4C2P1 spline curve PIIu in which the curve segment PII,iuu∈ui,ui+1 is the line segment AB, too. Note that the whole spline curve PIIu is still C2 continuous. From (22), the corresponding shape parameters are set as follows:(28)γi−1=Pi−2APi−2Pi−1,γi=Pi−1BPi−1Pi.Compared withPIu, the corresponding shape parameters in PIIu are fixed.


## 5. Problems with Clamped Knot Vector

In the B-spline model, clamped knot vectorU=u3,u3,u3,u4,⋯,un,un+1,un+1,un+1 can be used to interpolate the first control point P1 and the last control point Pn. The corresponding tangent direction at P1 or Pn is parallel to P1P2 or Pn−1Pn, respectively. Clamped knot vector is also suitable for the new basis functions constructed in Section 2.3.With a clamped knot vectorU, we can set α1=αn+1=1andα2=αn+2=0 for the W3D3C1P2 spline curve PIu as Property 7, and we have(29)PIu3=P1,PIun=Pn,PI′u3+P1P2,PI′un−=0.In Figure12(a), with a clamped knot vector, we can see that PIu interpolates the first control point P1 and the last control point Pn, and the tangential directs at P1 is parallel to P1P2. However, the tangent direction at Pn is not parallel to Pn−1Pn based on (29). In order to make the tangent direction at Pn is parallel to Pn−1Pn, we can construct PIu as follows:Step 1. Unclamp the right end of the knot vector U as U˜=u3,u3,u3,u4,…,un,un+1,un+2,un+3.Step 2. Choose the auxiliary end control point P˜n on the extension line of Pn−1Pn, which takes the place of Pn.Step 3. Set the shape parameter αn+1=Pn−1Pn/Pn−1P˜n from (18). Hence, we have PIun=Pn and PI′un−‖Pn−1Pn from (20) (Figure 12(b)).Figure 12
The improvement of theW3D3C1P2 spline curve (P)I (u): (a) knot vector (0, 0, 0, 1, 1, 1); (b) knot vector (0, 0, 0, 1, 2, 3) and control polygon P1P2P˜3
(a)(b)PIIu with a clamped knot vector also interpolates the first and the last control point, and the tangential direction at Pn is parallel to Pn−1Pn (Figure 13(a)). However, the tangent direction at P1 is not parallel to P1P2. The corresponding construction of PIIu is similar to the case of PIu (Figure 13(b)).Figure 13
The improvement of theW3D4C2P1 spline curve (P)II (u): (a) knot vector (0, 0, 0, 1, 1, 1); (b) knot vector (−2, −1, 0, 1, 2, 3) and control polygon P˜1P2P3.
(a)(b)


## 6. Conclusion

Two extensions of the quadratic nonuniformB-spline curve, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are constructed in the paper. Both of the extensions have the same structure as the quadratic nonuniform B-spline curve. The properties of the extensions are similar with the ones of the quadratic nonuniform B-spline curves (Table 2). Particularly, the quadratic nonuniform B-spline curve is the special case of the W3D3C1P2 spline curve when taking specified shape parameter series. The shapes of the extensions can be adjusted by the shape parameter series locally even the control points fixed. The effects of varying the shape parameters on the extensions are illustrated. With the shape parameters, the extensions can move locally toward or against the control polygon. Based on the properties of the extensions, two applications, the interpolation of the position and tangent direction and interpolation of a line segment, are presented without solving a system of linear functions.Table 2
Comparison between the quadratic nonuniformB-spline curve and the extensions.
PropertiesQuadratic nonuniformB-splineW3D3C1P2 spline curve PI (u)W3D4C2P1 spline curve PII (u)Geometric invariance√√√Affine invariance√√√Convex hull property√√√Shape-preserving property√√√Variation diminishing property√√√Width of the support intervals333Degree of polynomial basis functions234Continuity of the basis functionsC1C1C2Number of local shape parameter seriesNone2 series:αi, βi1 series:γiHowever, the extensions can not deal with the clamped knot vector very well. The tangent directions at the ends of control points are not parallel to the ends of control polygon. Although we construct the spline curve with the auxiliary points to avoid the defect, we will improve the basis functions in the case of multiple knots in our future research.



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