Construction and analysis of a new class of shape-preserving piecewise cubic polynomial curves

YAN Lanlan, FAN Jiqiu(College of Science, East China University of Technology, Nanchang 330013, China)

Construction and analysis of a new class of shape-preserving piecewise cubic polynomial curves

YAN Lanlan, FAN Jiqiu
(College of Science, East China University of Technology, Nanchang 330013, China)

This paper proposes a new class of shape-preserving piecewise cubic polynomial curves with both local and global shape control parameters. By presetting the properties of its basis functions and then solving equations, a set of polynomial basis functions with two shape parameters are derived, including the cubic uniform B-spline basis functions as a special case. Based on the relationship between the new basis functions and the cubic Bernstein basis functions, the totally positive property of the new basis functions is proved and a new class of piecewise cubic polynomial curves is therefore defined. The effect of the relative position of the control polygons’ side vectors onto the shape characteristic of the corresponding curve segments is analyzed. Necessary and sufficient conditions are obtained for the curve segments containing single or double inflection points, a loop or a cusp, or be locally or globally convex, which provide a theoretical guide for adjusting the shape of curve segments.

curve design; B-spline method; totally positive basis; shape parameter;shape analysis

保形分段三次多項(xiàng)式曲線的形狀分析.浙江大學(xué)學(xué)報(bào)(理學(xué)版)，2018,45(1)： 044-053

Constructing practical basis functions to generate free-form curves and surfaces is an important topic in computer aided geometric design (CAGD).As a unified mathematical model with many desirable properties, B-splines, particularly the cubic B-splines, have gained widespread application in CAGD[1].Piecewise cubic B-spline curves with four consecutive control points for each curve segment is flexible and can be used conveniently.However, the positions of the cubic B-spline curves are fixed relatively to their control polygon.Although the weights in the cubic non-uniform rational B-spline curves possess an effect on adjusting the shape of the curves, how to change the weights to adjust the shape of a curve is sometimes quite opaque to the user.

To enhance the flexibility of B-spline models, some researchers have suggested many types of curves with shape parameters incorporated into the basis functions.For instance, XU et al[2]proposed three kinds of extensions of cubic uniform B-spline. The advantage of the extensions is that they have shape parameters, which can be used to adjust the shape of the curves without shifting the control points.COSTANTINI et al[3]presented a method for the construction of cubic like B-splines with multiple knots.The proposed B-splines are equipped with tension parameters, associated to the knots, which permit a modification of their shape.HAN[4]constructed piecewise quartic polynomial curves with a local shape parameter.HAN[5]defined piecewise quartic spline curves with three local shape parameters.HU et al[6]presented B-spline curves with two local shape parameters.ZHU et al[7]defined B-spline-like curves with two local shape parameters.

Totally positive property is one of the most important properties of basis functions.Although the schemes given in[2, 4-6] improved the control of the shape of B-spline curves, whether the basis functions have total positivity is unknown, so whether the curves have variation diminishing is unknown.Although the curves in[3,7] have variation diminishing, the basis functions are not cubic polynomials.For many applications in geometric modeling, it is often necessary to detect singularities and inflection points on curves, and convexity is important as an intuitive geometric concept as well.There are many publications[8-13]on this topic from different points of view.

This paper is aimed at constructing a cubic polynomial basis functions with total positivity.The associated curves have local control, adjustable shape, and have variation diminishing thus have a good shape control.Considering that a curve with variation diminishing is suitable for conformal design, we analyze the shape feature of the new curves.We give conditions on the existence of cusp, loop and inflection point.The results are summarized in a shape diagram like the one in[8, 10-11].Furthermore, the influence of the shape parameters on the shape diagram and their ability for adjusting the shape of the curves is discussed.

The rest of the paper is organized as follows.Section 1 gives the basis functions and their properties.Section 2 defines the curves and gives their properties.Section 3 analyzes the shape feature of the curve segments.Section 4 defines the surfaces.Section 5 concludes the paper.

1 Basis functions and their properties

(1)

Theαβbasis can be rewritten in Bernstein basis functions form, i.e.,

(b0(t),b1(t),b2(t),b3(t))=

(2)

where |Jij,kl| denotes the minor formed by thei,jrows andk,lcolumns ofJ.The matrixJhas 16 third-order minors as follows：

where |Jijk,lmn| denotes the minor formed by thei,j,krows andl,m,ncolumns ofJ.Besides,

(3)

Theorem1Theαβbasis has the following properties：

(a) Degeneracy： Whenα=-1,β=0, theαβbasis is the cubic uniform B-spline basis.

(c) Symmetry：bi(1-t)=b3-i(t)(i=0,1,2,3).

(d) Property at the endpoints： For arbitraryαandβ, we have

and forα=-1,β=0, we have

(f) Non-negativity：bi(t)≥0(i=0,1,2,3).

(g) Total positivity： Theαβbasis forms a normalized totally positive basis of the space Ω={1,t,t2,t3}.

ProofWe shall prove (e)and(g).The remaining can be easily obtained by formula (1) or (2).

(g) The cubic Bernstein basis is the normalized B-basis of the space Ω. Thus, by formula (2), lemma 1, and the properties (b)and (e), we know theαβbasis is a normalized totally positive basis.

2 The αβ curves and their properties

Definition2Given control pointsPi(i=0,1,…,n)∈R2orR3and knotsu1

i=1,2,…,n-2,

i=1,2,…,n-2.

Theorem2From the properties of theαβbasis, we can obtain the following properties of theαβcurves.

(a) Affine invariance.

(b) Symmetry： When taking the sameαβbasis, the two polygons,Pi-1,Pi,Pi+1,Pi+2andPi+2,Pi+1,Pi,Pi-1,describe the sameαβcurve segment; The only thing that changes is the direction of traversal of the parameter.

(c) Local control： Changing one control point of anαβcurve, four segments will change mostly.

(d) Endpoint property： For arbitraryαandβi, we have

and forα=-1,βi=0, we have

RemarkBy the endpoint property, we can see that the position of the starting and ending points of the curve segment only related to the parameterα. The valueβi-αdecides to the length of the tangent vector at the starting and ending points, thus decides to the degree of blending of theith curve segment.

(e) Continuity： In general, theith and (i+1)thαβcurve segments areG1continuous at the junction, and they areG2continuous whenα=-1,βi=βi+1=0.

(f) Shape adjustable： The parametersαandβican be used to adjust the shape of theαβcurve without changing the control points.Theαis a global parameter, whileβiis a local parameter.The change ofβiwill only change the shape of theith segment.

(h) Variation diminishing and convexity-preserving.

Fig.1 The open αβ curves

Fig.2 The closed αβ curves

As can be seen in figs.1 and 2, the shape ofαβcurves can reflect the shape of the control polygon well.

3 Shape analysis of the curve segment

We consider anαβcurve segment

ai=Pi-Pi-1,i=1,2,3,

thenp(t)can be rewritten as

p(t)=P0+[1-b0(t)]a1+[b2(t)+
b3(t)]a2+b3(t)a3.

(4)

We first consider the case ofa1not parallel toa3. Sincea1anda3are linearly independent,a2can be represented by the linear combination ofa1anda3.Without loss of generality, leta2=ua1+va3, and substitute it into formula (4), and then we have

p(t)=P0+{1-b0(t)+u[b2(t)+b3(t)]}a1+
{b3(t)+v[b2(t)+b3(t)]}a3.

(5)

3.1 The case of cusp

The necessary condition that the curvep(t)has a cusp isp′(t)=0(0

Letp′(t)=0, fora1anda3are linearly independent, we obtain

and then the following parametric curveC?C(t)can be obtained.

(6)

To facilitate the analysis of the geometric properties ofC, we rewrite it in the form of quadratic rational Bézier curve, i.e,

(7)

(8)

Let (u0,v0)∈C, and 0α, sop″(t0)≠0. Then from

p′(t)=p″(t0)(t-t0)+O(t-t0),

we can see that the direction ofp′(t)is contravariant when it passes throught0.It shows thatCis the cusp curve, that is, the curvep(t)has cusp if and only if (u,v)∈C.

3.2 The case of loop

The sufficient and necessary condition that the curvep(t)has a loop is that there exists 0≤t1

{b0(t2)-b0(t1)+u[b2(t1)+b3(t1)-b2(t2)-

b3(t2)]}a1+{b3(t1)-b3(t2)+v[b2(t1)+

b3(t1)-b2(t2)-b3(t2)]}a3=0.

Fora1anda3are linearly independent, we obtain

(9)

where (t1,t2)∈Δ={(t1,t2)∈R2|0≤t1

The parametric equations ofL1andL2are as follows：

(10)

(11)

The curvesL1andL2can be rewritten in the form of quadratic rational Bézier curves

(12)

(13)

(14)

wheret∈(0,1). Eliminate the parametert, we obtain

3.3 The case of inflection point

The binormal vector ofp(t)isγ(t)=p′(t)∧p″(t), wherep′(t)∧p″(t)is the wedge product ofp′(t)andp″(t). If the direction ofγ(t)is changed when it passes throught0, thenp(t0)(0

The binormal vectorγ(t)can be represented asγ(t)=f(t;u,v)(a1∧a3), where

Sincea1×a3≠0,the direction ofγ(t)changes if and only if the sign off(t;u,v)changes.Hence, we only need to consider the sign change off(t;u,v).

S= {(u,v)|uv<0}∪{(u,0)|-1

{(0,v)|-1

Dis the open region surrounded by the curveCand the coordinate axes.

There is at least one straight linef(t0;u,v)=0 passing through every point (u0,v0)∈S∪D∪Ctangent to the curveC, heret0is the parameter corresponding to (u0,v0).

From

we can see that the sign off(t;u0,v0)does not change when it passes throught0. It meansp(t0)is not an inflection point, and there is no inflection point on the curvep(t).

we can see that the sign off(t;u0,v0)changes when it passes throught0. It indicates thatp(t0)is an inflection point.Furthermore, there is only one straight line tangent to the curveCand passing through (u0,v0)∈S, the corresponding curvep(t)has a single inflection point.There are two straight lines tangent toCand passing through (u0,v0)∈D,the corresponding curvep(t)has double inflection points.

3.4 The case of convexity

We discuss the case of (u,v)∈N=R2(C∪S∪D∪L), see fig.3.At this point, there is no cusp or loop or inflection point on the curvep(t). And, the direction of the binormal vectorγ(t)does not change.

Let

From eq. (5), we obtain

where

(15)

we can see that the sign ofφ(t;u0,v0)changes when it passes throught0. Sop(t)is local convex if (u,v)∈N1.

we can see that the sign ofψ(t;u0,v0)changes when it passes throught0. Hence,p(t)is local convex if (u,v)∈N2.

While if (u,v)∈N0=N(N1∪N2), the direction ofγ(t),m(t)andn(t)do not change, as a result the curvep(t)is global convex.

3.5 Result

Summarizing the discussion of section 3.1 to 3.4, we obtain the following conclusion.

Theorem3Forαβcurve segment, letai=Pi-Pi-1,i=1,2,3, ifa1not parallel toa3anda2=ua1+va3,then the shape characteristic ofp(t)is totally determined by the distribution of the point (u,v)onuv-plane (see fig.3), i.e.

Fig.3 gives the shape diagram of theαβcurve withα=-1,β=0. It is exactly the shape diagram of the traditional cubic uniform B-spline curve.

Fig.3 The shape diagram of the αβ curve

Examples of theαβapproximation curves with the six different shape characteristics are shown in fig.4.The settings of the parameters and the values of (u,v)are as follows：

Fig.4 The αβ curves with different shape characteristics

Ifa1parallel toa3, then after a similar discussion to section 3.1 to 3.4, we obtain the following conclusion.

Theorem4Whena1‖a3, the curvep(t)has no singularity.If and only if the direction ofa1is the same as that ofa3, excluding the four control points collinear, the curvep(t)has one and only one inflection point.

3.6 Regulatory effect of the shape parameters

Changingαandβ, the shape diagram of theαβcurve will change accordingly, see fig.5.In fig.5, the settings of the parameters are as follows：

Fig.5 The influence on shape diagram by α and β

As can be seen from fig.5(a) to (c), with the increase ofα, the regionDreduces and the regionN0expands,while the regionSremains unchanged.As can be seen from fig.5(d) to (f), with the increase ofβ, the regionDreduces and the regionsN1,N2,Lexpand, while the regionsN0andSremain unchanged.In addition, from fig.5(d), we see that whenβ→α, the regionsN1,N2,Lreduce to zero.From fig.5(c), we see that whenα→0 andβ=0, the regionsN1,N2,LandDreduce to zero.

From the above analysis, we can conclude that the regionScannot be changed by adjusting the shape parameters.That is to say, when there is only one inflection point on theαβapproximation curve, we cannot eliminate it by altering the values ofαandβ. However, when there is one cusp or one loop on the curve, or when the curve is local convex, it can be adjusted to a curve with two inflection points by takingβ→α. Further, the two inflection points can be removed by takingα→0 andβ=0. Meanwhile, the curve is adjusted to global convex.

4 The αβ surfaces

Definition3Given control pointsPij∈R3(i=0,1,…,m;j=0,1,…,n), two sets of knotsu1

wheres,t∈[0,1],i=1,2,…,m-2;j=1,2,…,n-2. All the patches constitute anαβsurface

u∈[ui,ui+1],v∈[vj,vj+1],

Theαβsurfaces have properties similar to theαβcurves.Fig.7 shows theαβsurfaces consisting of 7×4 patches defined by the same control net but different parameters.The settings of parameters are as follows：

Fig.7 The αβ surfaces

5 Conclusion

There are many extensions of cubic B-spline curves, but theαβcurves enjoy some advantageous properties from design.With variation diminishing and convexity-preserving, theαβcurves have a good shape control.Shape parameters are incorporated into theαβcurves, but we do not increase the degree of the basis polynomial, and therefore do not increase the amount of calculation.The shape diagram is useful for classifying and modifying the shape of theαβcurve.Notice that ifα=0, then theαβcurve segments interpolate to the two inner control points.Thus theαβcurves can be used to construct interpolation curves without solving equations.One of our future works is to discuss the shape parameter selection scheme of the interpolation curve.

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嚴(yán)蘭蘭, 樊繼秋
(東華理工大學(xué) 理學(xué)院, 江西 南昌 330013)

構(gòu)造了一種保形并且形狀可調(diào)的分段三次多項(xiàng)式曲線，并分析其形狀特征與控制多邊形之間的關(guān)系.首先，通過預(yù)設(shè)基函數(shù)的性質(zhì)再解方程組，構(gòu)造了一組帶2個(gè)形狀參數(shù)的多項(xiàng)式基函數(shù)，其包含三次均勻B樣條基函數(shù)作為特例.然后，借助基函數(shù)與三次Bernstein基函數(shù)之間的關(guān)系證明了基函數(shù)的全正性，由這組基函數(shù)定義了一種分段三次多項(xiàng)式曲線，使該曲線擁有一個(gè)局部和一個(gè)全局形狀參數(shù).最后，分析了控制多邊形邊變量之間的相對(duì)位置關(guān)系對(duì)曲線段形狀特征的影響，得到了曲線段擁有1個(gè)或2個(gè)拐點(diǎn)，1個(gè)二重點(diǎn)或1個(gè)尖點(diǎn)，為局部凸或全局凸時(shí)的充要條件.該結(jié)論為曲線段的形狀調(diào)整提供了理論基礎(chǔ).

曲線設(shè)計(jì); B樣條方法; 全正基; 形狀參數(shù); 形狀分析

TP 391

A

1008-9497(2018)01-044-10

date： April 5,2016.

Supported by the NSFC(11261003, 11761008 ), the Natural Science Foundation of Jiangxi Province ( 20161BAB211028) and the Science Research Foundation of Jiangxi Province Education Department ( GJJ160558).

AbouttheauthorYAN Lanlan (1982-),ORCID： http： //orcid.org/0000-0002-5472-9986, female, Ph.D,associate professor, the field of interest is CAGD, E-mail： yxh821011@aliyun.com.

10.3785/j.issn.1008-9497.2018.01.008