Li Jun-cheng

(College of Mathematics and Finance,Hunan University of Humanities,Science and Technology,Loudi,Hunan,417000)

Communicated by Ma Fu-ming

Abstract:In this paper,we focus on how to use strain energy minimization to obtain the optimal value of the free parameter of the planar Cardinal spline curves.The unique solution can be easily obtained by minimizing an appropriate approximation of the strain energy.An example is presented to illustrate the effectiveness of our method.

Key words:Cardinal spline,Catmull-Rom spline,minimization,strain energy,fair curve

1 Introduction

In computer aided design(CAD)and related application fields,the construction of fair curves is an important issues(see[1]–[3]).Although the fairness of a curve is difficult to be expressed in a quantitative way,the general ways to construct the fair curves are achieved by minimizing some energy functions(see[4]).The strain energy(also called bending energy)and curvature variation energy are two widely adopted metrics to describe the fairness of a curve(see[1]and[5]).For example,a lot of works on planar G1or G2Hermite interpolation via strain energy minimization or curvature variation minimization have been proposed,see e.g.[4]and[6]–[9].

As we know,the Cardinal spline(see[10])contains a free parameter.We can obtain different shapes of interpolating curves by altering the value of the free parameter even if the points are remained unchanged.Therefore,a natural idea drives us to find the optimal value of the free parameter by minimizing the strain energy or curvature variation energy to construct the fair Cardinal spline curves.In this paper,we apply the strain energy minimization to achieve this goal.We observe that the unique solution can be easily obtained by minimizing an appropriate approximation of the strain energy.

The rest of this paper is organized as follows.In Section 2,the main problem is presented.In Section 3,the strain energy minimization method is described.In Section 4,an example is outlined to show the effectiveness of the method.A short conclusion is given in Section 5.

2 Cardinal Spline

The Cardinal spline(see[10])is joined by a sequence of individual curves to form a whole curve that is specified by an array of points and a free parameter.The planar Cardinal spline curves can be expressed by

where 0≤ t≤ 1,pk∈ R2(k=0,1,···,n;n ≥ 3)are given points,and

with α is a free real parameter.

By a simple deduction from(2.1)and(2.2),we have

From(2.3)we know that each individual Cardinal spline curve interpolates the second and the third points,which means the whole curve passes all the given points except the first point p0and the end point pn.Because the curves are usually required to pass all the given points,it is necessary to add an auxiliary point p?1to the front of p0and an auxiliary point pn+1to the back of pn,which ensure that the whole curve can pass the first and the end points.Generally,we can repeat the first point and the end point as the two auxiliary points,viz.the two auxiliary points are taken as p?1=p0,pn+1=pn.

By the expression of the Cardinal spline curves,we can find that the free parameter α has a clear influence on the shape of the whole curve when all the points are remained unchanged.For example,suppose that the given points and the two auxiliary points are

Fig.2.1 shows the influence of the free parameter α on the shape of the whole curve,where the parameter of the dotted lines is taken as α=0.3,the parameter of the solid lines is taken as α=0.8.

Since the free parameter α has an obvious influence on the shape of the Cardinal spline curves,a natural idea derives us to find the optimal value of the free parameter to obtain the fair curves.In most cases,minimization of the strain energy(also called blending energy)can be applied to gain the fair curves(see e.g.[2],[7],[9]and[11]–[14]).

Fig.2.1 The influence of the free parameter on the shape of the whole curve

3 MSE-Cardinal Spline

The strain energy of the curve b(t)is defined by(see[1])

Because the strain energy(3.1)is highly nonlinear,one may use some approximate forms to linearize it.If we assume that the curve is approximately parameterized by arc-length,the strain energy could be approximated by(see[7])

By a deduction from(2.1)and(2.2),we can obtain that

where δpi:=pi+2?pi,?pi:=pi+1? pi.

Then,we have

According to(3.2)and(3.4),we can express the strain energy of each individual Cardinal spline curve as

Then,the strain energy of the whole curve bi(t)(i= ?1,0,···,n?2)can be expressed as

Since the free parameter α can completely determine the shapes of the Cardinal spline curves when the points pk(k= ?1,0,···,n+1)are fixed,the following problem could be naturally obtained if we want to minimize the strain energy of the curves,

The derivative of F can be calculated by

The following unique result can be solved fromwith the derivative expressed in(3.8),

Recall that δpi=pi+2?pi,?pi=pi+1?pi,then the following unique solution of(3.7)can be deduced from(3.9),

At last,we can obtain the Cardinal spline curves with minimum strain energy(called the MSE-Cardinal spline curves for short)by taking the optimal value of the free parameter calculated by(3.10)into(2.1).

4 Example

Dodgson[15]suggested that the best value of the free parameter α should be set to 0.5 when the Cardinal spline curves are used to deal with interpolation problems.In this case,the Cardinal spline curves are popularly called the Catmull-Rom spline curves.Here,we compare the MSE-Cardinal spline curves with the Catmull-Rom spline curves via the following example.

For the same points given in Fig.2.1,the optimal value of the free parameter calculated by(3.10)is α=0.6582.The strain energies of the MSE-Cardinal spline curves and the Catmull-Rom spline curves calculated by(3.6)are shown in Table 4.1.

The Catmull-Rom spline curves and the MSE-Cardinal spline curves are shown in Fig.4.1.

Fig.4.1 The Catmull-Rom spline curves(dotted lines)and the MSE-Cardinal spline curves(solid lines)

5 Conclusion

In this paper,we present a method for constructing the fair Cardinal spline curves by minimizing the strain energy.Our method focuses on how to determine the optimal value of the free parameter of the Cardinal spline curves,which is achieved by minimizing an appropriate approximation of the strain energy.The example shows that the Cardinal spline curves with minimum strain energy are better than the Catmull-Rom spline curves in dealing with interpolation problems.