# Adjustable Piecewise Quartic Hermite Spline Curve with Parameters

Authors: Jin Xie; Xiaoyan Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the quartic Hermite parametric interpolating spline curves are formed with the quartic Hermite basis functions with parameters, the parameter selections of the spline curves are investigated, and the criteria for the curve with the shortest arc length and the smoothest curve are given. When the interpolation conditions are set, the proposed spline curves not only achieveC1-continuity but also can realize shape control by choosing suitable parameters, which addressed the weakness of the classical cubic Hermite interpolating spline curves.


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## Body

## 1. Introduction

In CAGD&CG, it is always an important research topic to adjust and control the shape of fitting curves. Different kinds of spline curves with parameters were constructed to control the shape of interpolating curves. For example, the Bézier-type and B-spline curves with parameters are discussed in the articles [1–4].The classical cubic Hermite interpolation spline curves have been widely used in practical engineering problems in [5–7]. Nevertheless, when interpolation conditions are given, the shape of the cubic Hermite interpolation spline curves could not be changed. That is to say, we need to change interpolation conditions to modify the shape of the spline curves. However, when the interpolation conditions are taken from the real problems, this approach is not desirable. To overcome the limitations of the cubic Hermite interpolation spline curves in shape adaptability, the construction of Hermite interpolation spline with parameters has attracted the attention of many scholars. For example, the rational cubic Hermite interpolation spline with parameters is developed in [8–13], the quartic Hermite interpolation spline with parameters is developed in [14], and the cubic triangular Hermite interpolation spline with parameters is developed in [15]. Those articles proposed several interpolation spline functions with parameters, which have similar properties to the classical cubic Hermite interpolation, and they can push the curve to the designated area by modifying the parameters. However, the subject of how to select the parameters to obtain “good” fitting curves was not discussed in the abovementioned articles.In this paper, a class of polynomial Hermite interpolation spline with two parameters is constructed. The shape of the spline curve could be adjusted by amending the parameters when the interpolation conditions are satisfied. One could choose the appropriate parameters to fulfill the given criterion. More specifically, we inspected the techniques to determine the parameters such that the quartic Hermite spline curve has the shortest arc length or has the least curve energy value and makes the quartic Hermite spline curve the smoothest or achieves the minimal sum of the arc length and the curve energy value.


## 2. Basic Concepts of Cubic Hermite Interpolation Spline

Definition 1.
For0≤t≤1, the following four functions(1)a0t=1−3t2+2t3,a1t=3t2−2t3,b0t=t−2t2+t3,b1t=−t2+t3,are called the cubic Hermite basis functions, and its properties are as follows:(1)
End-points:(2)a00a10b00b10a01a11b01b11a0′0a1′0b0′0b1′0a0′1a1′1b0′1b1′1=1000010000100001.(2)
Quasisymmetry:a0t=a11−t,b0t=−b11−t.
The function’s graphs are shown in Figure1. From left to right are the graphs of the basis functions a0x, a1x, b0x, and b1x.Figure 1
Hermite basis function.Definition 2.
Given data pointspii=0,1,…,n and corresponding tangent vector pi′, for 0≤t≤1, piecewise cubic Hermite interpolant curves rit are defined as follows:(3)rit=a0tpi+a1tpi+1+b0tpi′+b1tpi+1′,i=0,1,…,n−1,where a0t,a1t,b0t, and b1t are given in (1).
By straightforward calculation, we have(4)ri0=pi,ri1=pi+1,ri′0=pi′,ri′1=pi+1′.


## 3. Quartic Interpolation Spline with Parameters

From the last section, we can see that for given interpolation and derivative values at designated interpolation points, the shape of the classical cubic Hermite interpolation curve is set. If you want to change the shape of the curve, you need a different set of data, which is not allowed in some situations. To modify the curve shape without changing the interpolation conditions we will construct a basis of quartic polynomial functions with parameters, when the parameters take different values, the shape of the corresponding new interpolating curves will be changed, which will be very convenient for interactive designing. If the appropriate parameter values are chosen, it can also improve the approximation effect of the interpolation curve.Definition 3.
For any given parametersαi,βi and 0≤t≤1, the following four functions(5)ea0t=1+αi−3t2+21−αit3+αit4,ea1t=3−αit2+2αi−1t3−αit4,eb0t=t+βi−2t2+1−2βit3+βit4,eb1t=−βi+1t2+2βi+1t3−βit4,are called quartic Hermite basis functions with parameters.
Whenαi=βi=0, equation (5) degenerates to equation (1), which are the cubic Hermite basis functions. When αi,βi take different values, the shape of the basis function will also be altered. Figure 2 shows the basis function of αi=βi=0 (solid line), αi=βi=1 (dashed line), and αi=βi=−2 (dot-dash line). From left to right are the graphs of the bases ea0x, eb1x, eb0x, and eb1x, respectively.Figure 2
Quartic Hermite basis functions with different parameters.Definition 4.
Leta=x0<x1<⋯<xn=b be a subdivision of the interval [a, b], and hi=xi+1−xi, t=x−xi/hi, then the following functions on the interval xi,xi+1(6)erix|xi,xi+1=ea0tpi+ea1tpi+1+eb0thipi′+eb1thipi+1′,i=0,1,…,n−1,are called quartic Hermite interpolation curves with parameters, where ea0t, ea1t, eb0t, and eb1t are quartic Hermite basis functions with parameters given in (3).
When all parameters areαi=βi=0, equation (6) degenerates to equation (3).


## 4. Properties of the Quartic Hermite Interpolation Spline

### 4.1. Continuity

The spline (6) is the piecewise polynomial curves. We need to show the continuity of the curves. For i=1,2,…,n−1, we have(7)erixi=pi,erixi+1=pi+1,erixi=pi′,erixi+1=pi+1′.Thus, we obtain(8)erkxj=pjk,k=0,1,j=i,i+1,i=0,1,…,n−1.That implies the spline (6) is C1-continuous. Also, the tangent line of curves erx at the point erxi is parallel to the tangent vector pi′ (for any αi,βi).


### 4.2. Total and Local Adjustment

By rewriting (6), for x∈xi,xi+1, we have(9)erxi|xi,xi+1=ea0t,αipi+ea1t,αipi+1+eb0t,βihipi′+eb1t,βihipi+1′,i=0,1,…,n−1.Obviously, the parametersαi,βi only affect the i-th curve segments erix without altering the other curve segments. So, we can adjust the shape of the interpolation curves by changing parameters αi,βi locally.Example 1.
Given data pointsp0=0,2, p1=2,2, p2=2,0, p3=2,−2, p4=0,−2, p5=−2,−2, p6=−2,0, and p7=−2,2 and the corresponding tangent vectors p0′=2,0, p1′=1,−1, p2′=0,−2, p3′=−1,−1, p4′=−2,0, p5′=−1,1, p6′=0,2, and p7′=1,1, Figure 3 shows the total and local adjustable interpolation curve. The solid line in Figure 3 is the classical cubic interpolation spline curve, which is equivalent to the parameter αi=βi=0 in equation (6). In Figure 3(a), the parameter values of the line of dashes are αi=βi=1 (i = 0, 1, 2, …, 7) and the parameter values of dot-dash lines are αi=βi=−2 (i = 0, 1, 2, …, 7). As we could see, the shape of the curve is adjusted as a whole. In Figure 3(b), the parameters are equal to 0, except for α0=0,β0=2,α7=2,β7=0, and the curve is adjusted locally in the first and 8-th segments.Figure 3
Total and local adjustment. (a) Total adjustment. (b) Local adjustment.


## 4.1. Continuity

The spline (6) is the piecewise polynomial curves. We need to show the continuity of the curves. For i=1,2,…,n−1, we have(7)erixi=pi,erixi+1=pi+1,erixi=pi′,erixi+1=pi+1′.Thus, we obtain(8)erkxj=pjk,k=0,1,j=i,i+1,i=0,1,…,n−1.That implies the spline (6) is C1-continuous. Also, the tangent line of curves erx at the point erxi is parallel to the tangent vector pi′ (for any αi,βi).


## 4.2. Total and Local Adjustment

By rewriting (6), for x∈xi,xi+1, we have(9)erxi|xi,xi+1=ea0t,αipi+ea1t,αipi+1+eb0t,βihipi′+eb1t,βihipi+1′,i=0,1,…,n−1.Obviously, the parametersαi,βi only affect the i-th curve segments erix without altering the other curve segments. So, we can adjust the shape of the interpolation curves by changing parameters αi,βi locally.Example 1.
Given data pointsp0=0,2, p1=2,2, p2=2,0, p3=2,−2, p4=0,−2, p5=−2,−2, p6=−2,0, and p7=−2,2 and the corresponding tangent vectors p0′=2,0, p1′=1,−1, p2′=0,−2, p3′=−1,−1, p4′=−2,0, p5′=−1,1, p6′=0,2, and p7′=1,1, Figure 3 shows the total and local adjustable interpolation curve. The solid line in Figure 3 is the classical cubic interpolation spline curve, which is equivalent to the parameter αi=βi=0 in equation (6). In Figure 3(a), the parameter values of the line of dashes are αi=βi=1 (i = 0, 1, 2, …, 7) and the parameter values of dot-dash lines are αi=βi=−2 (i = 0, 1, 2, …, 7). As we could see, the shape of the curve is adjusted as a whole. In Figure 3(b), the parameters are equal to 0, except for α0=0,β0=2,α7=2,β7=0, and the curve is adjusted locally in the first and 8-th segments.Figure 3
Total and local adjustment. (a) Total adjustment. (b) Local adjustment.


## 5. Parameter Selection for Constraint Conditions

As mentioned in Section4, the shape of the quartic Hermite interpolation spline curves can be changed by selecting suitable parameters. Theoretically, when the data points, interpolation values, and their tangent vectors remain unchanged, the shape of the interpolation curve can be adjusted arbitrarily by changing the values of parameters. However, in many practical engineering problems, one often needs to make the interpolation curve meet the corresponding requirements consistent with the certain specifications. At this time, it is necessary to determine the values of parameters according to the given criteria. Three criteria for determining the values of parameters are examined as follows.


### 5.1. Criteria I: The Arc Length of the Curve Is the Shortest

Generally, the arc length of the parametric curverxa≤x≤b can be expressed L=∫abr′xdx. Therefore, for given the data points pii=0,1,…,n and corresponding tangent vectors pi′i=0,1,…,n, the optimal values of parameters αi and βi should be determined so that the quartic Hermite spline curve has the shortest arc length and then the optimization model can be established as follows:(10)minLαi,βi=∑i=0n−1∫01ri′t2dt,s.t.αi,βi∈R.To find the value of the optimal parametersαi and βi, a system of equations relating to the partial derivatives of equations (10) can be found. We have the following equations:(11)∂Lαi,βi∂αi=0,∂Lαi,βi∂βi=0,i=0,1,…,n−1.The interpolation curve with the shortest arc length can be obtained by solving for the optimal parameter value from the above equations.Example 2.
For the given data pointspii=0,1,…,n and corresponding tangent vectors pi′i=0,1,…,n shown in example 1, we can obtain the parameter values αi=0,βi=−1.7i=0,1,…,7 according to equations (11). The acquired quartic Hermite spline curve has the shortest arc length, as shown in Figure 4.Figure 4
A closed plane curve with the shortest arc length.


### 5.2. Criteria II: The Curve Is the Smoothest

Generally, the smoothness of an interpolation curverxa≤x≤b can be approximated by its energy value E=∫abr″x2dx, and the smaller the energy value is, the smoother the curve is. Therefore, for the given data points pii=0,1,…,n and corresponding tangent vectors pi′, the optimal parameters are determined to minimize the curve energy value and make the quartic Hermite spline curve the smoothest. Then, the following optimization model can be obtained:(12)minEαi,βi=∑i=0n−1∫01ri″t2dt,s.t.αi,βi∈R.To find the optimal parametersαi and βi, a system of equations with the partial derivatives of the parameters αi and βi can be solved. The following equations are given:(13)∂Eαi,βi∂αi=0,∂Eαi,βi∂βi=0,i=0,1,…,n−1.By solving for the optimal parameter values from the above equations, the interpolation curve with the minimum energy can be obtained.Example 3.
For the given data pointspii=0,1,⋯,n and corresponding tangent vectors pi′ shown in example 1, we can attain the parameter values αi=0,βi=0i=0,1,…,7 according to equation (13). That is to say, the classical cubic Hermite spline curve is the smoothest in this case, as the black curve shown in Figure 3(a).


### 5.3. Criteria III: The Shorter Arc Length and the Smoother Curve Are Both Considered

For the given data points and corresponding tangent vectors, there is an optimization model to make the sum of the arc length and the energy of the quartic Hermite spline curve to be the smallest as follows:(14)minFαi,βi=Lαi,βi+Eαi,βi,s.t.αi,βi∈R.The partial derivatives of the above equations concerning parametersαi and βi are obtained as follows:(15)∂Fαi,βi∂αi=0,∂Fαi,βi∂βi=0.i=0,1,…,n−1.Example 4.
For the given data pointspii=0,1,…,7 and corresponding tangent vector pi′ given in example 1, we can obtain the parameter values αi=0,βi=−0.0407i=0,1,…,7 according to equation (15). Thus, the sum of the arc length and the energy of the quartic Hermite interpolation spline curve is the smallest, as shown in Figure 5.Figure 5
The plane closed curve with the least sum of the arc length and energy.


## 5.1. Criteria I: The Arc Length of the Curve Is the Shortest

Generally, the arc length of the parametric curverxa≤x≤b can be expressed L=∫abr′xdx. Therefore, for given the data points pii=0,1,…,n and corresponding tangent vectors pi′i=0,1,…,n, the optimal values of parameters αi and βi should be determined so that the quartic Hermite spline curve has the shortest arc length and then the optimization model can be established as follows:(10)minLαi,βi=∑i=0n−1∫01ri′t2dt,s.t.αi,βi∈R.To find the value of the optimal parametersαi and βi, a system of equations relating to the partial derivatives of equations (10) can be found. We have the following equations:(11)∂Lαi,βi∂αi=0,∂Lαi,βi∂βi=0,i=0,1,…,n−1.The interpolation curve with the shortest arc length can be obtained by solving for the optimal parameter value from the above equations.Example 2.
For the given data pointspii=0,1,…,n and corresponding tangent vectors pi′i=0,1,…,n shown in example 1, we can obtain the parameter values αi=0,βi=−1.7i=0,1,…,7 according to equations (11). The acquired quartic Hermite spline curve has the shortest arc length, as shown in Figure 4.Figure 4
A closed plane curve with the shortest arc length.


## 5.2. Criteria II: The Curve Is the Smoothest

Generally, the smoothness of an interpolation curverxa≤x≤b can be approximated by its energy value E=∫abr″x2dx, and the smaller the energy value is, the smoother the curve is. Therefore, for the given data points pii=0,1,…,n and corresponding tangent vectors pi′, the optimal parameters are determined to minimize the curve energy value and make the quartic Hermite spline curve the smoothest. Then, the following optimization model can be obtained:(12)minEαi,βi=∑i=0n−1∫01ri″t2dt,s.t.αi,βi∈R.To find the optimal parametersαi and βi, a system of equations with the partial derivatives of the parameters αi and βi can be solved. The following equations are given:(13)∂Eαi,βi∂αi=0,∂Eαi,βi∂βi=0,i=0,1,…,n−1.By solving for the optimal parameter values from the above equations, the interpolation curve with the minimum energy can be obtained.Example 3.
For the given data pointspii=0,1,⋯,n and corresponding tangent vectors pi′ shown in example 1, we can attain the parameter values αi=0,βi=0i=0,1,…,7 according to equation (13). That is to say, the classical cubic Hermite spline curve is the smoothest in this case, as the black curve shown in Figure 3(a).


## 5.3. Criteria III: The Shorter Arc Length and the Smoother Curve Are Both Considered

For the given data points and corresponding tangent vectors, there is an optimization model to make the sum of the arc length and the energy of the quartic Hermite spline curve to be the smallest as follows:(14)minFαi,βi=Lαi,βi+Eαi,βi,s.t.αi,βi∈R.The partial derivatives of the above equations concerning parametersαi and βi are obtained as follows:(15)∂Fαi,βi∂αi=0,∂Fαi,βi∂βi=0.i=0,1,…,n−1.Example 4.
For the given data pointspii=0,1,…,7 and corresponding tangent vector pi′ given in example 1, we can obtain the parameter values αi=0,βi=−0.0407i=0,1,…,7 according to equation (15). Thus, the sum of the arc length and the energy of the quartic Hermite interpolation spline curve is the smallest, as shown in Figure 5.Figure 5
The plane closed curve with the least sum of the arc length and energy.


## 6. Conclusion

In this paper, we construct a set of adjustable quartic Hermite interpolation spline functions with two parameters, which inherits the main characteristics of the classical cubic Hermite interpolation spline functions and overcomes its limitation. When the data points and their tangent vectors are given, the shape of the classical cubic Hermite spline curve cannot be changed, which is a weakness. The shape of our proposed quartic Hermite interpolation spline curve can be adjusted by adapting the parameters, which is valuable when certain requirements of the interpolation effect in practical engineering designs need to be met. In addition, the quartic Hermite parameter spline still adopts the piecewise polynomial form, which not only has a relatively simple expression but also stays in line with the standard Bézier curve, B-spline curve, and other polynomial parameter curves in CAD/CAM system.



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