Adjustable Piecewise Quartic Hermite Spline Curve with Parameters

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 achieve Ccontinuity but also can realize shape control by choosing suitable parameters, which addressed the weakness of the classical cubic Hermite interpolating spline curves.


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 Béziertype and B-spline curves with parameters are discussed in the articles [1][2][3][4]. e classical cubic Hermite interpolation spline curves have been widely used in practical engineering problems in [5][6][7]. Nevertheless, when interpolation conditions are given, the shape of the cubic Hermite interpolation spline curves could not be changed.
at 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][9][10][11][12][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]. ose 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. e 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.

Basic Concepts of Cubic Hermite
Interpolation Spline are called the cubic Hermite basis functions, and its properties are as follows: (1) End-points: (2) e function's graphs are shown in Figure 1. From left to right are the graphs of the basis functions a 0 (x), a 1 (x), b 0 (x), and b 1 (x).

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 parameters α i , β i and 0 ≤ t ≤ 1, the following four functions are called quartic Hermite basis functions with parameters. When α i � β i � 0, equation (5) degenerates to equation (1), which are the cubic Hermite basis functions. When α i , β i take different values, the shape of the basis function will also be altered. Figure 2 shows the basis function of . From left to right are the graphs of the bases ea 0 (x), eb 1 (x), eb 0 (x), and eb 1 (x), respectively.

Continuity. e spline
us, we obtain at implies the spline (6) is C 1 -continuous. Also, the tangent line of curves er(x) at the point er(x i ) is parallel to the tangent vector p i ′ (for any α i , β i ). (6), for

Total and Local Adjustment. By rewriting
Obviously, the parameters α i , β i only affect the i-th curve segments er i (x) without altering the other curve segments. So, we can adjust the shape of the interpolation curves by changing parameters α i , β i locally.

Parameter Selection for Constraint Conditions
As mentioned in Section 4, the shape of the quartic Hermite interpolation spline curves can be changed by selecting suitable parameters. eoretically, 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, Mathematical Problems in Engineering 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.
ree criteria for determining the values of parameters are examined as follows.

Criteria I: e Arc Length of the Curve Is the Shortest.
Generally, the arc length of the parametric curve erefore, for given the data points p i (i � 0, 1, . . . , n) and corresponding tangent vectors p i ′ (i � 0, 1, . . . , n), the optimal values of parameters α i and β i 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: To find the value of the optimal parameters α i and β i , a system of equations relating to the partial derivatives of equations (10) can be found. We have the following equations: e interpolation curve with the shortest arc length can be obtained by solving for the optimal parameter value from the above equations.

Criteria II: e Curve Is the Smoothest.
Generally, the smoothness of an interpolation curve r(x)(a ≤ x ≤ b) can be approximated by its energy value E � b a (r ″ (x)) 2 dx, and the smaller the energy value is, the smoother the curve is. erefore, for the given data points p i (i � 0, 1, . . . , n) and corresponding tangent vectors p i ′ , the optimal parameters are determined to minimize the curve energy value and make the quartic Hermite spline curve the smoothest. en, the following optimization model can be obtained: To find the optimal parameters α i and β i , a system of equations with the partial derivatives of the parameters α i and β i can be solved. e following equations are given: Mathematical Problems in Engineering 5