GEOMETRIC PIECEWISE CUBIC BÉZIER INTERPOLATING POLYNOMIAL WITH C

. Bézier curve is a parametric polynomial that is applied to produce good piecewise interpolation methods with more advantage over the other piecewise polynomials. It is, therefore, crucial to construct Bézier curves that are smooth and able to increase the accuracy of the solutions. Most of the known strategies for determining internal control points for piecewise Bezier curves achieve only partial smoothness, satisfying the first order of continuity. Some solutions allow you to construct interpolation polynomials with smoothness in width along the approximating curve. However, they are still unable to handle the locations of the inner control points. The partial smoothness and non-controlling locations of inner control points may affect the accuracy of the approximate curve of the dataset. In order to improve the smoothness and accuracy of the previous strategies, а new piecewise cubic Bézier polynomial with second-order of continuity C 2 is proposed in this study to estimate missing values. The proposed method employs geometric construction to find the inner control points for each adjacent subinterval of the given dataset. Not only the proposed method preserves stability and smoothness, the error analysis of numerical results also indicates that the resultant interpolating polynomial is more accurate than the ones produced by the existing methods.


Introduction.
The Missing values of dataset are the common issues in many areas of sciences such as statistics, computer sciences, and geophysics [1][2][3]. Several interpolation methods employed piecewise polynomials to estimate missing values. One of them is Bezier curve which is a parametric polynomial used extensively in computer-aided design (CAD) [4,5], numerical analysis [6,7], hitch avoidance path determination of unicycle robots [8,9], lane changing [10,11], and roundabouts [12,13] due to its flexibility, stability, and simplicity in representation. By taking the advantages of Bézier curve, researchers started to construct a piecewise cubic Bézier curve at every subinterval of data points in order to improve the smoothness of the interpolating polynomial and consequently increase the accuracy.
Ge and Kang [14] proposed two algorithms of piecewise Bezier functions. The first algorithm produces an approximation function for a dataset, while, the resultant function in the second algorithm interpolates through a dataset. However, both algorithms only satisfy the second order geometric continuity (G 2 ). In Pollock [15], piecewise cubic Bézier curves with the second order of continuity (C 2 ) have been achieved by adopting the construction of a natural cubic spline strategy. Three years later, a geometric technique piecewise Bezier interpolating was proposed by Shemanarev [16]. The resultant polynomial seemed smooth at data points, behaving like the first order geometric continuity (G 1 ) although he did not test the order of continuity. Yau

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Each subinterval requires two inner control points and two end points for constructing a cubic Bézier spline. Since there are n subintervals, 2n inner control points are needed to construct n cubic Bézier splines. The control points for each cubic Bézier curve is given by: Several previous studies have found good strategies for locating the inner control points for piecewise Bézier curves. Most of them, however, achieve partial smoothness by satisfying the first order of continuity C 1 as in Saaban et al. [18,20], Stelia et al. [22], and Zulkifli et al. [23]. Although, some researchers achieved to construct interpolating polynomials with wider smoothness along the approximating curve, including Pollock [15], they are still unable to handle the locations of the inner control points. The partial smoothness and/or non-controlling locations of inner control points may affect the accuracy of the approximate curve of the dataset.
3. Proposed Piecewise Cubic Bézier Polynomial. In order to improve the smoothness and accuracy of the previous strategies, a new piecewise interpolating polynomial known as C 2 Geometric Bézier Polynomial (C2GBP) is proposed in this study. To construct this polynomial, the inner control points will be located geometrically depending on the polygon of the dataset.
3.1. Construction of C2GBP. In the section, the construction of the new piecewise interpolating polynomial is discussed. The procedure details for constructing C2GBP are as follows: Step 1. .
Step 2. Find the straight lines connecting be straight lines as shown in Figure 2 with slopes j a defined by: whose y-intercepts are: The equations for straight lines of Step 3. Find y-intercepts of the straight lines pass through 1 Therefore, the slope of  is 1 1 a  , while y-intercept is given by: The value of 0  is defined as: where m is the number of inner control points, and q is the degree of the polynomial in each sub-interval. In our construction, the values of q and m are 3 and 2, respectively.
Then, y-intercepts is defined by: where the positive/negative value of 0  is determined using algorithm below:
The straight line of Step 8. Find the inner control point 1 1 p at second subinterval.

Let 1
 is a distance between 0 2 p and 1 W defined by : whose slope is: and y-intercept: using quadratic formula to find the value of 1

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Step 9. Find y-intercept ) ( 1 h of the straight line 1  with slope 1  as described in the following algorithms:

End end End
Step 10. The slope 1  is defined as: and y-intercept is:     where: with y-intercept:  Output: y-intercepts 1 )  of the straight line with slope 1  . Step 11.

Start
In order to define the value of  we will use the following relation: Substituting (4), (5) in (6) gives: By solving Equation (7) using quadratic formula, we can find the value of ε which then substituted into (4), (5) to get the value of 0 1 P and 1 2 P .
The quadratic formula will be employed in Equation (8)

t y t x t x x y t y y t x t y t x t x x y t y y t F x t y t
x t x x y t y y t

By definition, the first condition (a) satisfied. In order to investigate the condition (b), the values (1) and (0) are substituted in the first derivative of Equation (1) which is given by:
Substituting the values (1) and (0) into Equation (9) Figure 5 illustrates the comparison of the test function with other approximate parametric polynomials.  Figure 3 demonstrates C2GBP is capable of preserving the curvature compared with the other five previous methods in Problem 1. An irregular inflexion curve was detected in Problem 2. The numerical results indicate that C2GBP manages to handle this situation better than the other methods by producing the smallest errors as displayed in Figure 4. Figure 5 presents the numerical results obtained in the employed method for solving the increase steep of the curve occured in Problem 3. The results show that C2GBP also excels in non-oscillating curve output since the inner control points are geometrically constructed. The advantage of C2GBP is that it is able to control the curvature at subintervals which increases accuracy.
The results of the approximate parametric interpolating polynomials for solving Problems 1-3 in terms of errors are also displayed in Tables 1 to 3, respectively. The errors in terms of SSE, MAE, RMSE in Tables 1 to 3 suggest that C2GBP is the best option to be applied to approximate dataset in all test problems. Figures 6-8 illustrate the error rates on all  along approximate curves by using the RMSE, in order to provide a more accurate description. Curvature amplitude reveals that the error ratio of the curves increases which means the closer the curve to x-coordinate, the less the error is. It is worth to mention that the error at the dataset is zero since the interpolating points are the dataset.    inner control point locations for each sub-interval. The proposed method gives an approximate cubic Bézier curve representing the dataset with interpolating at all data points. The proposed procedure succeeded in achieving the second-order of parametric continuity between every adjacent subinterval of the data points. The newly constructed parametric interpolating polynomial was then compared with the existing natural cubic spline, piecewise cubic Hermite interpolating polynomial, modified Akima piecewise cubic Hermite interpolation, rational cubic Ball interpolation, and natural cubic Bézier curve using the same datasets. Three different error testing methods have been used by taking (100) test points for each sub-interval. The numerical results show that the proposed method is more accurate than the other existing methods shown in this study. All the details of the comparison have been indicated in tables and graphs for each testing. The resulting curve is very appropriate to find a fit, smooth, and accurate representation of the data points. The proposed method can also be used in many applications, as in image processing and geographic information systems. As well, it is expandable to include many applications in two-dimension.