**The Approximation Cubic Bézier Curve** \*\*Double blind review, please do not include authors information in this version \*\* Received Date: \*date Accepted Date: \*date Published Date: \*date **HIGHLIGHTS** - Cubic Bezier curves are more frequently used in CAGD. - To study the properties of cubic Bezier and approximate the cubic Bézier curve. ABSTRACT *This study* *focuses on the approximation of cubic Bézier curve in Computer-Aided Geometric Design (CAGD). The objectives in this study are to review the properties of cubic Bézier curve and approximate the cubic Bézier curves. In this study, the expanding equations from the basis function of cubic Bezier is used to approximate the cubic Bézier curve. Future researchers can propose the higher degree of Bézier curves to approximate the curves and surfaces in Computer-Aided Design (CAD), CAGD and engineering. The next studies of Bezier curve are recommended as a contribution for further research.* *Keywords: Bézier curves,* *cubic Bézier curve, properties of cubic Bézier curve, approximation of cubic Bezier curves.* # INTRODUCTION This paper focused on a brief description of an approximation of cubic Bézier curve. Bézier curves were studied in 1962 by Pierre Bézier, a French engineer, who started to use them in designing bodies of an automobile. However, the first study of these curves was developed in 1959 by a mathematician named Paul de Casteljau, using de Casteljau's algorithm, a numerically stable curve to evaluate Bézier curves (Kilicoglu & Senyurt, 2020). The Casteljau's algorithm evaluates and subdivides the Bézier curve as shown in Figure 1. Curve subdivision is necessary to divide the curve into several segments which is very useful for another studies in Bezier curves such as curve fitting, interpolation, segmentation, and interpolation (Fitter et al., 2014). ![Diagram Description automatically generated](6298da6986900_media/media/image1.png) **Figure 1:** Subdivision by De Casteljau's algorithm (Source: Fitter et al., 2014) Curve fitting is a crucial task in image extraction, in which the extracted object contours are identified into number of small segments that are further described using lines and curves, and the extracted object regions are fitted using surface fitting techniques such as triangular patches, least squares, multistage methods and many others (Fitter et al., 2014). A Bézier curves is widely used in computer graphics, animation, Computer-Aided Geometric Design (CAGD) and many other related fields (Kilicoglu & Senyurt, 2020). Additionally, satellite path planning, robotics, highway or railway designing, creation of 3D tensor product surface models, image compression or font designing and shape preserving curves and surfaces are extensively used in the fields of engineering and technology (Bashir et al., 2013). Bézier curves are commonly used in Computer-Aided Geometric Design (CAGD) because of their numerical and geometric properties (Abbas et al., 2011). In addition, there are a lot of properties in the Bézier curves because it employs a special class of polynomial basis functions (Bashir et al., 2013). Cubic Bézier curve is the curve that is present to perform the study. By referring to the review in Bézier curves, this study presented the study in properties of cubic Bezier and the approximation of cubic Bézier curves. **Problem Statement** Nowadays, technology is advancing with time. The same goes in the geometric field, and the researchers play an essential role to build the system that can improve the study in that field. In the area of computer graphics, many techniques in CAGD are used to approximate the curves or surfaces. The technique can build a mathematical equation with many curves, such as Bézier curves. Most researchers commonly discuss this topic because it has various curves and equations that can be used. This study focuses on the properties of cubic Bezier and approximation of cubic Bézier curves. **The Objectives of the Study** The objectives in this study are to review the properties of cubic Bezier and approximate the cubic Bézier curves. **The Scope of the Study** The approximation of a curve in this study is focused on the cubic Bézier curves. This study is conducted to discuss the cubic Bézier curve with their properties and approximate a curve by using cubic Bézier, which is particularly useful in CAGD. **The Significance of the Study** This study will bring benefits to other researchers that desire to carry out similar research on this topic. It can give a motivation to future researchers because they can use it as references in conducting the research. The findings and results in this study may also operate as information and guidance to them. Furthermore, the designers may also have the benefits as this curve is frequently used in the CAGD. CAGD gives advantages to the designers in related fields such as engineering, science and technology, and geometric fields in their future design studies. Finally, the study contributes to the body of existing literature and knowledge in this field of research and provide information for further research. **LITERATURE REVIEW** The main element in Computer-Aided Geometric Design (CAGD) is the study of curves and surfaces in many years. The necessity for effective computer representation of practical curves and surfaces used in engineering design prompted the development of CAGD techniques (Bashir et al., 2013). The most basic modelling tools in CAD/CAM systems are the Bézier curves (Bashir et al., 2013). A set of independent variables is known as Bézier curves representing the coordinates of some points in curved lines between two or more points (Rizal & Kim, 2015). A Bézier curve has a set of control points which is \(P_{0}\) through \(P_{n}\), where n is the order (Kilicoglu & Senyurt, 2019). The end points of the curve always use the first and last control points: however, the intermediate control points (if any) it does not lie on the curve (Kilicoglu & Senyurt, 2019). The Bézier polygon or control polygon is the polygon to connect the Bézier points with lines, starting with \(P_{0}\) and finishing with \(P_{n}\). Bézier curves is inside the convex hull of the Bézier polygon (Kilicoglu & Senyurt, 2019). Most of the researchers used Bézier curves to curve fitting, which has become one of the essential things in computer graphics illustration programs and CAD systems. The curve can be applied in many applications, such as designing the curves and surfaces of automobiles and defining the shape of letters or characters in type fonts (Rusdi & Yahya, 2015). Bézier curves are the most stable of all the polynomial-based curves because it is the ideal standard for describing complex piecewise polynomial curves (Rusdi & Yahya, 2015). Thus, the Bézier curves can always be used when generating smooth curves because it can produce high quality results (Rusdi & Yahya, 2015). There are several types of Bézier curves to approximate a curve. But for this study, the focus is on the cubic Bézier curves. According to figure 2, there are four control points in the cubic Bézier curve, which are \(P_{0},\ P_{1},\ P_{2}\) and \(P_{3}\). The curves start from \(P_{0}\), moves away to \(P_{1}\) and \(P_{2}\) before end with \(P_{3}\). The convex hull of its control points contains the cubic Bézier curve (Rusdi & Yahya, 2015). ![](6298da6986900_media/media/image2.png) **Figure 2:** Cubic Bézier curves (Source: Rusdi & Yahya, 2015) Based on Bashir, Abbas, and Ali (2013), approximating a curve or surfaces is represented in CAGD. Although cubic B-splines and Bézier curves are mostly used in CAD and CAGD, Bashir et al. (2013) mentioned it is challenging to acquire the desired shape because of their polynomial nature. Therefore, it will bring a problem to the researchers in designing the curves or surfaces. Some researchers developed an alternative using shape parameters to have a better shape of the curves. This also can be supported by Hadi et al. (2013) in the study of Beta-spline that can control the shape of the curve because it has its shape parameters, bias and tension. Then, the comparison between the Bézier curves and Beta-spline has been made. All the studies have their advantages. Moreover, Hadi et al. (2013) mentioned the Bézier curves is faster to plot than Beta-spline. However, Beta-spline has a smoothness faster compared to the Bézier curves if this curve is built on \(G^{2}\) continuity condition. This can prove that Beta-spline has the control polygon that is better shape, bigger and messier than the Bézier curves (Hadi et al., 2013). Rizal and Kim (2015) conducted a study using Bézier curves to reconstruct images in order to interpolate data between sampling points to reduce the error. Using the cubic Bézier curve, the result shows that the curve proposed has better quality and higher similarity levels than the other image reconstruction methods (Rizal & Kim, 2015). The work presented by Rusdi and Yahya (2015) suggested the cubic Bézier curve to reconstruct generic shapes using Least Square Method (LSM). Moreover, Sum Square Error (SSE) was used to find the best fit curve and minimize error calculation. The curve of cubic Bézier curve is easily applied to the data extracted by bitmap images to obtain the best optimal curve. They conclude that the cubic Bézier curve is a good curve and suitable for other applications in the future (Rusdi & Yahya, 2015). The study of Bézier cubic by Neagos (2019) showed the best approximation to a circle and other plane curves when coming up with some numerical. They also have a problem developing numerical approximation when they do not utilize mathematical software (Neagos, 2019). It shows that mathematical software is essential to approximate the curve. Abbas, Jamal, and Ali (2011) proposed a study of Bézier curves interpolation constrained by a line. Then, they developed an algorithm for the quadratic and cubic Bézier curve constrained interpolation. The changing shape comes from the middle control points of the quadratic and cubic Bézier curve constrained by a line. There are two types of a constrained line, which is the x-axis and any straight line. For that, they developed simpler constraints on the middle Bézier ordinates (Abbas et al., 2011). As a result, they approximated a C-shape and S-shape curves in the middle cubic Bézier, which may benefit path planning, highway or railway route designing or car-like robot path planning (Abbas et al., 2011). This interpolation also may be helpful in robotic motion studies. The work presented by Hadi et al. (2013) proposed a comparative study on cubic Bézier and Beta-spline curves. They present a curve for evaluating Beta-spline control points and comparing Bézier and Beta-spline curves in terms of continuity and circle approximation. There are two types of Bézier and Beta-spline, which are rational, and non-rational (Hadi et al., 2013). Non-rational cubic Bézier and Beta-splines are discussed in this paper. The equation involved in this study is the cubic Bézier curve, defined as Bernstein's polynomial. The result shows that the Bézier curve's continuity depends on how its control polygon is connected, while the continuity of the Beta-spline curve can achieve to degree two (Hadi et al., 2013). From the evaluation control points for Bézier and Beta-spline, the result shows Beta-spline have a bigger control polygon than Bézier when it approximates the same curve (Hadi et al., 2013). Other than that, when users want to create a circle for Beta-spline, they cannot use the same control points as with Bézier. It is because Bézier and Beta-spline in the same control polygon show different positions. The user can convert the Bézier control points to Beta-spline control points in this situation (Hadi et al., 2013). The result of the circle approximation shows that the control polygon of Beta-spline is bigger and messier than Bézier. Beta-spline is important in computer-aided design because it can provide the appropriate smoothness. Next, Rusdi and Yahya (2015) proposed four boundary of images: Fork, At, Plane and Love to obtain the outline of the bitmap images and to minimize the distance between the boundary of the original image and parametric curve. In this paper, LSM is used to present an efficient algorithm for the approximation of the boundary of bitmap images. The error given by those two curves is calculated using SSE. They conclude that LSM is an efficient algorithm for approximating the image boundary since the result shows Fork bitmap image produces the least SSE. In conclusion, the cubic Bézier curve is used to find the best optimal curve for the data extracted by bitmap images (Rusdi & Yahya, 2015). Meanwhile, LSM is a suitable method and also can be used for other applications in future (Rusdi & Yahya, 2015). Hence, Rizal and Kim (2015) conducted a study using the Bézier curves to reconstruct images to interpolate data between sampling points to reduce the error. They compare the reconstructed images using compressive sampling, Discrete Fourier transform (DFT), and Bézier curves in this paper. Four sampling points in the cubic Bézier curve is used to interpolate and reconstruct five values of the image (100×100 pixels) (Rizal & Kim, 2015). This study found that the Bézier curves is the best curve for image reconstruction because it has a closer quality to the original image than compressive sampling and DFT. At the same time, it also can maintain the small size of reconstructed images without harming their quality (Rizal & Kim, 2015). Furthermore, the cubic Bézier curve, the result found that the proposed curve has better quality and higher similarity levels than the other image reconstruction methods (Rizal & Kim, 2015). Other than that, in 2017, Asghar et al. are presented a multi-disciplinary strategy to automatically improve images filtered using low and high pass Butterworth in the frequency domain by intensity transformation in the spatial domain using a cubic Bézier curve. A Bézier curves is a smooth curve that is often used to represent data. The cubic Bézier curve was described and used to simulate the contrast enhancement problem in various situations based on scene or image types. The control points of the cubic Bézier function can be adjusted to get the desired curve shape for contrast enhancement. According to the findings of the experiments, they have a better contrast enhancement in results from the proposed curve compared to an existing state-of-the-art method and provides a simpler way to remove noises than Histogram Equalization (HE) and a cubic spline based curve Bézier Curve for Contrast Enhancement (BCCE) (Asghar, Gilanie, Saddique & Habib, 2017). Du et al. (2019) used the cubic Bézier curve to define Chinese characters because these curves were accurate and continuity. It may define characters in a unique way without affecting the handwritten Chinese characters' original trajectory. Thus, there is no breakpoint in the middle of each Chinese character's glyph curve, and the curve is smoother. Handwritten Chinese Character Recognition (HCCR) is applied as a detection tool for evaluating beautifying in this study. As a result, Chinese characters that have been well-written can be identified. In conclusion, it may be possible to fulfil the purpose of beautifying font shape after gathering multi-dimensional information of handwritten Chinese characters. However, there is still a need for improvement because the recognition rate of handwritten Chinese characters has not reached 100% yet. It will improve the accuracy of handwritten Chinese character recognition results and make the evaluation curve more flexible in the future. Another proposed method for the curve in CAGD is presented by (Mistro et al., 2017) which introduce the quintic Trigonometric Bezier Curve with two shape parameter in order to construct the curve and surfaces. This study is focuses on the higher degree of trigonometric Bezier curve with composition of two curves that satisfy the ![C squared](6298da6986900_media/media/image3.png "{\"mathml\":\"\\\\C\\2\\\\\"}") Hermite condition. The technique of two shape parameters gives the advantages to the user in modifying the shape of the curve. # RESEARCH METHODOLOGY **This study focused on the properties of cubic Bezier curves and the approximation** **of cubic Bézier curves.** These properties of the cubic Bézier curve are based on Zhang and Feng (2006), Sarfraz and Masood (2007) and Rusdi and Yahya (2015). A cubic Bézier curve is defined as follows, with four control points \(P_{i}\) *=* \(P_{0}\)*,*\(\ P_{1}\)*,* \(P_{2}\) *and* \(P_{3}\)*:* > \(B\left( t \right) = \ \sum_{i = 0}^{3}{\left( \frac{3}{i} \right)t^{i}}\ {(1 - t)}^{3 - i}\ \left( t \right)\ \left\lbrack P_{i} \right\rbrack\), (1) Expanding Eq. (1) yields \(B\left( t \right)\ = {(1 - t)}^{3}\ P_{0} + 3t\ {(1 - t)}^{2}\ P_{1} + 3t^{2}\ \left( 1 - t \right)\ P_{2} + t^{3}P_{3}.\) (2) Figure 3 shows the basis function of cubic Bézier curve with four control points, \(P_{0}\) (blue line), \(P_{1}\ \)(red line), \(P_{2}\) (yellow line) and \(P_{3}\) (purple line). ![Chart Description automatically generated](6298da6986900_media/media/image4.png) **Figure 3:** Basis function of Cubic Bézier curve The properties of cubic Bezier curve are given by: - The curve is passing through the two endpoints \(P_{0}\) *and* \(P_{3}\)*.* - The curve is continuous, infinitely differentiable, also continuous for second derivatives (For a polynomial curve, this is automated). - The cubic Bézier blending functions are all positive, with their sum is always 1, \(\sum_{i = 0}^{3}{P_{i}\left( t \right) = 1}.\) - The convex hull of its control points contains the cubic Bézier curve. - Only if the curve is linear, both \(P_{1}\) and \(P_{2}\) are on the curve. - \(P_{0}\) and \(P_{3}\) are on the curve. - *Along the line* \(P_{0}P_{1}\) *and* \(P_{2}P_{3}\) *there are slopes at the beginning and end of the cubic Bézier curve, respectively.* **In the approximation of cubic Bezier curve, the first stage is to locate the control points of the curves. The equations used in this study are expanded from the basis function of cubic Bézier curve.** **Then, the cubic Bézier curve is approximated based on the cubic Bezier equation. The process of approximation cubic Bezier curve is illustrated in Figure 4.** ![Icon Description automatically generated](6298da6986900_media/media/image5.png) **Figure 4:** The process of approximation the cubic Bezier curves **RESULTS AND DISCUSSIONS** **The results of** **the approximation cubic Bézier curve are presented in this section****.** In Figure 5, the cubic Bézier curve is approximated with four control points, \(P_{0}\)*,*\(\ P_{1}\)*,* \(P_{2}\) *and* \(P_{3}\). These four control points will form in the control polygon. A sample of control points for this study is modified from the example in (Mistro et al., 2017) where only four control points are used which are \(P_{0}\)*,*\(\ P_{1}\)*,* \(P_{2}\) *and* \(P_{3}\). To approximate the cubic Bézier curve, the expanding of basis function for cubic Bézier curve from Eq. (4) were combined with the control points, \(P_{0}\) *= (10,0),* \(P_{1}\) *= (0,5),* \(P_{2}\) *= (5,8) and* \(P_{3}\) *= (20,3).* The curve starts from \(P_{0}\)*,* moves away to \(P_{1}\) *and* \(P_{2}\) *before end with* \(P_{3}\). The shape of curves for cubic Bézier curves is contained in the control polygon. ![Chart, line chart Description automatically generated](6298da6986900_media/media/image6.png) **Figure 5:** Approximation of Cubic Bézier curve **Discussions** This study is conducted to approximate cubic Bézier curve and study the shape of the curve with their properties. The findings and results in this study can moved the body of scientific knowledge forward because future researchers can use it as a reference in conducting similar research. **CONCLUSIONS AND RECOMMENDATIONS** This study started from the background of the study of a Bézier curves. For this study, it is important to have the control points for the curves. The properties of cubic Bézier curve are reviewed and the expanding equations are needed to approximate the curve of cubic Bézier. Cubic Bézier curve has four control points and the shape of curves contained in the control polygon. The curve is chosen because of their numerical and geometric properties. **Recommendations** Bézier curves is widely used in many related fields. The same goes to this study under the topic of the Bézier curves. Future studies on the current topic are recommended for future researchers who are interested in this study because they can used in the future. For example, it can be used for other application other than focusing on its properties. Other than that, the higher degree of Bézier curves can be proposed which is useful in CAD, CAGD and engineering. **REFERENCES** Abbas, M., Jamal, E., & Ali, J. M. (2011). 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