**The Approximation of Quadratic and 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** - Quadratic and cubic Bezier curves are more frequently used in CAGD. - To approximate the quadratic and cubic Bézier curves. - To compare the quadratic Bézier curve and cubic Bézier curve with their properties. ABSTRACT *The approximation of a curve in a topic of Bézier curves is one of the curves that are commonly discussed in Computer-Aided Geometric Design (CAGD). This study focuses on the two curves, which are* *quadratic Bézier curve and cubic Bézier curve. The objective in this study is to approximate the quadratic and cubic Bézier curves and study the properties between the quadratic Bézier curve and cubic Bézier curve. In this paper, the expanding equations from the basis function of two curves are used to approximate the quadratic Bézier curve and cubic Bézier curve. In conclusion, future studies on the current topic are therefore recommended. Also, future researchers can expand the degree of the Bézier curves, which is more useful in Computer-Aided Design (CAD), CAGD and engineering. Hence, future researchers also can try to use the Ball curves other than Bézier curves. They can compare the results between the Ball curves and Bézier curves, where the results will be the contributions for further research.* *Keywords: Bézier curves, Quadratic Bézier curve, Cubic Bézier curve, Properties.* # INTRODUCTION This paper focused on a brief description of the approximation of quadratic and 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). Hence, the theory is called Bézier, although Casteljau was the first person who created it since Casteljau never published his results (Neagos, 2019). This also can be supported by Fitter et al. (2014) in their study stating that Casteljau too utilised Bernstein Basis at the same time as Pierre Bézier, focusing on the property of non-negativity and partition of unity of the basis function associated with the control points in his work. However, it was too late to name after him as de Casteljau's algorithm for his algorithm, which evaluates and subdivides the Bézier curve as in Figure 1. ![](6227144645991_media/media/image1.png) **Figure 1:** Subdivision by De Casteljau's algorithm (Source: Fitter et al., 2014) Curve subdivision is necessary to break the curve into the number of small segments for various applications such as curve fitting, segmentation, interpolation and many others. 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 that are popularly used among researchers are the parametric quadratic and cubic Bézier curves. Both 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 many good properties in the Bézier curves because it employs a special class of polynomial basis functions (Bashir et al., 2013). In this paper, quadratic Bézier and cubic Bézier curve are the curves that are present to perform the study. By referring to some of the study Bézier curves, quadratic and cubic Bézier curves, this paper presented the study of the approximation of quadratic and cubic Bézier curves with their properties. **Problem Statement** Nowadays, technology is advancing with time. The same goes in the geometric field, and the designers play an essential role to build the system that can improve the research in that field. In the area of computer graphics, CAGD is used to approximate the curves or surfaces. The system 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. Many types of approximation of a curve can be done under the Bézier curves. Ideally, the approximation of a curve has done by comparing the two curves: quadratic Bézier and cubic Bézier in this study. Even though these curves under the same curves which is Bézier curves, however, the curves that are approximated will be different because of different polynomial. That is why this study is being carried out in order to approximate a curve using both curves. Despite, the basis function and equation for every curve are different between them. In this study, two types of curves, quadratic Bézier curve and cubic Bézier curve, are approximate and the properties of quadratic and cubic Bézier are observed. Therefore, this study is conducted to approximate a curve by using the quadratic and cubic Bézier curves as stated in the study's objectives. **The Objectives of the Study** The objectives in this study are: 1\. To approximate the quadratic and cubic Bézier curves. 2\. To study the quadratic Bézier curve and cubic Bézier curve with their properties. **The Scope of the Study** The study of the approximation of a curve are focused on the quadratic and cubic Bézier curves. This study aims to approximate a curve by using quadratic and cubic Bézier, which is particularly useful in CAGD. After that, this study is conducted to compare the quadratic Bézier curve and cubic Bézier curve with their properties. **The Significance of the Study** The approximate of a curve is applied to conduct a study on the approximation of a curve by using quadratic and cubic Bézier. This study will bring benefits to other researchers that desire to carry out similar research on this topic. It can give good benefits to future researchers because they can use it as a references in conducting similar 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 are 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. **Summary** This chapter gives an overview of the study that has been conducted. The summarised this chapter as follows: - Background of the study - Problem statement - The objectives of the study - The scope of the study - The significance of the study The study's literature review will be explained in the next chapter to better understand the study. **LITERATURE REVIEW** This chapter discuss the related works according to the topic conducted in this study. Furthermore, it explained the curves and findings of the associated outcomes. **Computer-Aided Geometric Design (CAGD)** 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). In most studies in this field, the researchers have similarly stated that the Bézier curves is one of the curves that are popular in the CAGD. **Bézier Curves** 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 curves 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 which are linear Bézier curves, quadratic Bézier curves, cubic Bézier curves, quartic Bézier curves and quintic Bézier curves. But in this paper, the focus is on the quadratic and cubic Bézier curves. Based on the following Figure 2, the function \(B\left( t \right)\) represents a quadratic Bézier curve given the three control points, \(P_{0},\ P_{1}\) and \(P_{2}\). This equation is acknowledged as the linear interpolant of corresponding points on the linear Bézier curve from \(P_{0}\) to \(\ P_{1}\) and \(P_{1}\) to \(\ P_{2}\) (Kasihmuddin et al., 2016). ![](6227144645991_media/media/image2.png) **Figure 2:** Quadratic Bézier curves (Source: Kasihmuddin et al., 2016) Meanwhile, 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). Figure 3 below shows the cubic Bézier curve. ![](6227144645991_media/media/image3.png) **Figure 3:** Cubic Bézier curves (Source: Rusdi & Yahya, 2015) Bashir, Abbas, and Ali (2013) said approximating a curve or surfaces is represented in CAGD. Although cubic B-splines and Bézier curves are most 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) study Beta-spline that can control the shape of the curve because it has its shape parameters, bias and tension. In the study, the comparison between the Bézier curves and Beta-spline has been made. Both of them has 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. That 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 the Bézier curves to reconstruct images 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 minimise 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 utilise mathematical software (Neagos, 2019). It shows that mathematical software is essential to approximate the curve. **Previous Study on the Quadratic Bézier Curve** In the previous study, Choi, Curry, and Elkaim (2011) analysed the problem of minimising the curvature of a quadratic Bézier curve within a boundary constraint determined by a tetragonal concave polygon. The curve is defined by the "control lengths", which are the length between its control points. A minimising the maximum curvature may be necessary for a curve design such as highway design and motion planning of autonomous vehicles. At last, the curve to smooth a piecewise linear path generated from a path search technique is demonstrated by numerical results. These results may be applied whenever a smooth transition between two intersecting straight lines is needed (Choi et al., 2011). Next, the arc length and bending energy of the quadratic Bézier curve were studied by Ahn, Hoffmann, and Rosen (2013). The formulas used are in terms of coordinates of the control point. The basic algorithm amounts to sampling candidate contact points along a specified tangent or tangency circle and verifying the arc length or bending energy. Local minima are discovered using this quick computation, which may be refined iteratively or by oversampling subregions. Then, they used the **Graphics Processing Unit** (GPU's) high parallelism to address the constraint problem. A locus method is used to solve the tangency constraints. For example, given the points \(b_{0}\) and \(b_{2}\) and the tangent line T, the locus of the middle control point \(b_{1}\) of all the quadratic Bézier curve tangent to T is parallel to T at a distance easily derived from the distances of \(b_{0}\) and \(b_{2}\) from T. They proved that the minimum length tangency problem is a unique solution. However, they cannot anticipate a unique minimum solution for tangency to a circle. When the end points are symmetrically aligned with the circle centre, for example, there will be two global arc length minima (Ahn et al., 2013). **Previous Study on the Cubic Bézier Curve** This literature review presented the previous study on applications that used the Bézier curves and cubic Bézier curve. At first, 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. Last but not least, Beta-spline is important in computer-aided design because it can provide the appropriate smoothness. The work presented by Rusdi and Yahya (2015) proposed four 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. The boundary of the bitmap image can detect using the Matrix Laboratory (MATLAB) function. 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). Moreover, to compare the time to reconstruct the image, they used MATLAB and Open Source Computer Vision Library (OpenCV) software. 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). Finally, Du, Liu, and Xun (2019) proposed a curve for beautifying Chinese characters and evaluating the beautification result. This study seeks to incorporate handwritten Chinese character beautifying with machine learning and makes a preliminary evaluation of handwritten Chinese characters by identifying handwritten Chinese characters with a high recognition rate, thereby overcoming the subjectivity of artificial evaluation. The data used in this study were 52 Chinese characters that covered 33 standard strokes and 19 typical structures of Chinese characters. The handwritten Chinese characters were improved primarily in two aspects, such as the global adjustment and the elimination of jitter. First, a set of two-dimensional (2D) data points are expanded into three-dimensional (3D) space. The data set is then fitted with a Gaussian Mixture Model (GMM), and the layout of handwritten Chinese characters is changed using a point set registration procedure. Secondly, they used the properties of the cubic Bézier curve function, identify and eliminate the jitter in each stroke using an interpolation algorithm. 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. **Previous Study on the Quadratic Bézier Curve and Cubic Bézier Curve** A study conducted by Song, Zhang, Wang, Ren, and Meng (2018) about the curvilinear flexible surgical robots' shape estimation in real-time. In their study, they are focusing on the Bézier curves based. They make a comparison and discuss the advantages and disadvantages for the cubic Bézier curve and the quadratic Bézier curve. The following aspects in the comparison both curves are the number of sensors required to perform the shape estimation method for an N-section flexible robot, the bending restriction, and the performance under external payload. The first aspect shown when fitting a two-section robot with using the two quadratic Bézier curve, it does not require a sensor. While, when using the cubic Bézier curve, each section requires a sensor to provide information about the tip's position and direction. In second aspect, they use circle arcs to test the shape estimation methods. The results shown the quadratic Bézier curve and the cubic Bézier curve become invalid after the radian surpassing \(\pi\)/2 and \(3\pi\)/2, respectively. In last aspect, the result found that the cubic Bézier curve is more suitable to use if the robot tends to 'S' shape or bends to different planes compared to the quadratic Bézier curve. When the curve is a 'C' shape, both of these curves can represent the shape well when the bending is small. However, the quadratic Bézier curve produces a larger error in each node's position estimation compared to the cubic Bézier curve. We can conclude from the results that the cubic Bézier curve based performs better when there is an unknown force acting on the robot (Song et al., 2018). **Summary** This chapter presented the related works according to and relating to the topic conducted in this study. In addition, this chapter summarised Computer-Aided Geometric Design (CAGD), Bézier curves, a previous study on the quadratic Bézier curve, a previous study on the cubic Bézier curve, and a previous study on the quadratic Bézier curve and cubic Bézier curve. # RESEARCH METHODOLOGY This chapter presented the methodology used to carry out this research. **This study focused on the two curves to approximate the curve****, quadratic and cubic Bézier curves. This study is began with a literature review of journal articles, or other reading materials were reviewed to provide supporting details and references in this study.** **Secondly, locate the control points of the curves. Next, the equations used are expanded from the basis function of quadratic Bézier curve and cubic Bézier curve.** **Then, the quadratic and cubic Bézier curve are approximated. When the curves were obtained, both curves were compared to see the difference in their properties. The method of data collection in this study is illustrated as shown in Figure 4.** **Figure 4:** Steps to approximate the curves For the quadratic Bézier curve, it used three control points while the cubic Bézier curve uses four control points, \(P_{0},\ P_{1},\ P_{2}\) and \(P_{0},\ P_{1},\ P_{2},\ P_{3},\ \)respectively. The basis function for the quadratic Bézier curve is taken from Bashir et al. (2012) and Kilicoglu & Senyurt (2019), given in Definition 1. Meanwhile, to approximate a curve by using the cubic Bézier curve, the basis function for the cubic Bézier curve is taken from Kilicoglu and Senyurt (2020), as given in Definition 2. **Definition 1**. A quadratic Bézier curve is defined as follows, with three control points \(P_{i}\) = \(P_{0}\), \(P_{1}\) and \(P_{2}\): \(B\left( t \right) = \ \sum_{i = 0}^{2}{P_{i}\left( \frac{2}{i} \right)}\ {(1 - t)}^{2 - i}\ t^{i},\ \ t \in \left\lbrack 0,1 \right\rbrack,\) (1) Expanding Eq. (1) yields (Kilicoglu & Senyurt, 2019) \(B\left( t \right)\ = {(1 - t)}^{2}\ P_{0} + 2t\ (1 - t)\ P_{1} + t^{2}P_{2}.\) (2) **Definition 2**. A cubic Bézier curve is defined as follows, with four control points \(P_{i}\) = \(P_{0}\),\(\ P_{1}\), \(P_{2}\) and \(P_{3}\): \(B\left( t \right) = \ \sum_{i = 0}^{3}{\left( \frac{3}{i} \right)t^{i}}\ {(1 - t)}^{3 - i}\ \left( t \right)\ \left\lbrack P_{i} \right\rbrack\), (3) Expanding Eq. (3) yields \(B\left( t \right)\ = {(1 - t)}^{3}\ P_{0} + 3t\ {(1 - t)}^{2}\ P_{1} + 3t^{2}\ \left( 1 - t \right)\ P_{2} + t^{3}P_{3}.\) (4) The control points are substituted into the equations. Then, mathematical algorithm is used to approximate the curve. Finally, the quadratic Bézier curve and cubic Bézier curve are approximated. The following figure contains graphs of basis function for the quadratic Bézier curve and cubic Bézier curve such as Figure 5 and Figure 6, respectively. ![](6227144645991_media/media/image4.png) **Figure 5:** Basis function of Quadratic Bézier curve As shown in Figure 5, there are three control points for the basis function of quadratic Bézier curve which are \(P_{0}\) (blue line), \(P_{1}\ \)(red line) and \(P_{2}\) (yellow line). The horizontal axis which is x is known as t, t goes between 0 to 1. The vertical axis which is y is known as basis function also goes between 0 to 1. When t = 0, has 1 times with first control point, \(P_{0}\), plus 0 times with second control point, \(P_{1}\), plus 0 times with third control point, \(P_{2}\) from Eq. (2). So, when t = 0, Bézier curves is at the first control point. Next, when t = 0.2, the blue line has about 0.65 times with \(P_{0}\), about 0.32 times with \(P_{1}\) and a little bit about 0.03 times with \(P_{2}\). So, as t increasing, the amount of sort of influence for \(P_{0\ }\)the first basis function has on the curve is reducing and the amount of influence for \(P_{1}\ \)has increasing. When t = 0.5, from the figure can see that \(P_{1}\ \)which has more effect on the position on the Bézier curves. As can see from the figure, when t towards 1, the \(P_{2}\ \)that has more of an effect and the other two, \(P_{0}\) and \(P_{1}\) have less and of effect (Mathematics of Computer Graphic and Virtual Environments, 2015). ![](6227144645991_media/media/image5.png) **Figure 6:** Basis function of Cubic Bézier curve In Figure 6, the basis function of cubic Bézier curve is defined as follows, with four control points, \(P_{0}\) (blue line), \(P_{1}\ \)(red line), \(P_{2}\) (yellow line) and \(P_{3}\) (purple line). The horizontal axis which is x is known as t, t goes between 0 to 1. The vertical axis which is y is known as basis function also goes between 0 to 1. When t = 0, has 1 times with first control point, \(P_{0}\), plus 0 times with second control point, \(P_{1}\), plus 0 times with third control point, \(P_{2}\) and plus 0 times with fourth control point, \(P_{3}\) from Eq. (4). When t = 0, only the \(P_{0}\) that has an effect. As t increasing that the amount of effect that the \(P_{0}\) has decreasing and the other control points, \(P_{1},\ P_{2},\ P_{3}\) are increasing. When t = 0.5, from the figure can see that \(P_{1}\ \)and \(P_{2}\ \)which has more effect on the position on the Bézier curves. Lastly, when t is close to 1, it is only the \(P_{3}\) that has an effect (Mathematics of Computer Graphic and Virtual Environments, 2015). **RESULTS AND DISCUSSIONS** **This study analysed the results and discussions to meet the objectives of the study. The results of** **the approximation of quadratic and cubic Bézier curve are presented in this section, and the graphs are compared.** **Quadratic Bézier Curve** **As shown in Figure 7, there are three control points needed for the quadratic Bézier curve which are** \(P_{0}\)**,** \(P_{1}\) **and** \(P_{2}\)**. These three control points will form in the control polygon. The expanding of basis function for quadratic Bézier curve from Eq. (2) were combined with the control points,** \(P_{0}\) **= (10,10),** \(P_{1}\) **= (30,55) and** \(P_{2}\) **= (60,50) to generate the curve. From the figure, it can be seen that** \(t\) **increases from 0 to 1, the curve turns away from** \(P_{0}\) **in the direction of** \(P_{1}\)**, then bends to arrive at** \(P_{2}\) **from the direction of** \(P_{1}\)**.** ![Chart, line chart Description automatically generated](6227144645991_media/media/image6.png) **Figure 7:** Graph of Quadratic Bézier curve **Cubic Bézier Curve** In Figure 8, the cubic Bézier curve is defined as follows, with four control points, \(P_{0}\),\(\ P_{1}\), \(P_{2}\) and \(P_{3}\). These four control points will form in the control polygon. To approximate cubic Bézier curve, the expanding of basis function for cubic Bézier curve from Eq. (4) were combined with the control points, \(P_{0}\) = (10,10), \(P_{1}\) = (30,55), \(P_{2}\) = (60,50) and \(P_{3}\) = (80,28). The curves start from \(P_{0}\), moves away to \(P_{1}\) and \(P_{2}\) before end with \(P_{3}\). ![Chart Description automatically generated](6227144645991_media/media/image7.png) **Figure 8:** Graph of Cubic Bézier curve From the result in Figure 7 and Figure 8, it can be concluded that the difference from the two curves: quadratic and cubic Bézier are the number of control points. As revealed by the result, the graph for quadratic and cubic Bézier curves have three and four control points, respectively. Other than that, the shape of curves for quadratic and cubic Bézier curves are contained in the control polygon. Subsequently, the comparison the quadratic Bézier curve and cubic Bézier curve with their properties, are discussed in next section. **Comparison the Quadratic Bézier Curve and Cubic Bézier Curve with Their Properties** This section shows the comparison between both curves by their properties. Firstly, the properties for the quadratic Bézier curve were taken from Zhang and Feng (2006). - The curve moves away through the control points \(P_{0}\) and \(P_{2}\) since \(B\left( 0 \right)\ \)= \(P_{0}\) and \(B(1)\) = \(P_{2}\). - \(B\left( t \right)\) is a continuous curve and has continuous derivatives in all orders. (For a polynomial, this is automated). - The degree-2 quadratic Bézier are the functions \({(1 - t)}^{2},\ \ 2t\ \left( 1 - t \right),\ \ t^{2}\) that are used to "mix" the control points \(P_{0}\), \(P_{1}\) and \(P_{2}\). All of the functions are non-negative and sum to one. - Within the triangle \(\bigtriangleup P_{0}P_{1}P_{2}\), the curve is contained. Since, - > \(B\left( t \right)\) is a convex combination of the points \(P_{0}\), \(P_{1}\) and \(P_{2}\). - > The triangle’s convex hull is the triangle itself. - The curve is a straight line if the points \(P_{0}\), \(P_{1}\) and \(P_{2}\) are collinear. - This curve contains all of the points created by the divide-and-conquer method. - B(t) can be differentiated with respect to t and obtain \[\ \frac{d}{\text{dt}}B\left( t \right) = - 2\left( 1 - t \right)P_{0} + \left\lbrack - 2t + 2\left( 1 - t \right) \right\rbrack P_{1} + 2tP_{2}\] \(= \ 2\left\lbrack \left( 1 - t \right)\left( P_{1} - P_{0} \right) + t\left( P_{2} - P_{1} \right) \right\rbrack.\) > Thus, \[\frac{d}{\text{dt}}B\left( 0 \right) = 2\left( P_{1} - P_{0} \right),\] \[\frac{d}{\text{dt}}B\left( 1 \right) = 2\left( P_{2} - P_{1} \right).\] The properties of the cubic Bézier curve were taken from Zhang and Feng (2006). Also, the properties of the cubic Bézier curve were taken from Sarfraz and Masood (2007) and Rusdi and Yahya (2015). - At two endpoints \(P_{0}\) and \(P_{3}\) have the curve pass by. - 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* \(P_{2}P_{3}\) *is the slopes at the beginning and end of the cubic Bézier curve, respectively.* It can be seen from the comparison above that the properties of the quadratic Bézier curve are almost the same to the cubic Bézier curve. The first comparison shows that both curves have curve passes through the control points at their own endpoints. Secondly, quadratic Bézier is a continuous curve. It has continuous derivatives in all orders same like cubic Bézier but cubic Bézier has infinitely differentiable also continuous for second derivatives. These properties for both curves were automated because they were a polynomial curve. Thirdly, the quadratic Bézier functions are non-negative and sum to one similarly to cubic Bézier, are all positive. Next, both curves have the convex hull of their control points containing the quadratic and cubic Bézier curves. Thus, the quadratic Bézier curve is a straight line if the points \(P_{0}\), \(P_{1}\) and \(P_{2}\) are collinear, while only if the cubic Bézier curve is linear, both \(P_{1}\) and \(P_{2}\) are on the curve. Although the quadratic and cubic Bézier curves have other properties, but this paper has stated a few. This paper refers to Fang (2014) to compare the graph between both curves by its properties. Fang (2014) stated that two end slopes of the quadratic Bézier curve will never be parallel. However, the cubic Bézier curve can be used to accomplish this. Furthermore, the cubic Bézier curve are allowed to control the two end slopes separately, which the quadratic Bézier curve do not. Thus, the quadratic Bézier curve do not have inflection points, while the cubic Bézier curve may have inflection points if the control points are not carefully chosen. To sum up, the cubic Bézier curve are more popular compared to the quadratic Bézier curve due to its flexibility (Fang, 2014). **Discussions** In this paper, two curves: quadratic Bézier curve and cubic Bézier curve are used. Due to the practical implications in this study, using the high degree Bézier curves are too complicated to process and approximated (Riskus & Liutkus, 2013). As a result, quadratic and cubic Bezier curves are more frequently used in CAD/CAM. They stated that the cubic Bézier curve can approximate a circle but not perfectly fit a circle. The most common approach is to divide a circle into four separate arcs. Errors in approximating a quarter of the circle (90 degree circular arc) have been studied (Riskus & Liutkus, 2013). The limitations of this study were referred to the objectives and scope of the study stated in the earlier chapter. Firstly, to approximate the quadratic and cubic Bézier curves. This study conducted to study the shape of the curve when using the different number of control points. The approximation for quadratic and cubic Bézier curves used three and four control points, respectively. Secondly, the quadratic Bézier curve and cubic Bézier curve with their properties are discussed. 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. **Summary** This chapter presented the results and discussions of approximation quadratic and cubic Bezier curve. In addition, this chapter summarised the properties of quadratic Bézier curve and cubic Bézier curve. **CONCLUSIONS AND RECOMMENDATIONS** This study started from the background of the study of a Bézier curves. Under a topic of Bézier curves, only two curves are proposed in this study: quadratic Bézier curve and cubic Bézier curve. For this study, it is important to have the control points for the curves. Also, the expanding equations are needed to approximate the curve of quadratic and cubic Bézier. The quadratic Bézier curve has three control points, and the cubic Bézier curve has four control points. Both of them had the shape of curves contained in the control polygon. Then, this study discussed the quadratic Bézier curve and cubic Bézier curve with their properties. Both curves are commonly used in CAD and CAGD because of their numerical and geometric properties. **Recommendations** As all know a Bézier curves is widely used in many related fields. There are many types of Bézier curves that can be found. The same goes to this proposed study under the topic of the Bézier curves. This paper has proposed the approximation of quadratic and cubic Bézier curve. Future studies on the current topic are therefore recommended. It may benefit to future researchers who are interested in this proposed 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, future researchers also can expand the degree of the Bézier curves, which is more useful in CAD, CAGD and engineering. Hence, future researchers also can try to use the Ball curves other than Bézier curves. Ball curves have distinct advantages over Bézier curves found in the study by Majeed et al. (2015). For instance, firstly, to evaluate the Ball curves in an interactive design environment, a robust algorithm has been developed. Secondly, the generalised Ball basis is much better suited in degree elevation and reduction, which improves data portability and curve approximation in CAD systems. Other benefits of Ball curves such as flexibility, easy shape tweaking by changing control points, and extra free parameters that can be used to meet design constraints. Refer to the paper of Hu et al. (2022), many future researchers generalized Ball curves of high degree, because basis functions of rational cubic Ball curves are limited to cubic. Future researchers also can increase the degree of basis function for the Ball curves. 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