Geometric Modeling of C-Bézier Curve and Surface with Shape Parameters

Abstract

In order to solve the problem of geometric design and architectural design of complex engineering surface, we introduce the parametric and geometric continuity constraints of generalized C-Bézier curves and surfaces with shape parameters. Firstly, based on C-Bézier basis with parameters, we study the constraints of the control points of the curves needed to be satisfied when connecting them. Moreover, we study the continuity conditions between two adjacent C-Bézier surfaces with parameters. By the continuity conditions and different shape parameters, the curve and surface can be changed easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve and surface still preserve its characteristics and geometrical configuration. Some graphical examples ensure that the proposed method greatly improves the ability to design complex curves and surfaces and easy to implement.

1. Introduction

With the increasingly high requirements for product design, many products have to carry out the corresponding geometric modeling design of curves and surfaces before manufacturing, such as car shell design, aircraft wing design and people wearing shoes, clothes, and so on daily. The study of curve and surface modeling has always been the core content of CAGD research. In practical application, complex curve and surface modeling are often encountered, which is difficult to be represented by a curve or a piece of surface. How to realize the splicing of curves and surfaces, so that they are convenient and flexible to be applied to various curves and surfaces modeling, is the problem we need to solve. Traditional Bézier curves, which is formed by the classical Bernstein basis functions and control points, have many excellent properties like symmetry, terminal properties, partition of unity, non-negativity, linear precision, integral property, convex hull property, etc. We can easily construct any shape by using parametric and geometric continuity constraints of the classical Bézier curve, but its drawback is that we cannot modify and cannot make a small adjustment in the shape of the curves design without changing the control points. To overcome this problem, we study those basis functions that possess shape parameters that help us to make small modifications in the shape of the curves according to the shape parameters. These shape parameters do not affect the physical and geometrical configuration of the curves. In addition, many practical applications, such as the modeling of industrial products, are quite complex and usually cannot be constructed with a single surface [1,2]. Therefore, by connecting multiple surface patches, we can design the complex engineering surfaces.

In [3], Hering defined continuous Bézier and B-spline curves with $C^2$ and $C^3$ and their tangent polygons. He considered dividing the segmented Bézier curves and B-spline curves to express their parameters and geometric continuities. Yan [4] proposed a specific family of Bézier curves with three different shape parameters, also called adjustable Bézier curves. Those curves have the same shape and structure as the traditional quartic Bézier curve. Schneider and Kobbelt [5] described the discrete smoothing of curves and surfaces based on linear curvature distribution. Geometric and parametric continuities with arc length parametrization and smoothness were given in [6]. In [7], Bashir and Abbas used rational quadratic triangular Bézier curves to give the continuity conditions of $C^2$ and $G^2$ and their applications. They also used the rational quadratic triangular Bézier curve to construct a conic section-like circle and ellipse. Qin and Hu gave the parameter continuity and geometric continuity conditions of the GE Bézier curve, and presented the geometric meaning of the shape parameters in [8]. Misro and Ramli [9] presented a new quintic trigonometric Bézier curve with two shape parameters. Shape parameters provide more control on the shape of the curve compared to the ordinary Bézier curve. This technique is one of the crucial parts in constructing curves and surfaces because the presence of shape parameters will allow the curve to be more flexible without changing its control points. The paper also discussed its parameters and curvature continuity.

BiBi and Abbas [10] proposed an important idea to tackle the problem in the construction of some engineering symmetric revolutionary curves and symmetric rotation surfaces by using the generalized hybrid trigonometric Bézier curve. In addition, they described an algorithm for constructing various symmetrical rotation curves in 2D plane and also symmetric rotation surfaces in 3D (space) by using the GHT-Bézier curve involving shape parameter $\lambda$. BiBi and Abbas [11] proposed a new $G^3$ continuous method of the GHT-Bézier curve with many practical applications.

Hu and Bo [12] described the $G^1$ and $G^2$ smooth continuity conditions between two adjacent Q-Bézier curves of degree n and analyzed the influence rules of shape parameters on the shapes of splicing curves, as well as the basic steps of smooth continuity. In [13], Han and Ma proposed a cubic triangular Bézier curve with two different shape parameters and its properties, and discussed continuity constraints through curve modeling. Hu and Wu constructed a SG-Bézier curve with multiple shape parameters, and discussed the modeling of various engineering surfaces based on SG-Bézier such as oscillating surface, swept surface, and rotating surface in [14]. In [15], Reenu Sharma constructed the quartic trigonometric Bézier (QTB) curve with two different shape parameters and discussed the properties of the QTB curve with shape modeling and the shape control of the curves.

Hu and Cao [16] constructed a kind of generalized Bézier-like surfaces associated with multiple shape parameters. The $G^2$ continuity conditions for the generalized Bézier-like surfaces of degree $(m,n)$ are derived, and the influence rules of the shape parameters on splicing surfaces are analyzed.

The following chapters of this paper are arranged as follows: In Section 2, it mainly introduces some theoretical knowledge about C-Bézier basis functions with n shape parameters and C-Bézier curves. The parametric and geometric continuity of C-Bézier curves with their mathematical and graphical results are given in Section 3. All figures in this paper are realized by software Matlab 2015a. In Section 4, some concrete examples are given to verify the effectiveness of curve connection. Then, Section 5 describes the geometric continuity ($G^k$), $k\le 2$ of C-Bézier surfaces in various directions with its graphical and mathematical representations. An algorithm for the construction of C-Bézier surfaces by $G^2$ continuity conditions is presented in Section 6. Finally, this paper summarizes the research content of this paper.

2. Basic Knowledge of C-Bézier Basis with $N$ Parameters

Zhang [17,18] constructed cubic C-curves and C-surfaces with one parameter $\alpha$ in the space span $\{1,$ t, $cost,sint\}$. Chen and Wang [19] investigated a new C-Bézier basis with degree n through the space span $\{1,t,t^2,\cdots ,t^{n-2},cost,sint\}.$ When the parameter $\alpha \to 0$, these bases have the same properties as Bernstein bases. Li and Zhu [20] constructed a new C-Bézier basis function with n parameters. First, the original functions are given as:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{0,1}(t;\alpha )& =\frac{sin\alpha (1-t)}{sin\alpha }\hfill \\ \hfill u_{1,1}(t;\alpha )& =\frac{sin\alpha t}{sin\alpha }\hfill \end{array}\end{array}$$

(1)

where $\alpha \in (0,\pi ),t\in [0,1].$ Then, we can get

Definition 1

([20]). C-Bézier basis functions with n parameters are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{0,n}(t;{\alpha }_1)& =1-{\int }_0^t{\delta }_{0,n-1}{u}_{0,n-1}(x;{\alpha }_1)dx\hfill \\ \hfill u_{i,n}(t;{\alpha }_i,{\alpha }_{i+1})& ={\int }_0^t[{\delta }_{i-1,n-1}{u}_{i-1,n-1}(x;{\alpha }_i)-{\delta }_{i,n-1}{u}_{i,n-1}(x;{\alpha }_{i+1})]dx\hfill \\ \hfill u_{n,n}(t;{\alpha }_n)& ={\int }_0^t{\delta }_{n-1,n-1}{u}_{n-1,n-1}(x;{\alpha }_n)dx\hfill \end{array}\end{array}$$

(2)

where $u_{i,n-1}(x;{\alpha }_{i+1})=u_{i,n-1}(x;{\alpha }_{i+1},{\alpha }_{i+1})$, ${\delta }_{i,n-1}={\left({\int }_0^t{u}_{i,n-1}(t;{\alpha }_{i+1})dx\right)}^{-1},i=0,1,2,\dots ,$$n-1$. If n = 2, ${\alpha }_2\in (0,\pi ]$, if $n\ge 3,{\alpha }_i\in (0,2\pi ]$.

Figure 1 shows the image of the cubic C-Bézier basis functions with different parameter values.

3. Continuity Constraints of C-Bézier Curves with N Parameters

In the CAD/CAM system, it is a very difficult process to use the continuous conditions of $C^2$ and $G^2$ of the traditional Bézier curves to construct complex curves and figures. While the C-Bézier curve has different shape parameters and great smoothness, it can be easily bent by adjusting the shape parameters according to our choice. It uses parameters and geometric continuity constraints to construct various complex curves, which cannot be executed by classical Bézier curves.

Considering any two adjacent C-Bézier curves, which can be defined as:

$${\displaystyle \left\{\begin{array}{c}{W}_1\left(t;{\alpha }_1,\dots ,{\alpha }_n\right)={\sum }_{i=0}^n{\mathit{P}}_i{u}_{i,n}\left(t\right),n\ge 3\hfill \\ W_2\left(t;{\beta }_1,\dots ,{\beta }_m\right)={\sum }_{j=0}^m{\mathit{Q}}_j{u}_{j,m}\left(t\right),m\ge 3,\hfill \end{array}\right.}$$

(3)

where ${\mathit{P}}_i(i=0,1\dots ,n)$ and ${\mathit{Q}}_j(j=0,1,\dots ,m)$ are the control points of these two adjacent C-Bézier curves, ${\mathrm{u}}_{i,n}\left(t\right)$ and ${\mathrm{u}}_{j,m}\left(t\right)$ are C-Bézier basis functions of degree $n$ and $m$, respectively, ${\alpha }_1,\dots ,{\alpha }_n$ and ${\beta }_1,\dots ,{\beta }_m$ are the shape parameters of curves.

3.1. Parametric Continuity of C-Bézier Curves with Parameters

Given two C-Bézier curves $W_1\left(t\right)$ and $W_2\left(t\right)$ of the same degree, the necessary and sufficient conditions for parametric continuity at the joints are given as follows:

1. For $C^0$ continuity:

$${\mathit{Q}}_0={\mathit{P}}_n.$$

(4)

2. For $C^1$ continuity:

$$\left\{\begin{array}{c}{\mathit{Q}}_0={\mathit{P}}_n,\hfill \\ {\mathit{Q}}_1={\mathit{P}}_n+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-1}\left({\beta }_1\right)}\left({\mathit{P}}_n-{\mathit{P}}_{n-1}\right).\hfill \end{array}\right.$$

(5)

3. For $C^2$ continuity:

$$\left\{\begin{array}{c}{\mathit{Q}}_0={\mathit{P}}_n,\hfill \\ {\mathit{Q}}_1={\mathit{P}}_n+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-1}\left({\beta }_1\right)}\left({\mathit{P}}_n-{\mathit{P}}_{n-1}\right),\hfill \\ {\mathit{Q}}_2=\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\delta }_{n-1,n}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)}{\mathit{P}}_{n-2}-\left(\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\delta }_{n-1,n}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)}+\right.\hfill \\ \left.\frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\gamma }_{0,n-2}\left({\beta }_1\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-1}\left({\beta }_2\right)}\right){\mathit{P}}_{n-1}+\hfill \\ \left(1+\frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\gamma }_{0,n-2}\left({\beta }_1\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-1}\left({\beta }_2\right)}\right){\mathit{P}}_n.\hfill \end{array}\right.$$

(6)

Proof. 

For the $C^0$ continuity of C-Bézier curves, we keep both the first and second curves equal at the final and initial point of the domain respectively as $W_1\left(1\right)=W_2\left(0\right)$. We can obtain ${\mathit{Q}}_0={\mathit{P}}_n$.

Similarly, for $C^1$ and $C^2$ continuity conditions, we consider the first and second derivative of both curves equal like $W_1^{{}^{\prime }}\left(1\right)=W_2^{{}^{\prime }}\left(0\right)$ as in [10] to obtain the control points ${\mathit{Q}}_1$, ${\mathit{Q}}_2$ given in Equations (5) and (6), respectively. □

3.2. Geometric Continuity of C-Bézier Curves

Like parametric continuity, geometric continuity also helps us to construct different complex figures. It is better than parametric continuity because the scale factor gives us more smoothness.

Given two C-Bézier curves $W_1\left(t\right)$ and $W_2\left(t\right)$ of the same degree, the necessary and sufficient conditions for geometric continuity of $W_1\left(t\right)$ and $W_2\left(t\right)$ are given as follows:

1. For $G^0$ continuity:

$${\mathit{Q}}_0={\mathit{P}}_n.$$

(7)

2. For $G^1$ continuity:

$$\left\{\begin{array}{c}{\mathit{Q}}_0={\mathit{P}}_n,\hfill \\ {\mathit{Q}}_1={\mathit{P}}_n+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{\varphi {\gamma }_{0,n-1}\left({\beta }_1\right)}\left({\mathit{P}}_n-{\mathit{P}}_{n-1}\right).\hfill \end{array}\right.$$

(8)

3. For $G^2$ continuity:

$$\left\{\begin{array}{c}{\mathit{Q}}_0={\mathit{P}}_n,\hfill \\ {\mathit{Q}}_1={\mathit{P}}_n+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{\varphi {\gamma }_{0,n-1}\left({\beta }_1\right)}\left({\mathit{P}}_n-{\mathit{P}}_{n-1}\right),\hfill \\ {\mathit{Q}}_2=-\frac{{\delta }_{n-2,n-2}\left({\alpha }_{n-1}\right){\delta }_{n-2,n-1}\left({\alpha }_{n-1}\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1n-1}\left({\beta }_2\right){\varphi }^2}{\mathit{P}}_{n-2}+\left(\frac{{\delta }_{n-2,n-2}\left({\alpha }_{n-1}\right){\delta }_{n-2,n-1}\left({\alpha }_{n-1}\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right){\varphi }^2}-\right.\hfill \\ \frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right){\varphi }^2}-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\gamma }_{0,n-2}\left({\beta }_1\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)\varphi }+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right){\varphi }^2}\left.-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-1}\left({\beta }_1\right)\varphi }\right){\mathit{P}}_{n-1}+\hfill \\ \left(\frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right){\varphi }^2}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\gamma }_{0,n-2}\left({\beta }_1\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right)\varphi }-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-2}\left({\beta }_2\right){\gamma }_{1,n-1}\left({\beta }_2\right){\varphi }^2}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-1}\left({\beta }_1\right)\varphi }+1\right){\mathit{P}}_n,\hfill \end{array}\right.$$

(9)

where $\varphi$ is any positive real number.

Proof. 

For the $G^0$ continuity of C-Bézier curves, we keep both the first and second curves equal at the final and initial point of the domain, respectively, as $W_1\left(1\right)=W_2\left(0\right)$. We can obtain ${\mathit{Q}}_0={\mathit{P}}_n$. Similarly, for $G^1$ continuity conditions, we keep both the first and second curves equal at the final and initial point and we also consider the first derivative of both curves involving a scale factor such as

$$W_1\left(1\right)=W_2\left(0\right)$$

(10)

$$W_1^{\prime }\left(1\right)=\varphi W_2^{\prime }\left(0\right),\varphi >0.$$

(11)

Then, we can get:

$${\mathit{Q}}_1={\mathit{P}}_n+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\gamma }_{0,n-1}\left({\beta }_1\right)}\left({\mathit{P}}_n-{\mathit{P}}_{n-1}\right).$$

(12)

Now, for $G^2$ continuity, besides $G^1$ continuity, we need to satisfy the $G^2$ continuity condition that is the curvature of the first curve at the last point and the second curve at the first point should be equal, that is

$${\kappa }_1\left(1\right)={\kappa }_2\left(0\right),$$

therefore, the normal vector $L_1=W_1^{\prime }\left(1\right)\times W_1^{″}\left(1\right)$ of $W_1\left(t\right)$ and the normal vector $L_2=W_2^{\prime }\left(1\right)\times W_2^{″}\left(0\right)$ of $W_2\left(t\right)$ have the same direction. Therefore, these four vectors $W_1^{\prime }\left(1\right)$, $W_2^{\prime }\left(0\right)$, $W_1^{″}\left(1\right)$, $W_2^{″}\left(0\right)$ are in the same plane, so we have $W_1^{″}\left(1\right)=\xi W_2^{″}\left(0\right)+\varphi W_2^{\prime }\left(0\right)$:

$$\begin{array}{c}\hfill {\kappa }_1\left(1\right)=\frac{|W_1^{\prime }\left(1\right)\times W_1^{″}\left(1\right)|}{{\left|W_1^{\prime }\left(1\right)\right|}^3}=\frac{|\xi \varphi \left(W_2^{\prime }\left(0\right)\times W_2^{\prime }\left(0\right))\right.|}{{\varphi }^3{\left|W_2^{\prime }\left(0\right)\right|}^3}=\frac{|W_2^{\prime }\left(0\right)\times W_2^{″}\left(0\right)|}{{\left|W_2^{\prime }\left(0\right)\right|}^3}={\kappa }_2\left(0\right)\end{array}$$

(13)

and we can get $\xi ={\varphi }^2$; then, the $G^2$ continuity condition can be described as Equation (9). □

4. Examples

4.1. Algorithm for the Construction of Curves by Parametric Continuity Constraints

In this section, we present an algorithm for constructing complex curves with parametric continuity constraints, we know that smooth curves can be easily obtained by using continuity conditions, and shape parameters can be adjusted to modify the shape of curves according to our needs.

The procedure for the construction of complex figures by parametric continuity between two C-Bézier curve segments is given as follows:

I.

For C-Bézier curve of degree n, we consider the first curve with shape parameters like $W_1\left(t;{\alpha }_1,\dots ,{\alpha }_n\right)$ and its $n+1$ control points ${\mathit{P}}_0,{\mathit{P}}_1,\dots ,{\mathit{P}}_n$.

II.

For $C^0$ continuity by keeping $W_1\left(1;{\alpha }_1,\dots ,{\alpha }_n\right)=W_2\left(0;{\beta }_1,\dots ,{\beta }_n\right)$, we have new point ${\mathit{Q}}_0$ and the remaining control points are left to choice.

III.

Similarly, for $C^1$ continuous, the tangent vectors of the first curve at the end point and the second curve are equal, we obtain $W_1^{\prime }\left(1\right)=W_2^{\prime }\left(0\right)$. Therefore, the new control point ${\mathit{Q}}_1$ of the second curve is obtained, and the remaining control points of the second curve are free to choose.

IV.

Finally, for $C^2$ continuity, the $C^1$ continuity condition of the two curves is first guaranteed, and the second derivative of the initial curve and the second curve is also guaranteed to be equal at the end point, that is, $W_1^{″}\left(1\right)=W_2^{″}\left(0\right)$; then, we get the new control point ${\mathit{Q}}_2$ of the second curve, and the remaining control points of the second curve are free to choose.

Hence, by using the above algorithm, figures can be obtained by using continuity conditions. Some of the constructions of C-Bézier curve are given below.

1.

$C^1$ continuity of cubic C-Bézier curves with parameters.

Because the cubic C-Bézier curve has three shape parameters, and we can construct various figures by using the continuity of any two curves. Therefore, consider any two cubic C-Bézier curves named $W_1\left(t\right)$ and $W_2\left(t\right)$ containing shape parameters ${\alpha }_1,{\alpha }_2,{\alpha }_3$ and ${\beta }_1,{\beta }_2,{\beta }_3$, respectively:

$$\left\{\begin{array}{c}{W}_1\left(t;{\alpha }_1,{\alpha }_2,{\alpha }_3\right)={\sum }_{i=0}^3{\mathit{P}}_i{u}_{i,3}\left(t\right),t\in [0,1];\hfill \\ W_2\left(t;{\beta }_1,{\beta }_2,{\beta }_3\right)={\sum }_{j=0}^3{\mathit{Q}}_j{u}_{j,3}\left(t\right),t\in [0,1].\hfill \end{array}\right.$$

(14)

Example 1.

In Figure 2, control points ${\mathit{P}}_0=(0.04,0.2),{\mathit{P}}_1=(0.05,0.25),{\mathit{P}}_2=(0.075,0.26)$ and ${\mathit{P}}_3=(0.1,0.24)$ were selected to construct curves. Through the $C^1$ continuity condition, ${\mathit{Q}}_0$ and ${\mathit{Q}}_1$ could be obtained. The last two control points ${\mathit{Q}}_2$ and ${\mathit{Q}}_3$ could be freely selected according to our needs. All these multiple thin and dotted curves could be attained by the variation of shape parameters. The different values of shape parameters are mentioned underneath the figures.The shape parameters in the graph appear in the form of array. The first four groups $({\alpha }_1,{\alpha }_2,{\alpha }_3)$ and the last four groups $({\beta }_1,{\beta }_2,{\beta }_3)$ correspond to the curve colors in the graph: black, green, purple, and red.

2.

$C^2$ continuity of cubic C-Bézier curves with parameters.

We elaborate on the $C^2$ continuity of the curves, and, for $C^2$ continuity, we again consider two cubic C-Bézier curves given with three different shape parameters in Equation (14). We can also use the $C^2$ continuity constraints to construct various complex curves.

Example 2.

In Figure 3, control points ${\mathit{P}}_0=(0.04,0.2),{\mathit{P}}_1=(0.05,0.25),{\mathit{P}}_2=(0.075,0.26),$ and ${\mathit{P}}_3=(0.1,0.24)$ were selected to construct curves. Now, by using $C^2$ continuity conditions, the graphical representation of curves is presented. The last one control points of the second curve have to be taken according to our own choice. The different values of shape parameters are mentioned underneath the figures. The shape parameters in the graph appear in the form of array. The first four groups $({\alpha }_1,{\alpha }_2,{\alpha }_3)$ and the last four groups $({\beta }_1,{\beta }_2,{\beta }_3)$ correspond to the curve colors in the graph: black, green, purple, and red.

4.2. Algorithm for the Construction of Curves by Geometric Continuity Constraints

Like parametric continuity, we present an algorithm for constructing complex curves with geometric continuity constraints. The procedure for constructing the geometric continuity between two C-Bézier curves segments is as follows:

I.

For C-Bézier curve of degree n, we consider the first curve with shape parameters like $W_1\left(t;{\alpha }_1,\dots ,{\alpha }_n\right)$ and its $n+1$ control points.

II.

By keeping $W_1\left(1;{\alpha }_1,\dots ,{\alpha }_n\right)$ and $W_2\left(0;{\beta }_1,\dots ,{\beta }_n\right)$ equal, we obtain the control point ${\mathit{Q}}_0$ for $G^0$ continuity, i.e., ${\mathit{P}}_n={\mathit{Q}}_0$, and the remaining control points of the second curve will be chosen according to the designer’s choice.

III.

Similarly, for $G^1$ continuous, both the first and final curve segments with their tangent vectors will be equal at the last and first point of the domain, respectively. An extra positive scale factor will be added with the tangent vector of the second curve as $W_1^{\prime }\left(1\right)=\varphi W_2^{\prime }\left(0\right)$ to obtain ${\mathit{Q}}_1$ for $G^1$ continuity. The remaining control points will be left to the designer’s choice, and a new curve will be obtained smoothly by using this condition.

IV.

Finally, for $G^2$ continuity, $G^1$ continuity is first guaranteed, and then control point ${\mathit{Q}}_2$ is obtained through $W_1^{″}\left(1\right)={\varphi }^2{W}_2^{″}\left(0\right)+\varphi W_2^{\prime }\left(0\right)$. Meanwhile, the remaining control points of the second curve are freely selected.

1.

$G^1$ continuity of cubic C-Bézier curves with parameters.

Example 3.

Figure 4 depicts the graphical representation of the $G^1$ smooth continuity between two cubic C-Bézier curves (the same as defined above for parametric continuity).

 

In Figure 4, control points ${\mathit{P}}_0=(0.04,0.2),{\mathit{P}}_1=(0.05,0.25),{\mathit{P}}_2=(0.075,0.26)$ and ${\mathit{P}}_3=(0.1,0.24)$ were selected to construct curves. In addition, $\varphi$ is the scale factor, which has a positive value, and it is well worth modifying the shape of the curve. Through the $G^1$ continuity condition, ${\mathit{Q}}_0$ and ${\mathit{Q}}_1$ could be obtained, while the remaining control points would be taken according to our own will. All of these multiple thin and dotted curves could be attained by the variation of shape parameters. The different values of shape parameters are mentioned underneath the figures. The shape parameters in the graph appear in the form of array. The first four groups $({\alpha }_1,{\alpha }_2,{\alpha }_3)$ and the last four groups $({\beta }_1,{\beta }_2,{\beta }_3)$ correspond to the curve colors in the graph: black, green, purple and red, where $\varphi =0.8$ in the figure. Therefore, by varying the values of shape parameters, we can see the changes in the curves given in Figure 4.

2.

$G^2$ continuity of cubic C-Bézier curves with parameters.

Example 4.

The $G^2$ continuity of the curve has much more freedom compared to the $C^2$ continuity. Figure 5 represents the $G^2$ smooth continuity between two cubic C-Bézier curves. In this figure, the control points ${\mathit{P}}_0=(0.04,0.2),{\mathit{P}}_1=(0.05,0.25),{\mathit{P}}_2=(0.075,0.26)$ and ${\mathit{P}}_3=(0.1,0.24)$ were chosen to construct the thin colored lines of Figure 5. Now, by using $G^2$ continuity conditions, the graphical representation of curves is presented. The last control points of the second curve have to be taken according to our own choice. The different values of shape parameters are mentioned underneath the figures. The parameter ϕ in all figures is all selected as 0.8. Multiple shape parameters were used to construct the various curves given in Figure 5.

5. Geometric Continuity of C-Bézier Surface with Parameters

Similar to the classical Bézier surface, a form of C-Bézier tensor product Bézier surface with parameters can be obtained by blending a C-Bézier curve which was associated with multiple different parameters.

Definition 2.

For control mesh points ${\mathit{P}}_{i,j}\in R^3(i=0,1,2,\dots ,n,j=0,1,2,\dots ,m)$, such that the C-Bézier surface of order $n,m$ in the form of tensor product can be demonstrated as:

$$R(s,t;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)=\sum _{i=0}^n\sum _{j=0}^m{\mathit{P}}_{i,j}{u}_{i,n}\left(s\right)u_{j,m}\left(t\right),0\le s,t\le 1,m,n\ge 2$$

(15)

where $u_{i,n}\left(t\right)$ and $u_{j,m}\left(t\right)$ are C-Bézier basis functions associated with multi valued shape parameters ${\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m\in (0,\pi ]$ (${\alpha }_i,{\beta }_i\in (0,2\pi ],i\ge 3$).

For designing of complex figures and various modeling purposes, we discussed $G^2$ continuity constraints between two adjacent C-Bézier curves in Section 3. Now, it is time to elaborate $G^2$ continuity for C-Bézier surfaces. The need to reach $G^2$ continuity after $G^1$ continuity is to get high smoothness for construction of complex figures. Thus, $G^2$ continuity must possess $G^0$ and $G^1$ continuity first. In order to facilitate our discussion, we consider two $n\times m$ adjacent C-Bézier surfaces with shape parameters ${\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m$ as follows:

$$\left\{\begin{array}{c}{R}_1(s,t;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)={\sum }_{i=0}^n{\sum }_{j=0}^m{\mathit{P}}_{i,j}{u}_{i,n}\left(s\right)u_{j,m}\left(t\right),\hfill \\ R_2(s,t;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})={\sum }_{i=0}^{\hat{n}}{\sum }_{j=0}^{\hat{m}}{\mathit{Q}}_{i,j}{u}_{i,\hat{n}}\left(s\right)u_{j,\hat{m}}\left(t\right),\hfill \end{array}\right.$$

(16)

where ${\mathit{P}}_{i,j}$ and ${\mathit{Q}}_{i,j}(i=0,1,2,\dots ,n,j=0,1,2,\dots ,m)$ are the control points of C-Bézier surfaces $R_1(s,t;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)$ and $R_2(s,t;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}}),$ respectively. $G^k(k\le 2)$ continuity conditions between two adjacent C-Bézier surfaces can be discussed in two different ways as follows.

Smooth G² Continuity for C-Bézier Surfaces with Parameters in the S Direction

Theorem 1.

Consider two C-Bézier surfaces $R_1(s,t;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)$ and $R_2(s,t;{\hat{\alpha }}_1,\dots ,$${\hat{\alpha }}_{\hat{n}},$${\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})$, and they meet the $G^2$ smooth continuity condition in the s direction if it satisfies all the following continuity constraints:

$$\left\{\begin{array}{c}m=\hat{m},\hfill \\ {\beta }_1={\hat{\beta }}_1,{\beta }_2={\hat{\beta }}_2,\dots ,{\beta }_m={\hat{\beta }}_{\hat{m}},\hfill \\ {\mathit{Q}}_{i,0}={\mathit{P}}_{i,n},(i=0,1,\dots ,m),\hfill \\ {\mathit{Q}}_{i,1}={\mathit{P}}_{i,n}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{\varphi {\delta }_{0,n-1}\left(\widehat{{\alpha }_1}\right)}\left({\mathit{P}}_{i,n}-{\mathit{P}}_{i,n-1}\right),(i=0,1,\dots ,m),\hfill \\ {\mathit{Q}}_{i,2}=-\frac{{\delta }_{n-2,n-2}\left({\alpha }_{n-1}\right){\delta }_{n-2,n-1}\left({\alpha }_{n-1}\right)}{{\alpha }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}{\mathit{P}}_{i,n-2}+\left(\frac{{\delta }_{n-2,n-2}\left({\alpha }_{n-1}\right){\delta }_{n-2,n-1}\left({\alpha }_{n-1}\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}\right.\hfill \\ -\frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\delta }_{0,n-2}\left({\beta }_1\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right)\varphi }+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}\hfill \\ \left.-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-1}\left(\widehat{{\alpha }_1}\right)\varphi }\right){\mathit{P}}_{i,n-1}+\left(\frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\delta }_{0,n-2}\left(\widehat{{\alpha }_1}\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right)\varphi }\right.\hfill \\ \left.-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-1}\left(\widehat{{\alpha }_1}\right)\varphi }+1\right){\mathit{P}}_{i,n}\hfill \\ (i=0,1,\dots ,m)\hfill \end{array}\right.$$

(17)

hold, where ϕ is any positive real number.

Proof. 

When two adjacent C-Bézier surfaces satisfy the $G^2$ smooth continuity condition, the $G^0$ and $G^1$ continuity conditions must be satisfied at the joint first. In brief, two C-Bézier surfaces must have a common boundary and common tangent plane. For the $G^0$ continuity condition of C-Bézier surfaces, we have

$$R_1(s,1;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)=R_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}}).$$

After simplifying the above conditions, the following boundary points ${\mathit{Q}}_{i,0}={\mathit{P}}_{i,n}$ are obtained. Now, for the smooth continuity condition of $G^1$, any two adjacent C-Bézier surfaces have a common tangent plane at a joint point of the common boundary, i.e., there is a continuous tangential derivative at the boundary point, and the following conditions should be satisfied:

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}{R}_1(s,1;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)\times \frac{\partial }{\partial s}{R}_1(s,1;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)\\ \hfill =\varphi \frac{\partial }{\partial t}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})\times \frac{\partial }{\partial s}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}}).\end{array}$$

(18)

By further simplification according to [21], Equation (18) can be simplified as follows:

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}{R}_1(s,1;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)=\varphi \frac{\partial }{\partial t}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})\end{array}$$

(19)

where $\varphi$ is any real constant. By calculating the above values, we have

$$\begin{array}{c}\hfill {\mathit{Q}}_{i,1}={\mathit{P}}_{i,n}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{\varphi {\delta }_{0,n-1}\left(\widehat{{\alpha }_1}\right)}\left({\mathit{P}}_{i,n}-{\mathit{P}}_{i,n-1}\right),(i=0,1,\dots ,m),\end{array}$$

(20)

which are the values required by the continuous condition of smooth $G^1$ in the s direction.

Similarly, for the continuity of $G^2$ in the s direction, the two surfaces must possess the same normal curvature at common boundary [22,23] and satisfy

$$\begin{array}{cc}\hfill \frac{{\partial }^2}{\partial t^2}& R_1(s,1;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)={\varphi }^2\frac{{\partial }^2}{\partial t^2}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})\hfill \\ & +2\varphi f\left(s\right)\frac{{\partial }^2}{\partial t\partial s}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})+f^2\left(s\right)\frac{{\partial }^2}{\partial s^2}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})\hfill \\ & +\xi \frac{\partial }{\partial t}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})+h\left(s\right)\frac{\partial }{\partial s}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})\hfill \end{array}$$

(21)

where $\varphi$ and $\xi$ are any real constants, and $h\left(s\right)$ and $f\left(s\right)$ are linear functions. For our convenience and useful calculation, we consider $h\left(s\right)=f\left(s\right)=\xi =0$. Thus, Equation (21) can be simplified as follows:

$$\frac{{\partial }^2}{\partial t^2}{R}_1(s,1;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)={\varphi }^2\frac{{\partial }^2}{\partial t^2}{R}_2(s,0;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}}).$$

(22)

By calculating the above values, we have

$$\begin{array}{cc}\hfill {\mathit{Q}}_{i,2}=& -\frac{{\delta }_{n-2,n-2}\left({\alpha }_{n-1}\right){\delta }_{n-2,n-1}\left({\alpha }_{n-1}\right)}{{\alpha }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}{\mathit{P}}_{i,n-2}+\left(\frac{{\delta }_{n-2,n-2}\left({\alpha }_{n-1}\right){\delta }_{n-2,n-1}\left({\alpha }_{n-1}\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}\right.\hfill \\ & -\frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\delta }_{0,n-2}\left({\beta }_1\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right)\varphi }+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}\hfill \\ & \left.-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-1}\left(\widehat{{\alpha }_1}\right)\varphi }\right){\mathit{P}}_{i,n-1}+\left(\frac{{\delta }_{n-2,n-2}\left({\alpha }_n\right){\delta }_{n-1n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right){\delta }_{0,n-2}\left(\widehat{{\alpha }_1}\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right)\varphi }\right.\hfill \\ & \left.-\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-2}\left(\widehat{{\alpha }_2}\right){\delta }_{1,n-1}\left(\widehat{{\alpha }_2}\right){\varphi }^2}+\frac{{\delta }_{n-1,n-1}\left({\alpha }_n\right)}{{\delta }_{0,n-1}\left(\widehat{{\alpha }_1}\right)\varphi }+1\right){\mathit{P}}_{i,n}\hfill \\ & (i=0,1,\dots ,m)\hfill \end{array}$$

(23)

which satisfies the $G^2$ continuity conditions of the C-Bézier surface in the s direction and gives the values of the required control mesh points. □

6. Examples for the Construction of C-Bézier Surfaces with Parameters by G² Continuity

By using the continuity of C-Bézier surfaces, various figures can be constructed. The influences of parameters are shown in the figures. In this section, we discuss the construction of surfaces by $G^2$ continuity conditions between any two adjacent C-Bézier surfaces in the s direction (the t direction can also be discussed in a similar way).

By concluding the proof of Theorem 1, the steps are given as follows:

1.

Consider any two C-Bézier surfaces such as $R_1(s,t;{\alpha }_1,\dots ,{\alpha }_n,{\beta }_1,\dots ,{\beta }_m)$ and $R_2(s,t;{\hat{\alpha }}_1,\dots ,{\hat{\alpha }}_{\hat{n}},{\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\hat{m}})$.

2.

Let $m=\hat{m},{\beta }_1={\hat{\beta }}_1,{\beta }_2={\hat{\beta }}_2,\dots ,{\beta }_m={\hat{\beta }}_{\hat{m}},{\mathit{Q}}_{i,0}={\mathit{P}}_{i,n},(i=0,1,\dots ,m)$; both surfaces possess a common boundary and satisfy the $G^0$ continuity condition.

3.

For any value of $\varphi >0$, and by having multiple shape control parameter values, Equation (20) can be used to calculate the second row of control mesh points to meet the $G^1$ continuity requirement. The remaining control mesh points can be taken according to the designer’s choice.

4.

For any constant value of $\varphi >0$, the control mesh points in the third row can be calculated using Equation (23), which are the required control points for $G^2$ continuity. Furthermore, for the $G^2$ continuity condition, the previous two conditions ($G^0$ continuity and $G^1$ continuity) must be satisfied.

Example 5.

Consider any two adjacent C-Bézier surfaces of order $(m,n),$ where $m=n=3$. These two surfaces satisfy $G^1$ continuity conditions if they have a common boundary and common tangent plane. The first eight control points can be obtained by using the above steps. The control mesh points (as in Equation (23)) of a common boundary in Figure 6 can be obtained by using the procedure of step 1 above. Similarly, the control points for common tangent plane can also be obtained by using the third step given in step 2 above, while the remaining control points depend on designer’s choice. Different shape parameters are given under each graph and, by varying these shape parameters in their domain, the influence on the shapes can be shown (where ${\beta }_1={\hat{\beta }}_1=\frac{3\pi }{8},{\beta }_2={\hat{\beta }}_2=\frac{5\pi }{8},{\beta }_3={\hat{\beta }}_3=\frac{\pi }{8}$).

Example 6.

Figure 7 represents the $G^2$ continuity between two adjacent C-Bézier surfaces. These four figures can be obtained by varying the values of shape control parameters in their domain, and are mentioned under each figure (where ${\beta }_1={\hat{\beta }}_1=\frac{3\pi }{8},{\beta }_2={\hat{\beta }}_2=\frac{5\pi }{8},{\beta }_3={\hat{\beta }}_3=\frac{\pi }{8}$). The first 12 control mesh points can be obtained by using Equation (23), and the remaining four control mesh points can be taken according to the designer’s choice.

7. Conclusions

We know that C-Bézier curves and classical Bézier curves are very useful for image processing, graphics, and font designing. We can only draw a straight line with two points. However, when the number of control points is increased, we can obtain any curve shape. Moreover, when it comes to the modeling of complex figures and font designing, only a single C-Bézier curve is not enough. In order to solve this issue, the parametric continuity and geometric continuity conditions ($C^2$ and $G^2$ continuity) between any two C-Bézier curves are derived. The changes of figures under multiple shape parameters are also given by us. This method can obtain curves without increasing mathematical complexity, which is more valuable in practical applications.

Moreover, this paper also proposed a C-Bézier surface with the shape parameters. C-Bézier surfaces have multiple different shape parameters, so they can be modified not only by changing the control points but also by varying the values of shape parameters in their domain. Compared with other surface formation techniques, the method proposed in this study is more practical, effective, flexible, and efficient in mathematical modeling. Finally, $G^2$ continuity constraints between two adjacent C-Bézier surfaces in different directions have been presented. Surface modeling by continuity conditions has also been given.