# A Novel Generalization of Trigonometric Bézier Curve and Surface with Shape Parameters and Its Applications

Authors: Sidra Maqsood; Muhammad Abbas; Gang Hu; Ahmad Lutfi Amri Ramli; Kenjiro T. Miura
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Adopting a recurrence technique, generalized trigonometric basis (or GT-basis, for short) functions along with two shape parameters are formulated in this paper. These basis functions carry a lot of geometric features of classical Bernstein basis functions and maintain the shape of the curve and surface as well. The generalized trigonometric Bézier (or GT-Bézier, for short) curves and surfaces are defined on these basis functions and also analyze their geometric properties which are analogous to classical Bézier curves and surfaces. This analysis shows that the existence of shape parameters brings a convenience to adjust the shape of the curve and surface by simply modifying their values. These GT-Bézier curves meet the conditions required for parametric continuity (C0, C1, C2, and C3) as well as for geometric continuity (G0, G1, and G2). Furthermore, some curve and surface design applications have been discussed. The demonstrating examples clarify that the new curves and surfaces provide a flexible approach and mathematical sketch of Bézier curves and surfaces which make them a treasured way for the project of curve and surface modeling.


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## Body

## 1. Introduction

The study of curves and surfaces plays a very significant role in computer-aided geometric design (CAGD) and computer graphics (CG). CAGD deals with the composition and representation of free-form curves and surfaces. Numerous applications which demand free-form curves and surfaces arise in science and engineering, e.g., car bodies, ship hulls, airplane, and propeller blades. The use and effectiveness of curves in modeling are resolute by the type of input data and their consequence on the control of the resulting curve. Curves representation, related to the control points, flexible enough to bend and twist or change the curve shape by changing one or more control points as the designers' requirements are big challenges in curve modelling. For these reasons, parametric representation of curves and surfaces is mostly used to handle these challenges.Bézier curves and surfaces have been extensively used in CG and CAGD because of their valuable properties. Bézier curves are parametric curves and are constructed by using Bernstein polynomials as basis functions. To change the shape of the classical Bézier curves, their control points are essential to be adjusted because they have no shape parameters. In CAD/CAM technology fields, creating more suitable skills of scheming and amending of Bézier curve is an important research subject. Since traditional Bézier curves can be obtained by control points and Bernstein basis functions, after creating Bézier curves and surfaces, we can construct different shapes by using parametric and geometric continuities which fulfill our design requirements. Since shape designing is a time-consuming process, usually we cannot execute our required design in one step even though by using continuity conditions, especially, when, we are going to establish some complex curves and surfaces by the help of those Bézier curves. In order to overcome this cumbersome problem, we construct GT-Bézier curves/surfaces based on GT-basis functions with two shape parameters instead of any bunch of parameters [1]. Since trigonometric Bézier curve with the shape parameters is more continuous as compared to polynomial Bézier curve, by the variation of shape parameters, we can modify the shape according to our own choice.In practical applications, the appearance design of many products is relatively complex and cannot be illustrated by a single curve/surface. Therefore, there is a necessity to model such products using adjacent curves/surfaces with some continuities. The authors in [2] derived the G1,G2 geometric continuity conditions only for H-Bézier curves of degree n. Hu et al. [3–5] also derived the continuity conditions between generalized Bézier-like surfaces with multiple shape parameters in order to resolve the issue of shape design for complex surfaces in engineering. In industrial applications, computer-aided machines have the ability to cut the shapes like circles or helices or the shapes of many objects designed with the help of straight lines and circle arcs. Therefore, it is useful to have curves piecewisely spanned by linear polynomials, sine and cosine, i.e., cycloidal splines (also called helix splines) to approximate such objects [6].Many authors have constructed numerous varieties of Bézier curve/surface design. A class of extension of Bézier curve is constructed by Han and Liu [7], Wu et al. [8, 9], and Liu [10]. Due to the presence of one shape control parameter in all the above cited work, the control on the shape of Bézier curve has made more desirable, but still it is limited. Han et al. [11] suggested shape modification of cubic quasi-Bézier curve to enhance the shape adjustability of the curves. In [12], Qin et al. described the extension of cubic Bézier curve with different shape parameters and also its continuity conditions and applications. An extension of quartic Bézier curve with three shape parameters is presented by Zhu et al. [13], which is a continuation to a forth degree Bézier curve with a single parameter which improved the shape control of the curve. In [14], an extension of quartic Bézier curve is presented by Zhang et al. which not only inherits the outstanding properties of quadric Bézier curve but also fits the control polygon. Graphical examples with valuable design of curves and surfaces are also given in this literature. Yan and Liang [15] presented an extension of the Bézier model with all the properties. The newly created curves and surfaces by Qin et al. [16] not only have most properties of the corresponding classical Bézier curves and surfaces of order n but they also help to modify the shapes by using various shape parameters. Hu et al. [17, 18] described the rotation surfaces by using polynomial basis and shape modification of various curves by using shape parameters. The generalized B-spline (GB-spline) functions of arbitrary order having all the basic properties of the curve are presented by Ksasov and Sattayatham in [19]. Zhang and Krause [20] presented the unified trigonometric basis and the hyperbolic basis with a shape parameter. The functional B-splines (FB-splines) and subdivision B-splines (SB-splines) are also presented with a geometric proof of curvature continuity for SB-splines. Lü et al. [21] constructed the trigonometric polynomial B-spline curves in which they have many similar properties to traditional B-splines. Based on the explicit representation of the curves, the subdivision formulae for this new kind of curve are also presented in this literature. Wang et al. [22] described the subdivision formulae of the new kind of NAUT B-spline curves. The generation of tensor product surfaces by these new splines and unified and extended form of three types of splines are also studied in this work.In [23, 24], Han presented the cubic and quadratic trigonometric polynomial curves with the shape parameters. The author used them for various designing purposes with the help of continuity conditions. Nikolis and Seimenis [25] presented the special nonlinear dynamical systems by using cubic trigonometric splines. The existence of the unique spline approximation is proved, and the convergence order of the method is shown to be cubic. The methods based on various spline techniques for planning and fast modifications of a trajectory for robot manipulators are investigated by Dyllong and Visioli in [26]. Su and Zou [27] designed the manipulator trajectory using algebraic-trigonometric Hermite polynomial curves and interpolated the data points for the manipulator of these curves.Schweikert in [28] presented the use of a linearized mathematical spline for interpolation between given points which occasionally yields extraneous inflection points for some applications. Mazure [29] defined the Chebyshev–Bernstein basis as the dual bases of the linear functional giving the control points in which they shared the same properties as the Bernstein bases in polynomial spaces. Xu and Wang [30] presented the two unified mathematic models of conics and polynomial curves by AHT Bézier curves and NUAHT B-spline curves. Zhu and Liu [31] presented a class of trigonometric Bernstein-type basis functions with four different shape parameters. The continuity conditions and various modeling are also presented in this work. Mainer et al. [32] discussed several alternatives to the rational Bézier model based on using curves generated by mixing polynomial and trigonometric functions and expressing them in bases with optimal shape preserving properties (normalized B-bases). In [33], Wang and Fang unified and extended the polynomial, trigonometric, and hyperbolic splines by a new kind of UE-spline over a space.Bosner and Rogina in [34] proposed numerically stable algorithms for cycloidal splines. They developed a corner cutting algorithm for lower-order cycloidal curves through a straightforward generalization. Costantini et al. [35] presented the approximation power, the existence of a normalized B-basis, and the structure of a degree-raising process for spaces of the form requiring suitable assumptions on the functions. Mainer and Peña [36] presented the quadratic cycloidal curves associated to equally spaced knots and the properties of the generated curves for a specific domain. Gang et al. [37] constructed a unified approach, the generalized nonuniform B-splines, and studied the corresponding isogeometric analysis framework for solving the partial differential equation. The proposed frameworks have several advantages such as high accuracy and easy-to-compute derivatives and integrals due to the nonrational form when compared with the NURBS-IGA method.This paper defines the work based on the construction of new GT-basis functions of ordernn>2 which are described by taking a set of basis functions of degree 2 with two shape parameters, with identical characteristics to the classical Bernstein basis functions. As an alternative technique of representing curves/surfaces, GT-Bézier curves/surfaces not only demonstrate the valuable characteristics of Bézier curves/surfaces but also allow efficient shape modification by altering the values of shape parameters. In order to resolve the problem of not being able to construct complex curves/surfaces using a single curve/surface, we study the parametric and geometric continuity conditions for GT-Bézier curves/surfaces of degree n. The continuity conditions of C3 and G2 between two adjacent GT-Bézier curves and surfaces are proposed using terminal properties of GT-Bézier curves/surfaces of degree n. The present GT-basis functions and GT-Bézier curves/surfaces of degree n are novel for the smooth connection between two adjacent GT-Bézier curves by C3 continuity conditions and as far as we are aware, it has never been employed for thispurpose before.This paper is laid out as follows: in Section2, generalized trigonometric basis functions are constructed, and their properties are discussed. The GT-Bézier curves and surfaces are proposed with their properties in Sections 3 and 7, respectively. The geometric implication of the shape parameters will be examined in Section 4. Sections 5 and 6 provide the continuity conditions, some design applications, and examples of GT-Bézier curves and surfaces. A summarized conclusion is given in Section 8.


## 2. GT-Basis Function

We construct the GT-basis functions by using a recursive relation in this section.Definition 1.
For−1≤α,β≤1 and 0≤z≤1, the functions(1)w0,2z=1−sinπ2z1−αsinπ2z,w1,2z=1−w0,2z−w2,2z,w2,2z=1−cosπ2z1−βcosπ2z,are known as GT-basis functions of degree 2. The function wi,mzi=0,1,…,m described recursively for any integer mm≥3 as(2)wi,mz=1−sinπ2zwi,m−1z+sinπ2zwi−1,m−1z,is GT-basis function of the mth order. In situation, when i=−1 or i>m, the function wi,mz=0.


### 2.1. Properties of GT-Basis Functions

The GT-basis functions enjoy many properties as follows:(1)
Partition of unity:(3)∑i=1mwi,mz=1.(2)
Nonnegativity: forα,β∈−1,1, wi,mz≥0i=0,1,…,m.(3)
Terminal property:∀i=0,1,2,3,…,mm≥2, w0,m0=1, wi,m0=0i=1,2,…,m, wi,m1=0i=0,1,…,m−1, and wm,m1=1.(4)
Derivative at the corner points:(4)wi,m′0=−π2m1+α,i=0,π2m1+α,i=1,0,other,wi,m′1=−π21+β,i=m−1,π21+β,i=m,0,other,wi,m″0=π24m1m2+2m2+1α,i=0,−π242m1m2+22m2+1α+1−β,i=1,π24m1m2+2m2α+1−β,i=2,0,other,wi,m″1=−π24α−1,i=m−2,π24m2−2β+α−1,i=m−1,−π24m2−2β,i=m,0,other,wi,m″0=−π38m33−2m2−1+3m1m2−1α,i=0,π38m33−2m2+2m1m2m3−1+3m1m2+6m22−1α−3m21−β,i=1,−π383m1m2m3+m23m2−1α+6m21−β,i=2,π38m1m2m3+3m2m3α+3m21−β,i=3,0,other,wi,m′1=−π383m2+11+β,i=m,π386m2+11+β,i=m−1,−π383m21+β,i=m−2,0,other,where mk=m−kk=1,2,3.Using Definition 3.1, the GT-basis functions form=3,4, and 5 can be defined as follows:(1)
Form=3, we have(5)w0,3z=1−sinπ2zw0,2z,w1,3z=sinπ2zw0,2z+1−sinπ2zw1,2z,w2,3z=1−sinπ2zw2,2z+sinπ2zw1,2z,w3,3z=sinπ2zw2,2z.Figure1(a) exhibits the curves generated by cubic GT-basis functions for α,β=−1 (blue dotted), −0.5 (green), 0.5 (red), and 1 (black dashed).(2)
Form=4, we have(6)w0,4z=1−sinπ2z2w0,2z,w1,4z=1−sinπ2z2sinπ2zw0,2z+1−sinπ2zw1,2z,w2,4z=sinπ2z2w0,2z+1−sinπ2z2w2,2z+2sinπ2z1−sinπ2zw1,2z,w3,4z=sinπ2z21−sinπ2zw2,2z+sinπ2zw1,2z,w4,4z=sinπ2z2w2,2z.Figure1(b) depicts the graphs of quartic GT-basis functions with α,β=−1 (blue), −0.5 (red dotted), 0.5 (orange), and 1 (black dashed).(3)
Form=5, we have(7)w0,5z=1−sinπ2z3w0,2z,w1,5z=1−sinπ2z23sinπ2zw0,2z+1−sinπ2zw1,2z,w2,5z=1−sinπ2z3sinπ2z2w0,2z+3sinπ2z1−sinπ2zw1,2z+1−sinπ2z2w2,2z,w3,5z=sinπ2zsinπ2z2w0,2z+3sinπ2z1−sinπ2zw1,2z+31−sinπ2z2w2,2z,w4,5z=sinπ2z231−sinπ2zw2,2z+sinπ2zw1,2z,w5,5z=sinπ2z3w2,2z.Figure 1
GT-basis functions of multiple degrees with multiple shape parameters. (a)m=3. (b) m=4. (c) m=5. (d) m=10.
(a)(b)(c)(d)The graphs of5th and 10th degree GT-basis functions with shape parameters α,β=−1 (blue dotted), −0.3 (red), 0.5 (green), and 1 (black dashed) are given in Figures 1(c) and 1(d), respectively.


## 2.1. Properties of GT-Basis Functions

The GT-basis functions enjoy many properties as follows:(1)
Partition of unity:(3)∑i=1mwi,mz=1.(2)
Nonnegativity: forα,β∈−1,1, wi,mz≥0i=0,1,…,m.(3)
Terminal property:∀i=0,1,2,3,…,mm≥2, w0,m0=1, wi,m0=0i=1,2,…,m, wi,m1=0i=0,1,…,m−1, and wm,m1=1.(4)
Derivative at the corner points:(4)wi,m′0=−π2m1+α,i=0,π2m1+α,i=1,0,other,wi,m′1=−π21+β,i=m−1,π21+β,i=m,0,other,wi,m″0=π24m1m2+2m2+1α,i=0,−π242m1m2+22m2+1α+1−β,i=1,π24m1m2+2m2α+1−β,i=2,0,other,wi,m″1=−π24α−1,i=m−2,π24m2−2β+α−1,i=m−1,−π24m2−2β,i=m,0,other,wi,m″0=−π38m33−2m2−1+3m1m2−1α,i=0,π38m33−2m2+2m1m2m3−1+3m1m2+6m22−1α−3m21−β,i=1,−π383m1m2m3+m23m2−1α+6m21−β,i=2,π38m1m2m3+3m2m3α+3m21−β,i=3,0,other,wi,m′1=−π383m2+11+β,i=m,π386m2+11+β,i=m−1,−π383m21+β,i=m−2,0,other,where mk=m−kk=1,2,3.Using Definition 3.1, the GT-basis functions form=3,4, and 5 can be defined as follows:(1)
Form=3, we have(5)w0,3z=1−sinπ2zw0,2z,w1,3z=sinπ2zw0,2z+1−sinπ2zw1,2z,w2,3z=1−sinπ2zw2,2z+sinπ2zw1,2z,w3,3z=sinπ2zw2,2z.Figure1(a) exhibits the curves generated by cubic GT-basis functions for α,β=−1 (blue dotted), −0.5 (green), 0.5 (red), and 1 (black dashed).(2)
Form=4, we have(6)w0,4z=1−sinπ2z2w0,2z,w1,4z=1−sinπ2z2sinπ2zw0,2z+1−sinπ2zw1,2z,w2,4z=sinπ2z2w0,2z+1−sinπ2z2w2,2z+2sinπ2z1−sinπ2zw1,2z,w3,4z=sinπ2z21−sinπ2zw2,2z+sinπ2zw1,2z,w4,4z=sinπ2z2w2,2z.Figure1(b) depicts the graphs of quartic GT-basis functions with α,β=−1 (blue), −0.5 (red dotted), 0.5 (orange), and 1 (black dashed).(3)
Form=5, we have(7)w0,5z=1−sinπ2z3w0,2z,w1,5z=1−sinπ2z23sinπ2zw0,2z+1−sinπ2zw1,2z,w2,5z=1−sinπ2z3sinπ2z2w0,2z+3sinπ2z1−sinπ2zw1,2z+1−sinπ2z2w2,2z,w3,5z=sinπ2zsinπ2z2w0,2z+3sinπ2z1−sinπ2zw1,2z+31−sinπ2z2w2,2z,w4,5z=sinπ2z231−sinπ2zw2,2z+sinπ2zw1,2z,w5,5z=sinπ2z3w2,2z.Figure 1
GT-basis functions of multiple degrees with multiple shape parameters. (a)m=3. (b) m=4. (c) m=5. (d) m=10.
(a)(b)(c)(d)The graphs of5th and 10th degree GT-basis functions with shape parameters α,β=−1 (blue dotted), −0.3 (red), 0.5 (green), and 1 (black dashed) are given in Figures 1(c) and 1(d), respectively.


## 3. Construction of GT-Bézier Curves

Definition 2.
For any given control pointsPi∈R2 or R3i=0,1,…,m and shape parameters α,β, the GT-Bézier curve can be constructed as(8)Sz=∑i=0mPiwi,mz,where wi,mz are GT-basis functions (2).


### 3.1. Properties of GT-Bézier Curves

In this section, we examine the geometric properties of GT-Bézier curves which are identical to classical Bézier curves.(1)
Terminal properties:(9)S0=P0,S1=Pm,S′0=π2m1+αP1−P0,S′1=π21+βPm−Pm−1.Ifm=3,(10)S″0=π2421+2αP0−5+6α−βP1+3+2α−βP2,S″1=π241−αP1+α−2βP2−1−2βP3,S′0=π382−5αP0+1+11α−3βP1−32+2α−2βP2+3−3βP3,S′1=π381+β7P2−3P1−4P3.Ifm≥3,(11)S″0=π24m1m2+2m2αP2−2P1+P0+1−βP2−P1−2αP1−P0,S″1=π24α−1Pm−1−Pm−2−m2−2βPm−Pm−1,S″0=π38m33−2m2−1+3m1m2−1αP1−P0+3m21−βP3−2P2+P1+m1m2m3P3−3P2+2P1+m2α3m3P3−3m2−1P2+6m2P1,S″1=π381+βPm−1−Pm−2−3m21+βPm−2Pm−1+Pm−2.(2)
Convex hull property: the whole GT-Bézier curve segment absolutely lies inward its control polygon.(3)
Geometric invariance: the shape of the GT-Bézier curve does not depend on the choice of the chosen coordinate. This property can be visualized by Figure2. Figure 2(a) shows that black is the first curve Sz with control points Pi, and the blue curve is obtained by adding a point O′=3,9 in Sz. Yellow-dotted circle on the blue circle is obtained by the control points Pi+O′. Similarly, Figure 2(b) shows that black is the first curve Sz with control points Pi, the blue curve is represented by the control points Pi∗Y′, and the yellow-dotted curve printed on the blue curve is obtained by multiplying Sz with a matrix Y′=1231.Figure 2
Geometric invariance of GT-Bézier curves. (a) Geometric invariance with respect to addition. (b) Geometric invariance with respect to multiplication.
(a)(b)


## 3.1. Properties of GT-Bézier Curves

In this section, we examine the geometric properties of GT-Bézier curves which are identical to classical Bézier curves.(1)
Terminal properties:(9)S0=P0,S1=Pm,S′0=π2m1+αP1−P0,S′1=π21+βPm−Pm−1.Ifm=3,(10)S″0=π2421+2αP0−5+6α−βP1+3+2α−βP2,S″1=π241−αP1+α−2βP2−1−2βP3,S′0=π382−5αP0+1+11α−3βP1−32+2α−2βP2+3−3βP3,S′1=π381+β7P2−3P1−4P3.Ifm≥3,(11)S″0=π24m1m2+2m2αP2−2P1+P0+1−βP2−P1−2αP1−P0,S″1=π24α−1Pm−1−Pm−2−m2−2βPm−Pm−1,S″0=π38m33−2m2−1+3m1m2−1αP1−P0+3m21−βP3−2P2+P1+m1m2m3P3−3P2+2P1+m2α3m3P3−3m2−1P2+6m2P1,S″1=π381+βPm−1−Pm−2−3m21+βPm−2Pm−1+Pm−2.(2)
Convex hull property: the whole GT-Bézier curve segment absolutely lies inward its control polygon.(3)
Geometric invariance: the shape of the GT-Bézier curve does not depend on the choice of the chosen coordinate. This property can be visualized by Figure2. Figure 2(a) shows that black is the first curve Sz with control points Pi, and the blue curve is obtained by adding a point O′=3,9 in Sz. Yellow-dotted circle on the blue circle is obtained by the control points Pi+O′. Similarly, Figure 2(b) shows that black is the first curve Sz with control points Pi, the blue curve is represented by the control points Pi∗Y′, and the yellow-dotted curve printed on the blue curve is obtained by multiplying Sz with a matrix Y′=1231.Figure 2
Geometric invariance of GT-Bézier curves. (a) Geometric invariance with respect to addition. (b) Geometric invariance with respect to multiplication.
(a)(b)


## 4. Geometric Implication of Shape Parameters

### 4.1. Correlation amid Shape Parameters and Stationary Points on the Curves

From the definition of GT-Bézier curve (8), we confirm that it is a linear function for every shape parameter, and(12)∂Sz∂α=m−2i1−sinπ2zm−1−isinπ2zi+1Pi+1−Pi,i=0,1,…,m−2,∂Sz∂β=m−2i−1cosπ2z1−cosπ2z1−sinπ2zm−1−isinπ2zi−1Pi−Pi+1,i=1,2,…,m−1.Therefore, there is no relationship among∂Sz/∂α and α and ∂Sz/∂β and β. Modifying one shape parameter α or β, the point Sz on the curve changed linearly for an unmovable control polygon and defined value of z. The route modification is as follows:(13)PiPi+1,i=0,1,…,m−2forα,Pi+1Pi,i=1,2,…,m−1forβ.Figures3 and 4 depict the graphs of cubic and quartic GT-Bézier curves, respectively. Points marked on the cubic GT-Bézier curves relate to S0.2 in red, S0.4 in blue, S0.6 in black, and S0.8 in green, as well as on quartic GT-Bézier curves correlate to S0.2 in orange, S0.4 in blue, S0.6 in red, and S0.8 in green. From these figures, it can be concluded that, by just altering one shape parameter, the points on these curves change in a linear manner.Figure 3
Modifying sequel of shape parameters on the cubic GT-Bézier curve. (a)α=−1. (b) α=1. (c) α=β. (d) β=−1.
(a)(b)(c)(d)Figure 4
Effect of shape parameters on quartic GT-Bézier curves. (a)α=β. (b) α=0.5. (c) β=−0.5. (d) α=−1.
(a)(b)(c)(d)


### 4.2. Affiliation among the Shape Parameters and the Shape of the Curves

The above defined characteristics of the GT-basis functions make the shape of GT-Bézier curves extraordinarily easier to modify. The appearance of the GT-Bézier curves can be attuned by changing the values of shape parameters. Figure5 displays the graphs of cubic, quartic, and quintic GT-Bézier curves. Figure 5(a) shows the cubic GT-Bézier curves with α,β=−0.5,−1 (purple), 0.5,−0.5 (blue dotted), 1,0 (blue), 1,0.5 (red dashed), 1,1black. The second flower presented in Figure 5(b) is designed by cubic GT-Bézier curves when α,β=0,0 (red), 1,1 (green), −1,−1 (blue), 0.5,0.5 (black dashed), −0.5,−0.5red dotted. Figure 5(c) shows the flower created by quartic GT-Bézier curves with shape parameters α,β=−1,−1 (red dashed), −1,−0.5 (blue), 1,1 (black dotted), −1,0.5 (blue dashed), −1,1orange. By using different values of shape parameters, α,β=0,0 (black dotted), 1,1 (red), −1,−1 (blue dashed), 0.5,0.5 (orange), −0.5,−0.5purple, the quintic GT-Bézier flowers are generated in Figure 5(d). Figure 6(a) represents the shape modification of an apple by changing the values of shape parameters as for α,β=1,1 (purple), 0.5,0.5 (blue dotted), 0.0 (red dashed dashed), −0.5,−0.5black, whereas Figure 6(b) illustrates different shapes of the butterfly by applying these values of shape parameters α,β=0.9,0.9 (black), 0.5,0.5 (orange), 0.1,0.1blue.Figure 5
Flower design by GT-Bézier curves with different shape parameters. (a)m=3. (b) m=3. (c) m=4. (d) m=5.
(a)(b)(c)(d)Figure 6
GT-Bézier curve design with different shape parameters.
(a)(b)


## 4.1. Correlation amid Shape Parameters and Stationary Points on the Curves

From the definition of GT-Bézier curve (8), we confirm that it is a linear function for every shape parameter, and(12)∂Sz∂α=m−2i1−sinπ2zm−1−isinπ2zi+1Pi+1−Pi,i=0,1,…,m−2,∂Sz∂β=m−2i−1cosπ2z1−cosπ2z1−sinπ2zm−1−isinπ2zi−1Pi−Pi+1,i=1,2,…,m−1.Therefore, there is no relationship among∂Sz/∂α and α and ∂Sz/∂β and β. Modifying one shape parameter α or β, the point Sz on the curve changed linearly for an unmovable control polygon and defined value of z. The route modification is as follows:(13)PiPi+1,i=0,1,…,m−2forα,Pi+1Pi,i=1,2,…,m−1forβ.Figures3 and 4 depict the graphs of cubic and quartic GT-Bézier curves, respectively. Points marked on the cubic GT-Bézier curves relate to S0.2 in red, S0.4 in blue, S0.6 in black, and S0.8 in green, as well as on quartic GT-Bézier curves correlate to S0.2 in orange, S0.4 in blue, S0.6 in red, and S0.8 in green. From these figures, it can be concluded that, by just altering one shape parameter, the points on these curves change in a linear manner.Figure 3
Modifying sequel of shape parameters on the cubic GT-Bézier curve. (a)α=−1. (b) α=1. (c) α=β. (d) β=−1.
(a)(b)(c)(d)Figure 4
Effect of shape parameters on quartic GT-Bézier curves. (a)α=β. (b) α=0.5. (c) β=−0.5. (d) α=−1.
(a)(b)(c)(d)


## 4.2. Affiliation among the Shape Parameters and the Shape of the Curves

The above defined characteristics of the GT-basis functions make the shape of GT-Bézier curves extraordinarily easier to modify. The appearance of the GT-Bézier curves can be attuned by changing the values of shape parameters. Figure5 displays the graphs of cubic, quartic, and quintic GT-Bézier curves. Figure 5(a) shows the cubic GT-Bézier curves with α,β=−0.5,−1 (purple), 0.5,−0.5 (blue dotted), 1,0 (blue), 1,0.5 (red dashed), 1,1black. The second flower presented in Figure 5(b) is designed by cubic GT-Bézier curves when α,β=0,0 (red), 1,1 (green), −1,−1 (blue), 0.5,0.5 (black dashed), −0.5,−0.5red dotted. Figure 5(c) shows the flower created by quartic GT-Bézier curves with shape parameters α,β=−1,−1 (red dashed), −1,−0.5 (blue), 1,1 (black dotted), −1,0.5 (blue dashed), −1,1orange. By using different values of shape parameters, α,β=0,0 (black dotted), 1,1 (red), −1,−1 (blue dashed), 0.5,0.5 (orange), −0.5,−0.5purple, the quintic GT-Bézier flowers are generated in Figure 5(d). Figure 6(a) represents the shape modification of an apple by changing the values of shape parameters as for α,β=1,1 (purple), 0.5,0.5 (blue dotted), 0.0 (red dashed dashed), −0.5,−0.5black, whereas Figure 6(b) illustrates different shapes of the butterfly by applying these values of shape parameters α,β=0.9,0.9 (black), 0.5,0.5 (orange), 0.1,0.1blue.Figure 5
Flower design by GT-Bézier curves with different shape parameters. (a)m=3. (b) m=3. (c) m=4. (d) m=5.
(a)(b)(c)(d)Figure 6
GT-Bézier curve design with different shape parameters.
(a)(b)


## 5. Continuity of GT-Bézier Curves

The continuity conditions for connecting two GT-Bézier curve segments are described as follows:Lemma 1.
(see [16]). For two given GT-Bézier curves Sz=∑i=0mPiwi,mz and S1z=∑j=0nP1jwj,nz with control points P0,P1,P2,…,Pm, m≥3, and P10,P11,P12,…,P1n, n≥3, respectively, the necessary and sufficient constraints for parametric continuity are given by(1)
Pm=P10 for C0 continuity(2)
Pm=P10 and S′1=S1′0 for C1 continuity(3)
Pm=P10, S′1=S1′0, and S″1=S1″0 for C2 continuity(4)
Pm=P10, S′1=S1′0, S″1=S1″0, and S″′1=S1″′0 for C3 continuityLemma 2.
(see [16]). Given control points P0,P1,P2,…,Pm, m≥3, and P10,P11,P12,…,P1n, n≥3, the necessary and sufficient constraints for connecting two GT-Bézier curve segments Sz=∑i=0mPiwi,mz and S1z=∑j=0nP1jwj,nz are defined by(1)
ForG0 continuity: Pm=P10.(2)
ForG1 continuity: Pm=P10 and S′1=γS1′0, γ>0.(3)
ForG2 continuity: Pm=P10, S′1=γS1′0, γ>0, and the curvature(14)κ1=S′1×S″1S′13=S1′0×S1″0S1′03=κ10.
Parametric and geometric continuity constraints for connecting two GT-Bézier curve segments are given in the following two theorems.Theorem 1.
For two GT-Bézier curve segmentsSz;α,β=∑i=0mPiwi,mz and S1z;α1,β1=∑j=0nP1jwj,nz with control points P0,P1,P2,…,Pm,m≥3, and P10,P11,P12,…,P1n,n≥3, respectively, the necessary and sufficient constraints of parametric continuity are derived as(1)
C0 continuity, P10=Pm.(2)
ForC1 continuity,(15)P10=Pm,P11=Pm+1+βPm−Pm−1n1+α1.(3)
ForC2 continuity,(16)P10=Pm,P11=Pm+1+βPm−Pm−1n1+α1,P12=Pm+1n1n2+2n2α1+1−β1α−1Pm−1−Pm−2−m2−2β−a2n1n2+2n2−12α1+1−β1Pm−Pm−1.(4)
ForC3 continuity,(17)P10=Pm,P11=Pm+1+βPm−Pm−1n1+α1,P12=Pm+1n1n2+2n2α1+1−β1α−1Pm−1−Pm−2−m2−2β−a2n1n2+2n2−12α1+1−β1Pm−Pm−1,P13=Pm+1a11+βPm−1−Pm−2−3m21+βPm−2Pm−1+Pm−2+aa2−2n1n2n3−3n21−β1+6n22α1Pm−Pm−1+6n21−β1+3n1n2n3+n23n2−1α1a3,where a=1+β/n1+α1, a1=n231−β1+n1n3+3n3α1, a2=n33−2n2−1+3n1n2−1α, a3=1/n1n2+2n2α1+1−β1α−1Pm−1−Pm−2−m2−2β−a2n1n2+2n2−12α1+1−β1Pm−Pm−1, and nk=n−kk=1,2,3.Proof.
(1)
C0 continuity is simple and straightforward from S1=S10.(2)
FromC0 continuity condition, Pm=P10 and S′1=S1′0. It is simple to obtain the required value of control P11 from S′1=π/21+βPm−Pm−1,S1′0=π/2n1+α1P11−P10.(3)
C2 continuity condition is C1 continuity condition in addition with S″1=S1″0, so from Pm=P10,P11=Pm+1+βPm−Pm−1/n1+α1, and end constraints (10), C2 continuity constraint (16) is achieved.(4)
C3 continuity can be achieved by using C2 continuity conditions along with S″′1=S″′0. From the terminal properties (9)–(11) of the GT-Bézier curves, C3 continuity condition (17) is obtained.Theorem 2.
For two segments of GT-Bézier curvesSz;α,β=∑i=0mPiwi,mz and S1z;α1,β1=∑j=0nP1jwj,nz with control points P0,P1,P2,…,Pm,m≥3, and P10,P11,P12,…,P1n,n≥3, the necessary and sufficient constraints of geometric continuity are defined as(1)
G0 continuity, P10=Pm.(2)
ForG1 continuity,(18)P10=Pm,P11=Pm+1+βPm−Pm−1n1+α1γ.(3)
ForG2 continuity,(19)P10=Pm,P11=Pm+1+βPm−Pm−1n1+α1γ,P12=Pm+1γ2n1n2+2n2α1+1−β1α−1Pm−1−Pm−2−m2−2β−b2n1n2+2n2−12α1+1−β1−2λπn1+α1Pm−Pm−1,where b=1+β/γn1+α1.Proof.
(1)
Using straightforward computations,G0 continuity is obvious.(2)
G1 continuity condition can be achieved if P10=Pm and S′1=γS1′0, γ>0. Using end constraints S′1=π/21+βPm−Pm−1 and, S1′0=π/2n1+α1P11−P10. From the terminal properties of the GT-Bézier curves, G1 continuity condition (18) is obtained.(3)
If both GT-Bézier curvesSz;α,β and S1z;α1,β1 connect by G2 continuity, they need to connect first by G1 at a common joint, which means(20)S1;α,β=Pm=Q0=S10;α1,β1,S′1=γS1′0,γ>0.
Suppose that the vice normal vectorN1=S′1×S″1 for Sz at z=1 and reverse normal vector N2=S1′0×S1″0 of S1z; then, we have(21)N1=S′1×S″1,N2=S1′0×S1″0.
Both vice normal vectorsN1 and N2 must have the same direction at the joint to achieve the G2 continuity. Four vectors S′1, S″1, S1′0, and S1″0 become coplanar by combining equations (20) and (21). Thus, we obtain(22)S″1=δS1″0+λS1′0,δ>0.
AsG2 continuity is archived, if the curvatures κ1t and κ2t for the GT-Bézier curves Sz;α,β and S1z;α1,β1, respectively, have the same value at the joint point, i.e., κ11=κ20, using equations (20)–(22), we obtain(23)κ11=S′1×S″1S′13=γS1′0×δS1″0+λS1′0γ3S1′03=δS1′0×S1″0γ2S1′03=κ20.
From equation (23), δ=γ2, and substituting this value into equation (22), we have(24)S″1=γ2S1″0+λS1′0.
From the terminal properties of the GT-Bézier curves,G2 continuity condition (19) is obtained.


## 6. Examples and Discussions

Some examples and discussions about parametric and geometric continuity of GT-Bézier curves are given in this section.(1)
Figure7(a) illustrates the C1 continuity of GT-Bézier curves of order 3, 4, and 5, respectively, with control points P0=0.1,0.4, P1=0.15,0.75, P2=0.25,0.9, P3=0.5,0.9, P4=0.5,0.6=P10, P11=0.5,0.4, P12=0.65,0.3, P13=0.8,0.35, P14=0.9,0.5, P15=0.85,0.7=P20, P21=0.8,0.9, P22=0.95,1.05, and P23=1.1,0.8 for different values of shape parameters α,β. Applying C1 continuity conditions (15) on these GT-Bézier curves, points P11 and P21 for every curve can be derived as 0.5,0.48, 0.5,0.5, 0.5,0.25, and 0.5,0.557 and 0.817,0.83, 0.82,0.82, 0.825,0.8, and 0.833,0.766, respectively, with decreasing values of α,β.(2)
For given control pointsP0=0,0, P1=1,0.8, P2=2,1, P3=3,0.5=P10, and P13=1,−2, Figure 7(b) represents C2 continuity of two cubic GT-Bézier curve segments. Using C2 continuity constraints (16) of two GT-Bézier curves, the calculated control points P11 and P12 for each curve according to the decreasing values of β are 3.66,0.16, 3.6,0.2, 3.45,0.27, and 3.36,0.3 and 4.9,−0.46, 4.5,−0.22, 4,−0.015, and 3.6,0.17, respectively.(3)
Figure7(c) displays the G2 continuity of cubic and quartic GT-Bézier curves with control points P0=0.1,0.4, P1=0.2,0.75, P2=0.35,0.9, P3=0.55,0.9, P4=0.7,0.6=P10, and P13=0.3,−0.2 and shape parameters α,β. Using conditions (19) of G2 continuity from Theorem 2, the computed control points P11 and P12 for each curve corresponding to the increasing value of γ are given as 0.89,0.2, 0.83,0.33, 0.79,0.4, 0.78,0.44, and 0.76,0.46 and 1.09,−0.23, 1,−0.03, 0.94,0.1, 0.89,0.2, and 0.86,0.26, respectively.(4)
Two cubic GT-Bézier curves exhibitC3 continuity in Figure 7(d) along with control points P0=0,0, P1=1,2, P2=2,2, and P3=3,1=P10 and shape parameters α,β. Using conditions (17) of C3 continuity from Theorem 1, calculated control point P11 for each curve is 3.8,0.2, 3.79,0.21, 3.78,0.22, 3.77,0.23, and 3.76,0.24, point P12 is 5,−1, 4.8,−0.86, 4.6,−0.7, 4.8,−0.5, and 4.3,−0.41, and point P13 is 7,1, 6.1,0.75, 5.6,0.77, 5,0.73, and 4.7,0.733 correspondingly.Figure 7
Parametric and geometric GT-Bézier continuity curves of different degrees. (a)C1 continuity. (b) C2 continuity. (c) G2 continuity. (d) C3 continuity.
(a)(b)(c)(d)


## 7. GT-Bézier Surfaces

### 7.1. Formation of the GT-Bézier Surface

The GT-Bézier surfaces of orderm×n for given points Pi,j∈R2 or R3i=0,1,…,m,j=0,1,…,n with shape parameters α,α1,β,β1 are defined as(25)Wm,nz,z1=∑i=0m∑j=0nPi,jwi,mzwj,nz1,z,z1∈0,1,where wi,mz,wj,nz1i=0,1,…,m,j=0,1,…,n are GT-basis functions (2). Figures 8(a)–8(d) represent the GT-Bézier vine glass, GT-Bézier vase diagrams, and GT-Bézier swept surface of different orders but with fixed values of shape parameters, whereas Figures 9(a)–9(c) demonstrate the GT-Bézier surfaces of different degrees with different values of shape parameters.Figure 8
GT-Bézier surfaces of different degrees. (a) 5th order GT-Bézier surface. (b)5×6th order GT-Bézier surface. (c) 8×5th order GT-Bézier surface. (d) 6th order GT-Bézier surface.
(a)(b)(c)(d)Figure 9
GT-Bézier surfaces of different degrees with multiple shape parameters. (a) Cubic GT-Bézier surfaces. (b) Quartic GT-Bézier surfaces. (c) Quintic GT-Bézier surfaces.
(a)(b)(c)


### 7.2.G2 Continuity of GT-Bézier Surfaces

In engineering applications,G2 continuity between two adjacent surface patches is required, that is, at every point where the curved surfaces meet, the surfaces must have a combined tangent plane or a collective normal line to the surface [3–5, 38]. There exists continuity in different four types of directions between two GT-Bézier surfaces Wm1,n1z,z1;α,α1,β,β1=∑i=0m1∑j=0n1Pi,jwi,m1zwj,n1z1 and W1m2,n2z,z1;α∗,α1∗,β∗,β1∗=∑k=0m2∑l=0n2P1k,lwk,m2zwl,n2z1.


### 7.3. Continuity inz Direction on Every Surface

Two regions of GT-Bézier surfaces should have a collective edge curve to attainG1 continuity, i.e.,(26)Wm1,n1z,1;α,α1,β,β1=W1m2,n2z,0;α∗,α1∗,β∗,β1∗,which results in(27)∑i=0m1Pi,n1wi,m1z;α,β=∑k=0m2P1k,0wk,m2z;α∗,β∗,implying(28)∑i=0mPi,n1wi,mz;α,β=∑i=0mP1i,0wi,mt;α∗,β∗.Ifα=α∗,β=β∗, then(29)Pi,n1=P1i,0.The normal direction at the boundary is uninterrupted for two regions of the surfaces, i.e., the joining boundary should have a mutual tangent plane to meet theG1 continuity between two regions of the surfaces. So, we have the following condition:(30)∂∂zWm,n1z,1;α,α1,β,β1×∂∂z1Wm,n1z,1;α,α1,β,β1=fz1∂∂zW1m,n2z,0;α∗,α1∗,β∗,β1∗×∂∂z1,W1m,n2t,0;α∗,α1∗,β∗,β1∗,where fz1 is the scaling factor among their normal vectors such that fz1>0. Equivalently,(31)∂∂z1Wm,n1z,1;α,α1,β,β1=f⋅∂∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗.Since(32)∂∂z1Wm,n1z,1;α,α1,β,β1=π2∑i=0mwi,mz;α,β1+β1Pi,n1−Pi,n1−1,(33)∂∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗=π2∑i=0mwi,mz;α∗,β∗n2−1+α1∗P1i,1−P1i,0.Substituting (32) and (33) into equation (31), we have(34)π2∑i=0mwi,mz;α,β1+β1Pi,n1−Pi,n1−1=f⋅π2∑i=0mwi,mz;α∗,β∗n2−1+α1∗P1i,1−P1i,0.Ifα=α∗,β=β∗, then(35)1+β1Pi,n1−Pi,n1−1=f⋅n2−1+α1∗P1i,1−P1i,0.Continuity conditions on each surface are described in equations (29) and (35) which are the general conditions for G1 continuity in z direction. In accumulation, under G1 continuity condition, the two surfaces need to retain the similar normal curvature at any point on their joint boundary, so the two surfaces also have to fulfil the following G2 continuity constraints:(36)∂2∂z12Wm,n1z,1;α,α1,β,β1=f2∂2∂z12W1m,n2z,0;α∗,α1∗,β∗,β1∗+2fgz∂∂z∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗+g2z∂2∂z1∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗+c∂∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗+dz∂∂zW1m,n2z,0;α∗,α1∗,β∗,β1∗,where gz and dz are linear functions of z and c is a random constant. For the simplification of operation and calculation in applied submissions, generally, set gz=dz=c=0 [32], so the above equation can be further simplified as(37)∂2∂z12Wm,n1z,1;α,α1,β,β1=f2∂2∂z12W1m,n2z,0;α∗,α1∗,β∗,β1∗.Since(38)∂2∂z12Wm,n1z,1;α,α1,β,β1=π24∑i=0mwi,mz;α,β1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1,(39)∂2∂z12W1m,n2z,0;α∗,α1∗,β∗,β1∗=π24∑i=0mwi,mz;α∗,β∗n2−1n2−2+2α1∗P1i,0−P1i,1−n2−1n2−2+2n2−2α1∗+1−β1∗P1i,1−P1i,2.Substituting (38) and (39) into equation (37), we get(40)π24∑i=0mwi,mz;α,β1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2π24∑i=0mwi,mz;α∗,β∗n2−1n2−2+2α1∗P1i,0−P1i,1−n2−1n2−2+2n2−2α1∗+1−β1∗P1i,1−P1i,2.Ifα=α∗,β=β∗, we have(41)1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2n2−1n2−2+2α1∗P1i,0−P1i,1−n2−1n2−2+2n2−2α1∗+1−β1∗P1i,1−P1i,2.To sum up, if the two surfacesWm,n2z,1;α,α1,β,β1 and W1m,n2z,0;α∗,α1∗,β∗,β1∗ satisfy (29), (35), and (41) at the same time, the G2 continuity in z direction is achieved.


### 7.4. Continuity inz and z1 Direction

The condition of holding a common boundary when the surfaceWm1,n1z,z1;α,α1,β,β1 has continuity in z direction and the surface W1m2,n2z,z1;α∗,α1∗,β∗,β1∗ has continuity in z1 direction is(42)Wm1,n1z,1;α,α1,β,β1=W1m2,n20,z1;α∗,α1∗,β∗,β1∗,or(43)∑i=0m1Pi,n1wi,m1z;α,β=∑l=0n2P10,lwl,n2z1;α1∗,β1∗,implying(44)∑i=0mPi,n1wi,mz;α,β=∑i=0mP10,iwi,mz1;α1∗,β1∗.Ifα=α1∗,β=β1∗, then equation (44) is further simplified as(45)Pi,n1=P10,i.For having a common tangent plane, we have the following condition:(46)∂∂z1Wm1,n1z,1;α,α1,β,β1=f⋅∂∂zW1m2,n20,z1;α∗,α1∗,β∗,β1∗.Since(47)∂∂z1Wm1,n1z,1;α,α1,β,β1=π2∑i=0m1wi,m1z;α,β1+β1Pi,n1−Pi,n1−1,(48)∂∂zW1m2,n20,z1;α∗,α1∗,β∗,β1∗=π2∑l=0n2wl,n2z1;α1∗,β1∗m2−1+α∗P11,l−P10,l.Substituting (47) and (48) into equation (46), we obtain(49)π2∑i=0m1wi,m1z;α,β1+β1Pi,n1−Pi,n1−1=π2∑l=0n2P10,lwl,n2z1;α1∗,β1∗m2−1+α∗P11,l−P10,l.Ifα=α1∗,β=β1∗, we have(50)1+β1Pi,n1−Pi,n1−1=f⋅m2−1+α∗P11,i−P10,i.Hence, the general conditions forG1 continuity in z and z1 direction are described in equations (41) and (46); when α=α1∗,β=β1∗, then the continuity conditions are defined in equations (45) and (50).Moreover, fromG1 smooth continuity conditions, the two surfaces are also essential to have the same normal curvature at each point on their shared boundary, so for G2 continuity, they also required to satisfy(51)∂2∂z12Wm,n1z,1;α,α1,β,β1=f2∂2∂z2W1m,n20,z1;α∗,α1∗,β∗,β1∗.Substituting the boundary surfaces into equation (53), we get(52)π24∑i=0mwi,mz;α,β1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2π24∑i=0mwi,mz1;α1∗,β1∗m2−1m2−2+2α∗P10,i−P11,i−m2−1m2−2+2m2−2α∗+1−β∗P11,i−P12,i.Whenα=α1∗,β=β1∗, equation (52) yields(53)1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2m2−1m2−2+2α∗P10,i−P11,i+m2−1m2−2+2m2−2α∗+1−β∗P12,i−P11,i.Finally, if the two surfacesWm,n2z,1;α,α1,β,β1 and W1m,n20,z1;α∗,α1∗,β∗,β1∗ satisfy (45), (50), and (53), then both surfaces are connected by G2 continuity in z and z1 direction.


### 7.5. Continuity inz1 Direction

TheG1 and G2 continuity in z1 direction of two GT-Bézier surfaces is achieved in a similar fashion as the continuity in z direction of two GT-Bézier surfaces, which is discussed in Section 7.3.


### 7.6. Continuity inz1 and z Direction

TheG1 and G2 continuity in direction z1 and z is obtained in a similar way as the continuity in z and z1 direction which is proved in Section 7.4.TheG2 continuity between two GT-Bézier surfaces is also analyzed as for two GT-Bézier surfaces, the continuity in z direction is achieved; when α=α∗, β=β∗, then the G1 continuity between two regions of GT-Bézier surfaces has m1=m2=m mutual control points, and the control points Pi,n1−1,Pi,n1orP1i,0,P1i,1 are taken and varied in order. For continuity in z and z1 direction, when α=α1∗, β=β1∗, then the G2 continuity between two regions of GT-Bézier surfaces has m1=n2 mutual control points, and the control points Pi,n1−1,Pi,n1orP10,i,P11,i are in the same straight line and changeable in order.Figures10–15 depict the G2 continuity surfaces between two GT-Bézier surfaces of different degrees in z direction for different shape parameters and scaling factor f. For the fixed values of shape parameters, the control points P11,i (or P12,i) move closer to (away from) the control points P10,iorP11,i when thevalues of the scaling factor f increases (or decreases), and when f is fixed and the values of shape parameters are increased (or decreased), the surfaces move closer to (away from) the control polygon (control net). This analysis shows that the piecewise GT-Bézier surface is smooth and continuous at the joint, which significantly increases the abilities to solve problems in engineering appearance design by amending the position and shape of surfaces.Figure 10
The effect of changing scaling factor on cubic GT-Bézier surfaces. (a)f=0.75,α,β,α∗,β∗=1. (b) f=1,α,β,α∗,β∗=1. (c) f=1.5,α,β,α∗,β∗=1. (d) f=4,α,α∗=1,β,β∗=1.
(a)(b)(c)(d)Figure 11
The effect of shape parameters on cubic GT-Bézier surfaces. (a)f=2,α,α∗=0.5,β,β∗=1. (b) f=2,α,α∗=0.5,β,β∗=0.85. (c) f=2,α,α∗=−0.5,β,β∗=0.65. (d) f=2,α,α∗=−0.35,β,β∗=0.9.
(a)(b)(c)(d)Figure 12
The effect of changing scaling factor on quartic GT-Bézier surfaces. (a)f=0.35,α,β,α∗,β∗=1. (b) f=0.5,α,β,α∗,β∗=1. (c) f=1,α,β,α∗,β∗=1. (d) f=1.5,α,β,α∗,β∗=1.
(a)(b)(c)(d)Figure 13
The effect of shape parameters on quartic GT-Bézier surfaces. (a)f=0.5,α,β,α∗,β∗=0. (b) f=0.5,α,β,α∗,β∗=0.25. (c) f=0.5,α,β,α∗,β∗=1. (d) f=0.5,α,β,α∗,β∗=0.5.
(a)(b)(c)(d)Figure 14
The effect of changing scaling factor on quintic GT-Bézier surfaces. (a)f=0.25,α,β,α∗,β∗=1. (b) f=0.5,α,β,α∗,β∗=1. (c) f=1,α,β,α∗,β∗=1. (d) f=0.35,α,α∗=0,β,β∗=1.
(a)(b)(c)(d)Figure 15
The effect of shape parameters on quintic GT-Bézier surfaces. (a)f=0.5,α,β,α∗,β∗=0. (b) f=0.5,α,β,α∗,β∗=1. (c) f=0.5,α,α∗=−0.5,β,β∗=1. (d) f=0.5,α,α∗=0,β,β∗=1.
(a)(b)(c)(d)


## 7.1. Formation of the GT-Bézier Surface

The GT-Bézier surfaces of orderm×n for given points Pi,j∈R2 or R3i=0,1,…,m,j=0,1,…,n with shape parameters α,α1,β,β1 are defined as(25)Wm,nz,z1=∑i=0m∑j=0nPi,jwi,mzwj,nz1,z,z1∈0,1,where wi,mz,wj,nz1i=0,1,…,m,j=0,1,…,n are GT-basis functions (2). Figures 8(a)–8(d) represent the GT-Bézier vine glass, GT-Bézier vase diagrams, and GT-Bézier swept surface of different orders but with fixed values of shape parameters, whereas Figures 9(a)–9(c) demonstrate the GT-Bézier surfaces of different degrees with different values of shape parameters.Figure 8
GT-Bézier surfaces of different degrees. (a) 5th order GT-Bézier surface. (b)5×6th order GT-Bézier surface. (c) 8×5th order GT-Bézier surface. (d) 6th order GT-Bézier surface.
(a)(b)(c)(d)Figure 9
GT-Bézier surfaces of different degrees with multiple shape parameters. (a) Cubic GT-Bézier surfaces. (b) Quartic GT-Bézier surfaces. (c) Quintic GT-Bézier surfaces.
(a)(b)(c)


## 7.2.G2 Continuity of GT-Bézier Surfaces

In engineering applications,G2 continuity between two adjacent surface patches is required, that is, at every point where the curved surfaces meet, the surfaces must have a combined tangent plane or a collective normal line to the surface [3–5, 38]. There exists continuity in different four types of directions between two GT-Bézier surfaces Wm1,n1z,z1;α,α1,β,β1=∑i=0m1∑j=0n1Pi,jwi,m1zwj,n1z1 and W1m2,n2z,z1;α∗,α1∗,β∗,β1∗=∑k=0m2∑l=0n2P1k,lwk,m2zwl,n2z1.


## 7.3. Continuity inz Direction on Every Surface

Two regions of GT-Bézier surfaces should have a collective edge curve to attainG1 continuity, i.e.,(26)Wm1,n1z,1;α,α1,β,β1=W1m2,n2z,0;α∗,α1∗,β∗,β1∗,which results in(27)∑i=0m1Pi,n1wi,m1z;α,β=∑k=0m2P1k,0wk,m2z;α∗,β∗,implying(28)∑i=0mPi,n1wi,mz;α,β=∑i=0mP1i,0wi,mt;α∗,β∗.Ifα=α∗,β=β∗, then(29)Pi,n1=P1i,0.The normal direction at the boundary is uninterrupted for two regions of the surfaces, i.e., the joining boundary should have a mutual tangent plane to meet theG1 continuity between two regions of the surfaces. So, we have the following condition:(30)∂∂zWm,n1z,1;α,α1,β,β1×∂∂z1Wm,n1z,1;α,α1,β,β1=fz1∂∂zW1m,n2z,0;α∗,α1∗,β∗,β1∗×∂∂z1,W1m,n2t,0;α∗,α1∗,β∗,β1∗,where fz1 is the scaling factor among their normal vectors such that fz1>0. Equivalently,(31)∂∂z1Wm,n1z,1;α,α1,β,β1=f⋅∂∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗.Since(32)∂∂z1Wm,n1z,1;α,α1,β,β1=π2∑i=0mwi,mz;α,β1+β1Pi,n1−Pi,n1−1,(33)∂∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗=π2∑i=0mwi,mz;α∗,β∗n2−1+α1∗P1i,1−P1i,0.Substituting (32) and (33) into equation (31), we have(34)π2∑i=0mwi,mz;α,β1+β1Pi,n1−Pi,n1−1=f⋅π2∑i=0mwi,mz;α∗,β∗n2−1+α1∗P1i,1−P1i,0.Ifα=α∗,β=β∗, then(35)1+β1Pi,n1−Pi,n1−1=f⋅n2−1+α1∗P1i,1−P1i,0.Continuity conditions on each surface are described in equations (29) and (35) which are the general conditions for G1 continuity in z direction. In accumulation, under G1 continuity condition, the two surfaces need to retain the similar normal curvature at any point on their joint boundary, so the two surfaces also have to fulfil the following G2 continuity constraints:(36)∂2∂z12Wm,n1z,1;α,α1,β,β1=f2∂2∂z12W1m,n2z,0;α∗,α1∗,β∗,β1∗+2fgz∂∂z∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗+g2z∂2∂z1∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗+c∂∂z1W1m,n2z,0;α∗,α1∗,β∗,β1∗+dz∂∂zW1m,n2z,0;α∗,α1∗,β∗,β1∗,where gz and dz are linear functions of z and c is a random constant. For the simplification of operation and calculation in applied submissions, generally, set gz=dz=c=0 [32], so the above equation can be further simplified as(37)∂2∂z12Wm,n1z,1;α,α1,β,β1=f2∂2∂z12W1m,n2z,0;α∗,α1∗,β∗,β1∗.Since(38)∂2∂z12Wm,n1z,1;α,α1,β,β1=π24∑i=0mwi,mz;α,β1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1,(39)∂2∂z12W1m,n2z,0;α∗,α1∗,β∗,β1∗=π24∑i=0mwi,mz;α∗,β∗n2−1n2−2+2α1∗P1i,0−P1i,1−n2−1n2−2+2n2−2α1∗+1−β1∗P1i,1−P1i,2.Substituting (38) and (39) into equation (37), we get(40)π24∑i=0mwi,mz;α,β1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2π24∑i=0mwi,mz;α∗,β∗n2−1n2−2+2α1∗P1i,0−P1i,1−n2−1n2−2+2n2−2α1∗+1−β1∗P1i,1−P1i,2.Ifα=α∗,β=β∗, we have(41)1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2n2−1n2−2+2α1∗P1i,0−P1i,1−n2−1n2−2+2n2−2α1∗+1−β1∗P1i,1−P1i,2.To sum up, if the two surfacesWm,n2z,1;α,α1,β,β1 and W1m,n2z,0;α∗,α1∗,β∗,β1∗ satisfy (29), (35), and (41) at the same time, the G2 continuity in z direction is achieved.


## 7.4. Continuity inz and z1 Direction

The condition of holding a common boundary when the surfaceWm1,n1z,z1;α,α1,β,β1 has continuity in z direction and the surface W1m2,n2z,z1;α∗,α1∗,β∗,β1∗ has continuity in z1 direction is(42)Wm1,n1z,1;α,α1,β,β1=W1m2,n20,z1;α∗,α1∗,β∗,β1∗,or(43)∑i=0m1Pi,n1wi,m1z;α,β=∑l=0n2P10,lwl,n2z1;α1∗,β1∗,implying(44)∑i=0mPi,n1wi,mz;α,β=∑i=0mP10,iwi,mz1;α1∗,β1∗.Ifα=α1∗,β=β1∗, then equation (44) is further simplified as(45)Pi,n1=P10,i.For having a common tangent plane, we have the following condition:(46)∂∂z1Wm1,n1z,1;α,α1,β,β1=f⋅∂∂zW1m2,n20,z1;α∗,α1∗,β∗,β1∗.Since(47)∂∂z1Wm1,n1z,1;α,α1,β,β1=π2∑i=0m1wi,m1z;α,β1+β1Pi,n1−Pi,n1−1,(48)∂∂zW1m2,n20,z1;α∗,α1∗,β∗,β1∗=π2∑l=0n2wl,n2z1;α1∗,β1∗m2−1+α∗P11,l−P10,l.Substituting (47) and (48) into equation (46), we obtain(49)π2∑i=0m1wi,m1z;α,β1+β1Pi,n1−Pi,n1−1=π2∑l=0n2P10,lwl,n2z1;α1∗,β1∗m2−1+α∗P11,l−P10,l.Ifα=α1∗,β=β1∗, we have(50)1+β1Pi,n1−Pi,n1−1=f⋅m2−1+α∗P11,i−P10,i.Hence, the general conditions forG1 continuity in z and z1 direction are described in equations (41) and (46); when α=α1∗,β=β1∗, then the continuity conditions are defined in equations (45) and (50).Moreover, fromG1 smooth continuity conditions, the two surfaces are also essential to have the same normal curvature at each point on their shared boundary, so for G2 continuity, they also required to satisfy(51)∂2∂z12Wm,n1z,1;α,α1,β,β1=f2∂2∂z2W1m,n20,z1;α∗,α1∗,β∗,β1∗.Substituting the boundary surfaces into equation (53), we get(52)π24∑i=0mwi,mz;α,β1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2π24∑i=0mwi,mz1;α1∗,β1∗m2−1m2−2+2α∗P10,i−P11,i−m2−1m2−2+2m2−2α∗+1−β∗P11,i−P12,i.Whenα=α1∗,β=β1∗, equation (52) yields(53)1−α1Pi,n1−2−Pi,n1−1+n1−21+β1Pi,n1−1−Pi,n1=f2m2−1m2−2+2α∗P10,i−P11,i+m2−1m2−2+2m2−2α∗+1−β∗P12,i−P11,i.Finally, if the two surfacesWm,n2z,1;α,α1,β,β1 and W1m,n20,z1;α∗,α1∗,β∗,β1∗ satisfy (45), (50), and (53), then both surfaces are connected by G2 continuity in z and z1 direction.


## 7.5. Continuity inz1 Direction

TheG1 and G2 continuity in z1 direction of two GT-Bézier surfaces is achieved in a similar fashion as the continuity in z direction of two GT-Bézier surfaces, which is discussed in Section 7.3.


## 7.6. Continuity inz1 and z Direction

TheG1 and G2 continuity in direction z1 and z is obtained in a similar way as the continuity in z and z1 direction which is proved in Section 7.4.TheG2 continuity between two GT-Bézier surfaces is also analyzed as for two GT-Bézier surfaces, the continuity in z direction is achieved; when α=α∗, β=β∗, then the G1 continuity between two regions of GT-Bézier surfaces has m1=m2=m mutual control points, and the control points Pi,n1−1,Pi,n1orP1i,0,P1i,1 are taken and varied in order. For continuity in z and z1 direction, when α=α1∗, β=β1∗, then the G2 continuity between two regions of GT-Bézier surfaces has m1=n2 mutual control points, and the control points Pi,n1−1,Pi,n1orP10,i,P11,i are in the same straight line and changeable in order.Figures10–15 depict the G2 continuity surfaces between two GT-Bézier surfaces of different degrees in z direction for different shape parameters and scaling factor f. For the fixed values of shape parameters, the control points P11,i (or P12,i) move closer to (away from) the control points P10,iorP11,i when thevalues of the scaling factor f increases (or decreases), and when f is fixed and the values of shape parameters are increased (or decreased), the surfaces move closer to (away from) the control polygon (control net). This analysis shows that the piecewise GT-Bézier surface is smooth and continuous at the joint, which significantly increases the abilities to solve problems in engineering appearance design by amending the position and shape of surfaces.Figure 10
The effect of changing scaling factor on cubic GT-Bézier surfaces. (a)f=0.75,α,β,α∗,β∗=1. (b) f=1,α,β,α∗,β∗=1. (c) f=1.5,α,β,α∗,β∗=1. (d) f=4,α,α∗=1,β,β∗=1.
(a)(b)(c)(d)Figure 11
The effect of shape parameters on cubic GT-Bézier surfaces. (a)f=2,α,α∗=0.5,β,β∗=1. (b) f=2,α,α∗=0.5,β,β∗=0.85. (c) f=2,α,α∗=−0.5,β,β∗=0.65. (d) f=2,α,α∗=−0.35,β,β∗=0.9.
(a)(b)(c)(d)Figure 12
The effect of changing scaling factor on quartic GT-Bézier surfaces. (a)f=0.35,α,β,α∗,β∗=1. (b) f=0.5,α,β,α∗,β∗=1. (c) f=1,α,β,α∗,β∗=1. (d) f=1.5,α,β,α∗,β∗=1.
(a)(b)(c)(d)Figure 13
The effect of shape parameters on quartic GT-Bézier surfaces. (a)f=0.5,α,β,α∗,β∗=0. (b) f=0.5,α,β,α∗,β∗=0.25. (c) f=0.5,α,β,α∗,β∗=1. (d) f=0.5,α,β,α∗,β∗=0.5.
(a)(b)(c)(d)Figure 14
The effect of changing scaling factor on quintic GT-Bézier surfaces. (a)f=0.25,α,β,α∗,β∗=1. (b) f=0.5,α,β,α∗,β∗=1. (c) f=1,α,β,α∗,β∗=1. (d) f=0.35,α,α∗=0,β,β∗=1.
(a)(b)(c)(d)Figure 15
The effect of shape parameters on quintic GT-Bézier surfaces. (a)f=0.5,α,β,α∗,β∗=0. (b) f=0.5,α,β,α∗,β∗=1. (c) f=0.5,α,α∗=−0.5,β,β∗=1. (d) f=0.5,α,α∗=0,β,β∗=1.
(a)(b)(c)(d)


## 8. Conclusions

Newmth order GT-Bézier curves along with two shape parameters which share most properties with the nth order classical Bézier curve have been presented in this study. GT-Bézier surfaces of order m×n have also been proposed by using GT-Bézier curves of order m and n. The contour of GT-Bézier curves and surfaces can be transformed not by modifying the control points but it is by varying the values of the shape parameters. Since the GT-Bézier curves and surfaces have been described by using amendable shape parameters, the connection of such curves and surfaces is unwrinkled than that of traditional Bézier curves and surfaces. In contrast with other curve-forming techniques, the technique proposed here can develop curves whose degree and mathematical complication are not increased which make it more useful in practical submissions. Some curve and surface design examples exhibit that this scheme is more convenient, flexible, and effective for both curve and surface interaction modeling and has significant mathematical and applied applications. Additionally, this scheme can be applied to generate trigonometric surfaces over triangles with adjustable shape parameters [37].



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