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

Adopting a recurrence technique, generalized trigonometric basis (or GT-basis, for short) functions along with two shape parameters are formulated in this paper. 'ese basis functions carry a lot of geometric features of classical Bernstein basis functions and maintain the shape of the curve and surface as well. 'e 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. 'is 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. 'ese 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. 'e demonstrating examples clarify that the new curves and surfaces provide a flexible approach andmathematical sketch of Bézier curves and surfaces which make them a treasured way for the project of curve and surface modeling.


Introduction
e 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. e 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 timeconsuming 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. erefore, there is a necessity to model such products using adjacent curves/ surfaces with some continuities. e authors in [2] derived the G 1 , G 2 geometric continuity conditions only for H-Bézier curves of degree n. Hu et al. [3][4][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. erefore, 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. e 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. e 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. e 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. e 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. e 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. e existence of the unique spline approximation is proved, and the convergence order of the method is shown to be cubic. e 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. e 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. ey 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. e 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.
is paper defines the work based on the construction of new GT-basis functions of order n(n > 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. e continuity conditions of C 3 and G 2 between two adjacent GT-Bézier curves and surfaces are proposed using terminal properties of GT-Bézier curves/surfaces of degree n. e 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 C 3 continuity conditions and as far as we are aware, it has never been employed for thispurpose before.
is paper is laid out as follows: in Section 2, generalized trigonometric basis functions are constructed, and their properties are discussed. e GT-Bézier curves and surfaces are proposed with their properties in Sections 3 and 7, respectively. e 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. e graphs of 5th 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.

Construction of GT-Bézier Curves
Definition 2. For any given control points P i ∈ R 2 or R 3 (i � 0, 1, . . . , m) and shape parameters α, β, the GT-Bézier curve can be constructed as where w i,m (z) are GT-basis functions (2).

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: (2) Convex hull property: the whole GT-Bézier curve segment absolutely lies inward its control polygon.  Yellow-dotted circle on the blue circle is obtained by the control points P i + O ′ . Similarly, Figure 2(b) shows that black is the first curve S(z) with control points P i , the blue curve is represented by the control points P i * Y ′ , and the yellow-dotted curve printed on the blue curve is obtained by multiplying

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 erefore, there is no relationship among (zS(z)/zα) and α and (zS(z)/zβ) and β. Modifying one shape parameter α or β, the point S(z) on the curve changed linearly for an unmovable control polygon and defined value of z. e route modification is as follows:

Affiliation among the Shape Parameters and the Shape of the Curves.
e above defined characteristics of the GTbasis functions make the shape of GT-Bézier curves extraordinarily easier to modify. e appearance of the GT-Bézier curves can be attuned by changing the values of shape parameters. Figure 5 displays the graphs of cubic, quartic, and quintic GT-Bézier curves. Figure 5

Examples and Discussions
Some examples and discussions about parametric and geometric continuity of GT-Bézier curves are given in this section.

G 2 Continuity of GT-Bézier Surfaces.
In engineering applications, G 2 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][4][5]38].

Continuity in z and z 1 Direction.
e condition of holding a common boundary when the surface W m1,n1 (z, z 1 ; α, α1, β, β1) has continuity in z direction and the surface W1 m2,n2 (z, z 1 ; α * , α1 * , β * , β1 * ) has continuity in z 1 direction is If α � α1 * , β � β1 * , then equation (44) is further simplified as For having a common tangent plane, we have the following condition: Since Figures 10-15 depict the G 2 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 P1 1,i (or P1 2,i ) move closer to (away from) the control points P1 0,i (or P1 1,i ) when thevalues of the scaling factor f increases (or Figure 11: e effect of shape parameters on cubic GT-Bézier surfaces. 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). is 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.

Conclusions
New mth 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. e 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].

Data Availability
e experimental data used to support the findings of this study are available within this paper. Disclosure e second author presented this paper in C-2018-21st International Conference on Humans and Computers, March 28th and 29th, 2019, at Shizuoka University, Hamamatsu, Japan, as a keynote speaker.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.

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Mathematical Problems in Engineering