A Class of Sextic Trigonometric Bézier Curve with Two Shape Parameters

In this paper, we present a new class of sextic trigonometric Bernstein (ST-Bernstein, for short) basis functions with two shape parameters along with their geometric properties which are similar to the classical Bernstein basis functions. A sextic trigonometric Bézier (ST-Bézier, for short) curve with two shape parameters and their geometric characteristics is also constructed.)e continuity constraints for the connection of two adjacent ST-Bézier curves segments are discussed. Shape control parameters can provide an opportunity to modify the shape of curve as designer desired. Some open and closed curves are also part of this study.


Introduction
A Bézier curve is a parametric curve that is used to draw the shapes in the fields of computer graphics and computeraided geometric design. e Bézier curve is usually followed by the defining polygon. e tangent vectors direction at the end is the same as the vector defined by the first and last segment of the Bézier curves. It is useful in many industrial and engineering fields with various applications. Since Bézier curve always mimics the shape of its control polygon, designers can easily attain the required shape for designing purposes. e trigonometric Bézier curves got a lot of attention in the fields of computer graphics and computer-aided geometric design due to their construction of conic section. Bézier curve with two parameters and control point was introduced by Kun [1], but the behavior of curve was not symmetric. e generalized Bézier-like curve with all of its geometric characteristics and continuity conditions is described by Yan and Liang [2]. ey also applied the generalized Bézier-like curve for the tensor product surfaces to gain access for triangular surfaces as well. e modeling of innovative surfaces based on stream curves was described by Liu et al. [3]. By extending the concept of Bézier curve, Li [4] defined alpha-Bézier-like curves of degree n with shape control parameters. e properties and applications of alpha-Bézier-like curves are also given. Han et al. [5][6][7] introduced some cubic and quartic trigonometric Bézier curves. ey created an ellipse by using a cubic trigonometric Bézier curve and some designing and geometric modeling also made by continuity conditions. e behavior of shape control parameters also examined on these Bézier curves. Xiujuan et al. [8] investigated special revolution surfaces and their dramatic improvement. Bashir et al. [9] derived a class of quasi-quantic trigonometric Bézier curves with two shape parameters and proved their geometric features. e properties of the basis functions and curves are established, and the effect of the shape control parameters is also discussed. Practical applications of Bézier curves in geometric modeling and engineering are limited due to their shortcomings, and much work has been done to resolve these shortcomings [10][11][12][13][14][15][16].
BiBi et al. [17,18] suggested a new method for solving the problem of constructing symmetric curves and surfaces by using GHT-Bézier curves with four different shape parameters. e shape of curves can easily be modified by using different values of shape parameters. ey generated some free-form complex curves with parametric continuity conditions by using GHT-Bézier curves to demonstrate the efficiency of modeling. Yan and Liang [2] used the recursive technique to create the rectangular Bézier curve and surface based on a new class of polynomial basis functions with one shape parameter. Hu et al. [19] presented a novel scheme to generate free-form complex figures using shape-adjustable generalized (SG) Bézier curves with some geometric continuities conditions. ey constructed the necessary and sufficient constraints for G 1 and G 2 continuity for connection of two adjacent SG-Bézier curves to overcome the difficulty that most of the composite curves in engineering cannot often be constructed by using only a single curve. Majeed and Qayyum [20] presented the cubic and rational cubic trigonometric B-spline curves using new trigonometric functions with shape parameter. e proposed curves inherit the basic properties of classical B-spline and have been proved. Misro et al. [21] developed the general technique to construct S-and C-shaped transition curves using cubic trigonometric Bézier Curve with two shape parameters which satisfy G 2 Hermite condition. Misro et al. [22] constructed a new quintic trigonometric Bézier curve that has the potential to estimate the maximum driving speed allowed for safe driving on roads. e shape parameters used in this trigonometric Bézier function provided more flexibility for users in designing highways. e trigonometric Bézier curve of fifth degree with two shape parameters has been presented by Misro et al. [23]. Shape parameters provided more control on the shape of the curve compared to the classical Bézier curve. Juhász and Róth [24] presented a scheme for interpolating the given set of data points with C n continuous trigonometric spline curves of order n + 1 which are produced by blending elliptical arcs with global parameter α ∈ (0, π). Zhu and Han [25] constructed four new trigonometric Bernstein-like basis functions with two exponential shape parameters, based on which a class of trigonometric Bézier-like curves, similar to the cubic Bézier curves, have also been developed. e trigonometric Bézierlike curves corner cutting algorithm was also constructed. Yan and Liang [26] presented a new kind of algebraictrigonometric blended spline curve, called xyB curves. e proposed curves not only inherit most properties of classical cubic B-spline curves in polynomial space but also enjoy some other advantageous properties for modeling.
In this article, the research begins with the development of new ST-Bernstein basis functions with two shape control parameters. is study also provides a guarantee to construct a new ST-Bézier curve with two shape parameters. e newly constructed curves share all geometric properties of classical Bézier curves except the shape modification property, which is superior to the classical Bézier curve. e C 2 and G 2 continuity constraints are constructed to connect the two adjacent ST-Bézier curves segments. Moreover, in contrast with classical Bézier curves, our proposed scheme gives more shape adjustability in curve designing. Several examples are presented to show that the proposed method has high applied values in geometric modeling in terms of some closed and open curves. e remainder of the paper is organized as follows. In Section 2, the new ST-Bernstein basis functions with two parameters are presented, which possess all geometric properties. In Section 3, the graphical representation of ST-Bézier curve with all geometric properties is given. e parametric and geometric continuity conditions with shape control parameters are given in Section 5. Shape control on ST-Bézier curve via shape parameters is given in Section 4. In Section 6, some application to construction of some closed and open curves is given with multiple shape control parameters. Finally, concluding remarks of this work are given in Section 7.

Sextic Trigonometric Bernstein Basis Functions
In this section, the ST-Bernstein basis function with two shape parameters μ, ω and their geometric properties is discussed.

e ST-Bernstein basis functions in equation
Monotonicity: for the given value of the shape parameter μ, ω, Φ 0 (η) is monotonically decreasing and Φ 6 (η) is monotonically increasing (2) It is obvious by Definition 1.
For i � 1, From the figures of the functions f 1 (η) and f 2 (η), we can see erefore, Φ 0 (η) and Φ 6 (η) are monotonically decreasing and increasing about η, respectively. is can also be shown graphically in Figure 1.
, Φ 6 (π/2) � 1 and the first derivatives of these basis functions at their end points are given as follows: Similarly, the second derivatives of these basis functions at their end points are given as follows (see Figure 2):

Sextic Trigonometric Bézier Curves with Two Shape Parameters
In this section, the ST-Bézier curves with two shape parameters μ, ω and their geometric properties are discussed.
is called ST-Bézier curve, where Φ i (η)(i � 0, 1, 2, . . . , 6) are called ST-Bernstein basis functions and μ, ω are the shape parameters. Some graphical results of ST-Bézier curve are discussed as follows: when shape parameters vary equally, Figure 3 is generated, while, by keeping one parameter fixed to 1, Figure 4 is generated. In Figure 4(a), when μ � 1 and ω varies, then influence of shape parameters can be seen on the left side of the figure. Meanwhile when we consider ω � 1 and parameter μ varies, the influence of these parameters can be observed on the right side of Figure 4(b).

Continuity Conditions for ST-Bézier Curve Segments
In this section, the continuity conditions are derived for smooth joining of two ST-Bézier curves segments.

C 1 Continuity of ST-Bézier Curve.
For parametric continuity of degree 1, consider two adjacent ST-Bézier curve segments with shape control parameters. In this case, we should have common tangents of the two curve segments at joint point. e first two control points of second curve can be achieved as given in eorem 3. Figures 7(a)-7(c) can be obtained by varying the shape control parameters via C 1 continuity constraints.

C 2 Continuity of ST-Bézier
Curve. e C 2 continuity can be achieved by connecting two adjacent ST-Bézier curve segments if they fulfill the C 0 and C 1 continuity conditions. e second derivative of these segments at joint point must be the same as that of C 2 continuity. e control points of the first curve can be chosen according to the designer's requirement, while the control points for the second curve can be obtained from eorem 3. Different values of shape control parameters can be used to obtain different curves as given in Figure 8.

Geometric Continuity of ST-Bézier Curve
To get more smoothness between any two adjacent ST-Bézier curves, geometric continuity conditions have been derived. In geometric continuity conditions, one extra parameter is involved, which is used for the modification of the curve.

G 2 Continuity of ST-Bézier Curve.
e geometric continuity of degree 2 between any two adjacent ST-Bézier curves is given here. Different figures display the behavior of various shape parameters and scale factors. In Figures 9(a)-9(c), the shape parameters for both curves vary, while the scale factors remain fixed. In Figures 10(a)-10(d), the shape

Conclusions
In this research, a newly constructed ST-Bernstein basis and Bézier curve with two shape parameters has been proposed. It can be concluded that its geometric properties are similar to those of the classical Bézier curve. e shape of the curve can be regulated by changing the values of shape parameters. e suggested curve can be used to create open and closed curves with different values of shape parameter. e parametric and geometric continuities for two adjacent ST-Bézier curves are also presented, which demonstrate the efficiency of adjoining the ST-Bézier curves.

Data Availability
e experimental data used to support the findings of this study are included within the paper.