Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications

Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G^{1}$\end{document} and G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G^{2}$\end{document} (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.


Introduction
Due to the ease of engineering procedure, developable surfaces are especially fascinating and tempting. A developable surface is acquired by simply twisting a plane in the absence of any contraction or stretching. In the language of differential geometry, a smooth surface having zero Gaussian curvature at each point on it is known as a developable surface. Developable surfaces may be distort but have powerful isometric features. They can be calmly parameterized remarkably as to conserve arc lengths, they are marvelous applicants for structure mapping. Consequently, numerous products which utilize leather sheets, metal, paper, and other identical malleable materials are designed taking developable surfaces. Actual developable surfaces have an instinctive implementation in several fields of engineering and manufacturing as an aircraft architect uses them to form airplane wings and a tinsmith uses them to attach two tubes of different designs with flattened sections of metal sheets. In plat-metal-based manufacturing industries, the designing of developable surfaces is contemplated as a very imperative petition.
From an assortment of fabrication like computer animation, architecture, automotive, clothing, footwear, image processing, and shipbuilding, the development of developable surfaces has taken more consideration. Therefore, the research issue for the construction and designing of developable surfaces is consistently important in CAD/CAM [2,3] as it is concerned with modeling and invigorating objects which are examined in daily life. In this context, Chung et al. [1] suggested a technique to make shoe uppers by taking triangles and also to improve the surface to make it more developable.
The designing techniques for developable surfaces have two divisions: The first is the point geometric representation and the second is the line and plane geometric representation familiar as dual representation. Further two particular approaches are there in point geometric representation to bargain with this manifestation. First approach is to build up a developable surface on the support of the original direction and given directrix, and the second approach is to formulate it by two interpolating boundary curves. Aumann [4] designed some interpolating developable Bézier patches with some essential and adequate requirements to free them from singular points. Additionally, he also derived G 1 and G 2 continuity requirements among these patches. Algorithms that present the developable surfaces using Bézier curve of arbitrary shape and order were generated by Aumann [5,6] in which the control of singular points is insured. As a directrix of developable Bézier surface, Zhang et al. [7] used a space Bézier curve and explored the geometric design of developable Bézier surfaces. As a generality of Aumann's algorithm for Bézier developable surfaces to B-spline developable surfaces, Fernandez [8] provided a linear algorithm for construction of a random number of pieces and order B-spline control nets of spline developable surfaces. Chu et al. [9] introduced a CAGD technique to interpolate a strip in the conical form described by two space curves along with developable patches.
Hwang et al. [10] proposed developable surfaces by folding a planar segment around cylinders and cones, and also through successive mappings, he designed complicated developable surfaces taking various cones and cylinders of different shapes and sizes. However, the point geometric representation has some deficiencies such as the ambiguous description of a developable surface and the nonlinearity of characteristic equations, which results in tough computation. Hence, the aforementioned drawbacks limit its area of application. On the other hand, dual or plane geometric description presents a developable surface like a curve in a dual projective space, which removes the flaws of point geometric representation.
For the first time to create developable surfaces, Bodduluri and Ravani [11,12] suggested the dual B-spline and Bézier interpretations and made their practical and effective use to the engineering designs of the corresponding developable surface. In the meantime, explicit interpretation of developable surface was given, and further studies have been done in this context along with some important conclusions [13][14][15][16][17]. In [11][12][13][14][15][16], the structure of a developable surface was decided by its control planes only, which causes a problem for an engineering appearance model. For the sake of resolving this problem, rational developable surfaces [13,14] have been used. These surfaces can be adjusted by modifying their weight factors without disturbing control planes. Anyhow, usage of rational fractions further generates some other difficulties such as singularity and complex analysis formula [20].
To tackle the shape adjustment issue and maintain the advantages of developable surfaces, Zhou et al. and Hu et al. proposed developable surfaces manipulating C-Bézier and λ-Bézier basis functions along one shape parameter in [17] and [21] respectively. However, having one shape parameter the above constructed developable surfaces have limited shape control. Li and Zhu [18] developed G 1 connection of four pieces of developable surfaces with Bézier boundary curves using de Casteljau algorithm. Chu and Chen [19] constructed G 2 geometric design of developable surfaces that consist of consecutive Bézier patches In recent times, using multiple shape parameters, Hu et al. [22][23][24][25] introduced some straightforward schemes for computer-aided design of developable Bézier-like, H-Bézier, generalized quartic H-Bézier, and generalized C-Bézier developable surfaces sequentially. For a smooth continuous connection (G 1 and G 2 ) among the above-proposed surfaces, the author in [22][23][24][25] also computed the essential continuity requirements and described their application in geometric modeling. Kusno [26] constructed the regular developable Bézier patches. Recently, Li, Hu et al., and Ammad et al. proposed the designing approaches for developable C-Bézier [27], cubic developable C-Bézier surfaces [28], and generalized developable cubic trigonometric Bézier surfaces [29], respectively. These schemes bring a beneficial opportunity for the developable surfaces to the actual geometric modeling techniques.
In surface modeling, the construction of a developable surface using a trigonometric polynomial function space is a fascinating issue. With the progress of CAD/CAM application software, our proposed developable GBT-Bézier surfaces will come up with a contemporary set of mathematical theory, and its utilization area also comprises computer graphics, shipbuilding, automotive, architecture, clothing, footwear, computer animation, image processing, etc.
In this study, some technical contributions are made which are as follows: • Construction of GBT-Bézier surface by a new set of GBTB functions with two shape parameters. • Construction of some computer-based engineering surfaces using GBT-Bézier with shape parameters. • The complex computer-based developable surfaces using GBT-Bézier patches are composed by G k (k = 1, 2, 3) continuity conditions. • Our described developable GBT-Bézier surfaces take over the most advantageous features of the classical developable Bézier surfaces. Moreover, the shape adjustment property is additional to that of classical developable Bézier surfaces which makes it superior to the classical one. • The approach becomes more convenient and efficient because the exclusion of complex calculations comes from the nonlinearity of characteristic equations. • The local and overall shape of a composite developable GBT-Bézier surface with a smooth connection can be accommodated by modifying its shape parameters without re-establishing the control planes.  The 2nd degree GBTB basis functions with respect to x having shape parameters μ, ν are defined as follows: where μ, ν ∈ [-1, 1] and x ∈ [0, 1]. Also, for all integers k(k ≥ 3), the GBTB basis functions g i,k (x) (i = 0, 1, 2, . . . , k) can be recursively defined as follows: where g i,k (x) are known as GBTB basis functions of degree k [31].
Proof The authentication of all the above consequences is as demonstrated in [31].

Construction of GBT-Bézier curves with shape parameters
Definition 2 For any defined control points R i ∈ R k (k = 2, 3; i = 0, 1, . . . , k), the curves are familiar as GBT-Bézier curves corresponding to GBTB basis functions g i,k (x).
As the GBT-Bézier curves are defined on the bases of GBTB basis functions, so the aforementioned features of GBTB basis functions demonstrate that the GBT-Bézier curves have most dominant features of the traditional Bézier curves inclusively, end point con- , and G (1; μ, ν), convexity, symmetry, variation diminishing feature, shape adjustment feature, and geometric invariance feature. These features specify that the GBT-Bézier curves interpolate to the end points of its convex hull, and also by choosing appropriate values of shape parameters μ, ν in their respective value range μ, ν ∈ [-1, 1], the shape of GBT-Bézier curves can be adjusted easily according to design requirements [31].

Shape adjustability of GBT-Bézier curves
A group of GBT-Bézier curves can be derived from expression (2.2) by taking the distinct values of their shape parameters μ, ν in their corresponding value range having control points R 0 , R 1 , R 2 , . . . , R k . Owing to the reality that every GBT-Bézier curve assigns two local shape parameters μ, ν, hence the structure of a GBT-Bézier curve can be easily settled and modified by changing the values of these two shape parameters. From Definition 2, we acknowledge that the curve G(x; μ, ν) is a linear function of every shape parameter μ, ν and

Therefore from equations (2.3) and (2.4), it is obvious that there is no relationship among
∂G(x) ∂μ and μ, and ∂G(x) ∂ν and ν. Hence modifying one shape parameter μ or ν, the point R(x) on the curve moves linearly for a fixed convex hull and defined value of x. Also the alternation of direction is given by

Construction of developable GBT-Bézier surfaces with shape parameters 3.1 Dual generation of a single-parameter family of planes
As stated in the duality principle between points and planes, a single-parameter family of control points of a curve is dual to a single-parameter family of planes [11][12][13]. Thus by treating the control points of a GBT-Bézier curve as GBT-Bézier control planes, a singleparameter family of planes { x } can be developed. Therefore the expression of a singleparameter family of planes { x } of a GBT-Bézier curve is described by using expression (2.2) as follows: where Q i (i = 0, 1, 2, . . . , k) are control planes of { x }, μ, ν are shape parameters, and x is the are the coordinates of the control points of a GBT-Bézier curve in a 3D projective space.
In view of the duality principle and (3.1), vector form of expression (3.1) can be defined as follows: Thus, equation (3.3) can be demonstrated as

Description of enveloping developable GBT-Bézier surfaces with shape parameters
We are familiar with the definition and features of developable surfaces i.e. a developable surface is an envelope of a single-parameter family of planes. Consequently, a developable GBT-Bézier surface is an envelope of a single-parameter family of planes { x } of a GBT-Bézier curve. The crossing line of two successive planes analogous to any value of x in { x } will surely lie on the enveloping developable GBT-Bézier surface of { x }. Mathematically, the plane analogous to any value of x in (3.4) can be described in a subsequent linear equation: By taking the derivative of (3.5) with respect to x we attain where prime represents 1st derivative corresponding to x. The generator J(x; μ, ν) of the developable GBT-Bézier surface corresponding to x is the crossing line of planes (3.5) and (3.6), lies on the developable GBT-Bézier surface of { x }, and can be calculated in terms of its plucker coordinates as follows [11,12]: and can also be expressed as Let ψ(x) indicate the nearest point on the generator J(x; μ, ν) to the origin that can be calculated as follows [11,12,17,21]: Consequently, in a parametric form, the generator J(x; μ, ν) can be defined as where μ, ν are the shape parameters. For distinct values of family parameter x in its given value range, all generators J(x; μ, ν) construct an enveloping developable GBT-Bézier surface having μ, ν as shape parameters. Hence, by using equation (3.7), an enveloping developable GBT-Bézier surface of { x } can be described in its linear geometric representation.

Description of spine curve developable GBT-Bézier surfaces with shape parameters
A characteristic point S(x) in a single parameter family of planes { x } of a GBT-Bézier curve is a point where its three successive planes related to x coincide and locus of this point is a curve, identified as a spine curve of the developable GBT-Bézier surface. From the intersection of equations (3.5) and (3.6) and the second derivatives of (3.5), the characteristic point of a GBT-Bézier curve related to x can be achieved. The second derivative of (3.5) corresponding to x is Hence the coordinates of intersecting point S(x) of the three coinciding planes related to x, familiar as a characteristic point, can be expressed as [11,12,17,21] Modifying parameter x in the interval [0, 1], all characteristic points S(x) construct a space curve termed spine curve. From the definition of a developable surface, let us suppose that S(x) is a spine curve of developable GBT-Bézier surfaces, then the surface made up of the tangent lines of the spine curve S(x) is a spine curve developable GBT-Bézier surface. Thus, a parametric spine curve developable GBT-Bézier surface can be described as [17,21] V( (3.10) Finally, in formulating developable GBT-Bézier surfaces (3.7) and (3.10), a singleparameter family of planes { x } using GBTB basis as basis functions is acquired. Therefore, the developable surfaces introduced in this study are identified as developable GBT-Bézier surfaces (enveloping and spine curve). It is worth mentioning here that the developable GBT-Bézier surfaces are considered as a single-parameter family of planes, and there are numerous dissimilarities among developable GBT-Bézier surfaces and tensor product GBT-Bézier surfaces. Therefore, for their dual relationship, there are many identical features among GBT-Bézier curves and developable GBT-Bézier surfaces.

Some characteristics of developable GBT-Bézier surfaces
Geometric features of a GBT-Bézier curve are directly linked with its control points including ending and starting points, tangents and curvatures of the curve at two points. Hence, in the designing process, the shape of a curve can be handled in a well manner by using its control points. As stated in duality theory [11,17], the connection among resulting developable surfaces and control planes is identical to the connection among its control points and dual curves. Thus, from the definition of { x } and the 1st and 2nd order derivatives of expression (3.2), we can derive some meaningful results as at and at x = 1 we have (3.12) Following is the geometrical significance of (3.11) to (3.12). The first equations of expressions (3.11) and (3.12) demonstrate that the first and the last plane in { x } are defined by the architecture according to their own requirements and considered as the 1st and the last control plane sequentially. Also at x = 0 and x = 1, these two planes are tangent to the defined developable GBT-Bézier surface with its generators J(x; μ, ν). Furthermore, at x = 0, the generator J(0; μ, ν) of a developable surface obtained from the intersection of the first two equations of expression (3.11) is labeled as a starting generator. Consequently, the generator J(0; μ, ν) is the connection of the planes Q 0 and 1 2 (2(k -2) + π(1 + μ))(Q 1 -Q 0 ) and can be expressed in the following vector form: ⎧ ⎨ ⎩ t 0 .W = s 0 , (k -2 + π 2 (1 + μ))(t 1t 0 ).W = (k -2 + π 2 (1 + μ))(s 1s 0 ), (3.13) where t 0 = (p 0 , q 0 , r 0 ), t 1 = (p 1 , q 1 , r 1 ), W = (X, Y , Z).

Some designing pattern of developable GBT-Bézier surfaces
We can analyze from all the above discussion that a developable GBT-Bézier surface can be designed as far as its control planes are given, and its shape can be modified using different values of its shape control parameter μ, ν. Therefore, based on control planes, some modeling examples of cubic and quartic developable GBT-Bézier surfaces using multiple values of shape parameter are presented here to clear the influence role of shape parameters in designing features of developable GBT-Bézier surfaces. In all designing examples of enveloping developable GBT-Bézier surfaces, we assume that the midpoints of all planes of the enveloping developable GBT-Bézier surfaces lie in the same plane and have the same distance from the origin. To manifest the impact of shape parameters on the shape of a cubic enveloping developable GBT-Bézier surface apparently from the same viewpoint and coordinate system, Figs. 6-8 depict the graphs of cubic enveloping developable GBT-Bézier surfaces with multiple values of their shape parameters μ, ν. From these figures, we can conclude that with fixed control planes, the shape parameters affect the shape of the enveloping developable GBT-Bézier surface in the following manners:

Designing examples of enveloping developable GBT-Bézier surfaces
1. Whenever the values of μ stay unchanged, the location of the starting generator J(0; μ, ν) and the length and location of the ending generator J(1; μ, ν) will not modify with modifying the value of ν; however the length of the starting generator J(0; μ, ν) will modify. In other words, increasing the value of ν will bring an increase    The influence of parameters μ on cubic enveloping developable GBT-Bézier surface

Some designing examples of spine curve developable GBT-Bézier surfaces
To demonstrate the designing feature of a spine curve developable GBT-Bézier surface constructed from the tangent lines of the spine curve Sx, some design examples are given With different values of shape parameters μ and ν, a class of cubic spine curve developable GBT-Bézier surfaces can be designed on the requirement of provided control planes Q 0 , Fig. 13 and Fig. 14 respectively, with distinct shapes.

Continuity requirements among developable GBT-Bézier surfaces
The designing of free-form complicated surfaces is a major issue in product designing, graphics, and CAD/CAM. In practical applications, the appearing design of many products is relatively complex and cannot be presented by an individual surface. Therefore, there is a requirement to construct these surfaces by using adjoining surfaces. The evaluation criteria for unwrinkled joining among two adjoining developable GBT-Bézier surfaces are G 1 and G 2 (Farin-Boehm and beta) continuity, etc. [15,17,21,25,30]. Now, we Hence some geometric design methods of developable GBT-Bézier surfaces (like continuity requirements, tangent planes, and terminal properties) in a 4D homogeneous space are identical to those of GBT-Bézier curves. Therefore, for continuity requirements of developable GBT-Bézier surfaces, it is imagined that the single parameter families of planes of two developable GBT-Bézier surfaces H 1 (x; μ 1 , ν 1 ) and H 2 (x; μ 2 , ν 2 ) of order k and l, respectively, that require to be joined together are defined as follows: where μ i , ν i (i = 1, 2) are shape parameters and the control planes of { x,1 } and { x,2 } are Q i,1 (i = 0, 1, . . . , k) and Q j,2 (j = 0, 1, . . . , l), respectively.

Figure 15
Farin-Boehm G 2 continuity among two cubic enveloping developable GBT-Bézier surfaces Step 4. From the 3rd equation of (5.16), the control plane Q 2,2 of the developable GBT-Bézier surface H 2 (x; μ 2 , ν 2 ) can be determined on the bases of Step 2 and Step 3 for a defined value of constant λ.
Step 5. The last control plane Q 3,2 of H 2 (x; μ 2 , ν 2 ) can be selected openly for establishing G 2 beta smooth continuous connection among two or more adjoining developable GBT-Bézier surfaces.

Figure 16
Farin-Boehm G 2 continuity betwixt two cubic enveloping developable GBT-Bézier surfaces From iteration of the above steps among two developable GBT-Bézier surfaces, G 2 beta smooth continuity can be attained among multiple developable GBT-Bézier surfaces, and it can also be applied on other continuity conditions in the same manners.

Some modeling examples of smooth developable GBT-Bézier surfaces
This portion will give some designing examples of G 2 smooth connection (beta and Farin-Boehm) among two adjoining cubic developable GBT-Bézier surfaces sequentially. Additionally, the effect of shape parameters on combined surfaces is also examined.  The green surfaces are the secondary enveloping developable GBT-Bézier surfaces. For all spliced surfaces, the last control plane Q 3,2 is given openly, and the first three control planes are derived from Farin-Boehm G 2 continuity conditions. Figure 15 demonstrates the influence of shape parameter μ on combined developable GBT-Bézier surfaces with fixed value of ν. These four graphs indicate that without disturbing control planes, Figure 18 The effects of scale factor λ on composite developable GBT-Bézier surface with G 2 beta smooth continuity the shape of the combined developable GBT-Bézier surface can be simply modified by amending the shape parameters. Figure 16 represents the impact of shape parameter ν on the combined developable GBT-Bézier surface for a fixed value of μ.
Example 5.2 Geometric design of G 2 beta continuity among two contiguous cubic enveloping developable GBT-Bézier surfaces is illustrated in Fig. 17. In Fig. 17, the green surface represents first cubic enveloping developable GBT-Bézier surface H1(x; μ 1 , ν 1 ) with control planes described in (5.18), whereas red surface represents the 2nd cubic enveloping developable GBT-Bézier surface H2(x; μ 2 , ν 2 ) which fulfills the G 2 beta continuity requirements with H1(x; μ1, ν1). By setting all shape parameters μ 1 , ν 1 , μ 2 , ν 2 of the corresponding cubic enveloping developable GBT-Bézier surfaces equal to 1, the coordinates of the control planes of cubic enveloping developable GBT-Bézier surface H2(x; μ 2 , ν 2 ) are calculated as follows: where the last control plane Q 3,2 is given without restraint, and the control planes Q 0,2 , Q 1,2 , Q 2,2 are calculated according to (5.16) (γ = 1, λ = 0). Figure 17 represents the combined enveloping developable GBT-Bézier surface having G 2 beta connection among them after amending the values of its shape parameters, while Fig. 18 indicates the combined enveloping developable GBT-Bézier surface for multiple values of scale factor γ . Figure 18 indicates that a unified merged developable GBT-Bézier surface can be achieved by setting scaling factor γ = 1, unconcerned of modifying the shape parameter, and at connection a gap will be achieved on merged developable GBT-Bézier surfaces when γ = 1. We can use these characteristics to design a complex surface according to our needs.

Conclusions
In this research work, developable GBT-Bézier surfaces (enveloping developable and spine curve developable) along two shape parameters have been proposed. The geometric features and influence role of shape parameters on these newly constructed developable GBT-Bézier surfaces have been examined. Additionally, the geometric continuity requirements (G 1 , G 2 Farin-Boehm and G 2 beta) among two contiguous developable GBT-Bézier surfaces have been derived for the construction of complicated developable GBT-Bézier surfaces. These proposed developable GBT-Bézier surfaces have been shown to reduce the flaws of line and plane geometric description in modeling developable GBT-Bézier surface design, and we proved that they are more beneficial than the actual developable Bézier surfaces. In contrast with other Bézier curves and surfaces techniques having multiple shape parameters, our interpretation of basis functions determined in this research is simple and extra succinct.