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Shapeadjustable developable generalized blended trigonometric Bézier surfaces and their applications
Advances in Difference Equations volume 2021, Article number: 459 (2021)
Abstract
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 computeraided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBTBézier surfaces) with shape parameters. A developable GBTBézier surface is established by making a collection of control planes with generalized blended trigonometric Bernsteinlike (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 GBTBézier surfaces that preserves the features of the developable GBTBézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBTBézier surfaces, the necessary and sufficient \(G^{1}\) and \(G^{2}\) (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBTBézier 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 platmetalbased 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 Bspline developable surfaces, Fernandez [8] provided a linear algorithm for construction of a random number of pieces and order Bspline 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 Bspline 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 [13–17]. In [11–16], 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 CBézier and λBézier 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. [22–25] introduced some straightforward schemes for computeraided design of developable Bézierlike, HBézier, generalized quartic HBézier, and generalized CBézier developable surfaces sequentially. For a smooth continuous connection (\(G^{1}\) and \(G^{2}\)) among the aboveproposed surfaces, the author in [22–25] 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 CBézier [27], cubic developable CBézier 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 GBTBézier 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 GBTBézier surface by a new set of GBTB functions with two shape parameters.

Construction of some computerbased engineering surfaces using GBTBézier with shape parameters.

The complex computerbased developable surfaces using GBTBézier patches are composed by \(G^{k} (k=1, 2,3)\) continuity conditions.

Our described developable GBTBézier 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 GBTBézier surface with a smooth connection can be accommodated by modifying its shape parameters without reestablishing the control planes.
This research work is laid out: Definition and features of GBTBézier curves are given in Sect. 2. Section 3 presents the corresponding enveloping developable and spine curve developable GBTBézier surfaces along with their properties. To expose the potency of the scheme, modeling examples of the proposed enveloping developable and spine curve developable GBTBézier are illustrated in Sect. 4. In Sect. 5, continuity requirements among these developable GBTBézier surfaces with some practical examples are given. Finally, Sect. 6 gives a summarizing ending.
Some preliminaries
The generalized blended trigonometric Bernsteinlike basis functions with shape parameters
Definition 1
The 2nd degree GBTB basis functions with respect to x having shape parameters \(\mu,\nu \) are defined as follows:
where \(\mu,\nu \in [1,1]\) and \(x\in [0,1]\). Also, for all integers \(k(k\geq 3)\), the GBTB basis functions \(g_{i,k}(x)\ (i=0,1,2,\ldots,k)\) can be recursively defined as follows:
where \(g_{i,k}(x)\) are known as GBTB basis functions of degree k [31].
The GBTB basis functions \(g_{i,k}(x)\) become zero (\(g_{i,k}(x)=0\)) for \(i=1\) or \(i>k\). Different degrees GBTB basis functions are illustrated in Fig. 1 with different values of their shape parameters as \(\mu, \nu =1\) (green), −0.3 (purple), 0.3 (yellow), and 1 (pink).
Theorem 1
The GBTB basis functions have the subsequent features:

1.
Degeneracy: For \(\mu, \nu =1\) and \(\sin (\frac{\pi }{2}x), 1\cos (\frac{\pi }{2}x)=z\), the kth degree GBTB basis functions are just like the traditional kth degree Bernstein basis functions (\(g_{i,k}(x)=B_{i,k}(x)\), \(i=0,1,\ldots,k\); \(k\geq 2\)).

2.
Nonnegativity: \(g_{i,k}(x)\geq 0\) \((i=0,1,2,\ldots,k)\) for any value of \(\mu,\nu \in [1,1]\).

3.
Partition of unity: \(\sum^{k}_{i=0} g_{i,k}(x)=1\).

4.
Symmetry: For every \(\mu =\nu \), \(g_{i,k}(x)\ (i=0,1,2,\ldots,k)\) are symmetric i.e. \(g_{ki,k}(x, \mu, \nu )= g_{i,k}(1x, \mu, \nu )\).

5.
Linear independence: For any \(\mu,\nu \in [1,1]\), the \(kth\) order GBTB basis functions are linearly independent.

6.
Terminal properties: For all \(i=0,1,2,3,\ldots,k\); \(k\geq 2\), the GBTB basis functions \(g_{0,k}(0)=1\), \(g_{k,k}(1)=1\), \(g_{i,k}(0)=0\) \((i=1,2,\ldots,k)\), and \(g_{i,k}(1)=0\) \((i=0,1,\ldots,k1)\).

7.
Derivative at corner points: \(g'_{i,k}(0)\), \(g'_{i,k}(1)\), \(g''_{i,k}(0)\), \(g''_{i,k}(1)\), \(g'''_{i,k}(0)\), and \(g'''_{i,k}(1)\).
Proof
The authentication of all the above consequences is as demonstrated in [31]. □
Construction of GBTBézier curves with shape parameters
Definition 2
For any defined control points \(R_{i}\in \mathbf{R}^{k}\) \((k=2,3; i=0,1,\ldots,k)\), the curves
are familiar as GBTBézier curves corresponding to GBTB basis functions \(g_{i,k}(x)\).
As the GBTBézier curves are defined on the bases of GBTB basis functions, so the aforementioned features of GBTB basis functions demonstrate that the GBTBézier curves have most dominant features of the traditional Bézier curves inclusively, end point constrains \(G(0; \mu, \nu )\), \(G(1; \mu, \nu )\), \(G'(0; \mu, \nu )\), \(G'(1; \mu, \nu )\), \(G''(0; \mu, \nu )\), \(G''(1; \mu, \nu )\), \(G'''(0; \mu, \nu )\), and \(G'''(1; \mu, \nu )\), convexity, symmetry, variation diminishing feature, shape adjustment feature, and geometric invariance feature. These features specify that the GBTBézier curves interpolate to the end points of its convex hull, and also by choosing appropriate values of shape parameters \(\mu, \nu \) in their respective value range \(\mu,\nu \in [1,1]\), the shape of GBTBézier curves can be adjusted easily according to design requirements [31].
Shape adjustability of GBTBézier curves
A group of GBTBézier curves can be derived from expression (2.2) by taking the distinct values of their shape parameters \(\mu,\nu \) in their corresponding value range having control points \(R_{0},R_{1},R_{2},\ldots,R_{k}\). Owing to the reality that every GBTBézier curve assigns two local shape parameters μ, ν, hence the structure of a GBTBézier curve can be easily settled and modified by changing the values of these two shape parameters. From Definition 2, we acknowledge that the curve \(G(x;\mu,\nu )\) is a linear function of every shape parameter \(\mu,\nu \) as:
and
Therefore from equations (2.3) and (2.4), it is obvious that there is no relationship among \(\frac{\partial G(x)}{\partial \mu }\) and μ, and \(\frac{\partial G(x)}{\partial \nu }\) 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
Figure 2 portrays the graphs of cubic GBTBézier curves. The remarkable points on cubic GBTBézier curves related to \(G(0.2)\), \(G(0.4)\), \(G(0.6)\), \(G(0.8)\) are red, blue, black, and green respectively. When the shape parameters \(\mu,\nu \) take different values in their defined value range \([0, 1]\), Fig. 2 illustrates the adjustment role of shape parameters \(\mu,\nu \) on the shape of GBTBézier curves. Figure 2(a) represents the graph of GBTBézier curves with \(\mu,\nu =1\) (orange lines), 0.5 (purple lines), 0 (yellow lines), −0.5 (pink lines), and −1 (gray lines). Figure 2(b) displays the graph of GBTBézier curves with fixed μ (\(\mu =0\)) and modifying value of ν as \(\nu =1\) (orange lines), \(\nu =0.5\) (purple lines), \(\nu =0\) (yellow lines), \(\nu =0.5\) (pink lines), and \(\nu =1\) (gray lines). Figure 2(c) gives GBTBézier curves when μ takes different values as \(\mu =1\) (orange lines), \(\mu =0.5\) (purple lines), \(\mu =0\) (yellow lines), \(\mu =0.5\) (pink lines), \(\mu =1\) (gray lines), and the value of ν remains fixed (\(\nu =1\)). Figure 2(d) depicts the curves with \(\mu =1\) and \(\nu =1\) (orange lines), \(\nu =0.5\) (purple lines), \(\nu =0\) (yellow lines), \(\nu =0.5\) (pink lines), and \(\nu =1\) (gray lines). From all the above figures, we made the following two conclusions:

1.
By modifying the values of shape parameter either μ or ν or both, the points on the curves change linearly.

2.
With simultaneous increase in the values of shape parameters \(\mu,\nu \), the GBTBézier curves gradually move towards their convex hull.
Some curve design examples of GBTBézier curves with different shape parameters are shown in Figs. 3, 4 and 5.
Construction of developable GBTBézier surfaces with shape parameters
Dual generation of a singleparameter family of planes
As stated in the duality principle between points and planes, a singleparameter family of control points of a curve is dual to a singleparameter family of planes [11–13]. Thus by treating the control points of a GBTBézier curve as GBTBézier control planes, a singleparameter family of planes \(\{\Pi _{x}\}\) can be developed. Therefore the expression of a singleparameter family of planes \(\{\Pi _{x}\}\) of a GBTBézier curve is described by using expression (2.2) as follows:
where \(Q_{i}\ (i=0,1,2,\ldots,k)\) are control planes of \(\{\Pi _{x}\}\), \(\mu,\nu \) are shape parameters, and x is the family parameter of \(\{\Pi _{x}\}\). We imagine that \({Q_{i}=(p_{i},q_{i},r_{i},s_{i})}\), \(p_{i},q_{i},r_{i},s_{i}\in \mathbf{R}\ (i=0,1,2,\ldots,k)\) are the coordinates of the control points of a GBTBézier 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:
Let
Thus, equation(3.3) can be demonstrated as
Description of enveloping developable GBTBézier 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 singleparameter family of planes. Consequently, a developable GBTBézier surface is an envelope of a singleparameter family of planes \(\{\Pi _{x}\}\) of a GBTBézier curve. The crossing line of two successive planes analogous to any value of x in \(\{\Pi _{x}\}\) will surely lie on the enveloping developable GBTBézier surface of \(\{\Pi _{x}\}\). 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 \(J(x;\mu,\nu )\) of the developable GBTBézier surface corresponding to x is the crossing line of planes (3.5) and (3.6), lies on the developable GBTBézier surface of \(\{\Pi _{x}\}\), and can be calculated in terms of its plucker coordinates as follows [11, 12]:
where \(\mathbf{a}(x)=\{a_{0}(x),a_{1}(x),a_{2}(x)\}\), \(\mathbf{a}'(x)=\{a_{0}'(x),a_{1}'(x),a_{2}'(x)\}\). ϕ expresses the direction vector of the line \(J(x;\mu,\nu )\) and can also be expressed as
Let \(\psi (x)\) indicate the nearest point on the generator \(J(x;\mu,\nu )\) to the origin that can be calculated as follows [11, 12, 17, 21]:
Consequently, in a parametric form, the generator \(J(x;\mu,\nu )\) can be defined as
where \(\mu,\nu \) are the shape parameters. For distinct values of family parameter x in its given value range, all generators \(J(x;\mu,\nu )\) construct an enveloping developable GBTBézier surface having \(\mu,\nu \) as shape parameters. Hence, by using equation (3.7), an enveloping developable GBTBézier surface of \(\{\Pi _{x}\}\) can be described in its linear geometric representation.
Description of spine curve developable GBTBézier surfaces with shape parameters
A characteristic point \(S(x)\) in a single parameter family of planes \(\{\Pi _{x}\}\) of a GBTBézier 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 GBTBézier surface. From the intersection of equations (3.5) and (3.6) and the second derivatives of (3.5), the characteristic point of a GBTBézier 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]
where \(\mathbf{a}''(x)=\{a''_{0}(x),a''_{1}(x),a''_{2}(x)\}\).
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 GBTBézier surfaces, then the surface made up of the tangent lines of the spine curve \(S(x)\) is a spine curve developable GBTBézier surface. Thus, a parametric spine curve developable GBTBézier surface can be described as [17, 21]
Finally, in formulating developable GBTBézier surfaces (3.7) and (3.10), a singleparameter family of planes \(\{\Pi _{x}\}\) using GBTB basis as basis functions is acquired. Therefore, the developable surfaces introduced in this study are identified as developable GBTBézier surfaces (enveloping and spine curve). It is worth mentioning here that the developable GBTBézier surfaces are considered as a singleparameter family of planes, and there are numerous dissimilarities among developable GBTBézier surfaces and tensor product GBTBézier surfaces. Therefore, for their dual relationship, there are many identical features among GBTBézier curves and developable GBTBézier surfaces.
Some characteristics of developable GBTBézier surfaces
Geometric features of a GBTBézier 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 \(\{\Pi _{x}\}\) and the 1st and 2nd order derivatives of expression (3.2), we can derive some meaningful results as at \(x=0\) we have
and at \(x=1\) we have
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 \(\{\Pi _{x}\}\) 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 GBTBézier surface with its generators \(J(x;\mu,\nu )\). Furthermore, at \(x=0\), the generator \(J(0;\mu,\nu )\) 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 \(J(0;\mu,\nu )\) is the connection of the planes \(Q_{0}\) and \({\frac{1}{2}(2(k2)+\pi (1+\mu ))(Q_{1}Q_{0})}\) and can be expressed in the following vector form:
where \(t_{0}=(p_{0},q_{0},r_{0})\), \(t_{1}=(p_{1},q_{1},r_{1})\), \(\mathbf{W}=(X,Y,Z)\).
Clearly, from interpretation of 1st two planes of (3.13), the control plane \(Q_{1}\) can be attained. Thus \(J(0;\mu,\nu )\), the starting generator is the intersection of control plane \(Q_{0}\) and control plane \(Q_{1}\). Moreover, \(J(0;\mu,\nu )\) is tangent to the GBTBézier curve described by the control points \(Q_{i}\ (i=0,1,2,\ldots,k)\) as \(J(0;\mu,\nu )\) is dual to the line connecting the points \(Q_{0}\) and \(Q_{1}\) (as 3D Euclidean space control planes are treated as 4D homogeneous space control points). Similarly, at \(x=1\) the last two control planes, control plane \(Q_{k1}\) and control plane \(Q_{k}\), represent the ending generator of a developable GBTBézier surface denoted as \(J(1;\mu,\nu )\) obtained from the first two equations of expression (3.12). Consequently, expressions (3.11) and (3.12) can be interpreted in terms of matrices as follows:
and
where \(k_{1}=(k2)(k3+\pi (1+\mu ))\) and \(k_{2}=(k2)(k3+\pi (1+\nu ))\).
Owning to the fact that the characteristic point \(S(x)\) of spine curve developable GBTBézier surfaces is the intersection of the planes \(H(x; \mu,\nu )\), \(H'(x; \mu,\nu )\), \(H''(x; \mu,\nu )\), from expression (3.11), the coordinates of the control planes \(Q_{0}\), \(Q_{1}\), \(Q_{2}\) can be attained from the linear combination of \(H(0; \mu,\nu )\), \(H'(0; \mu,\nu )\), \(H''(0; \mu,\nu )\). Hence the intersection of the control planes \(Q_{0}\), \(Q_{1}\), \(Q_{2}\) is the characteristic point \(S(0)\) of the spine curve developable GBTBézier surfaces that occurs on the starting generator \(\mathbf{V}(x_{2},0; \mu,\nu )\). Identically the characteristic point \(S(1)\) of the spine curve lies on ending generator \(\mathbf{V}(x_{2},1; \mu,\nu )\), which can be achieved from the intersection of the control planes \(Q_{k2}\), \(Q_{k1}\), \(Q_{k}\) of expression (3.12).
Some designing pattern of developable GBTBézier surfaces
We can analyze from all the above discussion that a developable GBTBézier 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 \(\mu, \nu \). Therefore, based on control planes, some modeling examples of cubic and quartic developable GBTBézier surfaces using multiple values of shape parameter are presented here to clear the influence role of shape parameters in designing features of developable GBTBézier surfaces. In all designing examples of enveloping developable GBTBézier surfaces, we assume that the midpoints of all planes of the enveloping developable GBTBézier surfaces lie in the same plane and have the same distance from the origin.
Designing examples of enveloping developable GBTBézier surfaces
Example 4.1
Designing features of a cubic enveloping developable GBTBézier surface along various values of its shape parameters are demonstrated in this example. As described above in designing a cubic enveloping developable GBTBézier, it is assumed that the midpoints of the four control planes of a cubic enveloping developable GBTBézier surface have the same distance from the origin and are laid in the same plane. The four control planes of cubic enveloping developable GBTBézier surface are defined as follows:
To manifest the impact of shape parameters on the shape of a cubic enveloping developable GBTBézier surface apparently from the same viewpoint and coordinate system, Figs. 6–8 depict the graphs of cubic enveloping developable GBTBézier surfaces with multiple values of their shape parameters \(\mu, \nu \). From these figures, we can conclude that with fixed control planes, the shape parameters affect the shape of the enveloping developable GBTBézier surface in the following manners:

1.
Whenever the values of μ stay unchanged, the location of the starting generator \(J(0;\mu, \nu )\) and the length and location of the ending generator \(J(1;\mu, \nu )\) will not modify with modifying the value of ν; however the length of the starting generator \(J(0;\mu, \nu )\) will modify. In other words, increasing the value of ν will bring an increase in the length of the starting generator. Particularly at \(\nu =1\), the length of \(J(0;\mu, \nu )\) becomes maximum, see Fig. 6.

2.
For fixed values of ν, the location of \(J(0;\mu, \nu )\) and \(J(1;\mu, \nu )\) will not fluctuate with fluctuating values of μ, but the lengths of ending generators \(J(1;\mu, \nu )\) will modify. That is, the length of generators \(J(1;\mu, \nu )\) increases with increasing value of μ and will be maximum at \(\mu =1\), see Fig. 7.

3.
For identical values of \(\mu, \nu \), the length and location of both starting \(J(0;\mu, \nu )\) and ending \(J(1;\mu, \nu )\) generators will modify with modifying values of \(\mu, \nu \). With increasing values of \(\mu, \nu \), the length of both generators increases. At the same instant, the structure of the enveloping developable surface will also modify as the height enveloping developable GBTBézier surface increases with decreasing values of \(\mu, \nu \), see Fig. 8.
Example 4.2
In another construction of a developable GBTBézier surface, the control planes of the cubic enveloping developable GBTBézier surfaces are taken as follows:
We can analyze from Figs. 9–12 that, with fluctuating values of shape parameters \(\mu, \nu \), a class of enveloping developable GBTBézier surfaces can be designed on the requirement of provided control planes. A detailed effect of \(\mu, \nu \) on the shape of an enveloping developable GBTBézier surface is described as follows:

1.
When we fix the value of ν, the length of generator \(J(1;\mu, \nu )\) will modify, but the location of generator \(J(1;\mu, \nu )\) and the length and location of generator \(J(0;\mu, \nu )\) will not modify with fluctuating values of μ. In other words, the length of the ending generator will increase simultaneously with an increase in the value of μ, see Figs. 9–10.

2.
When we take positive values of \(\mu, \nu \), the length and location of \(J(0;\mu, \nu )\) and location of \(J(1;\mu, \nu )\) will not alter with the altering values of \(\mu, \nu \), but the length of the ending generator \(J(1;\mu, \nu )\) will modify, see Fig. 11.

3.
When we take negative values of \(\mu, \nu \), the location of \(J(0;\mu, \nu )\) and \(J(1;\mu, \nu )\) will not fluctuate with fluctuating values of \(\mu, \nu \); however, the length of the ending and starting generator will modify along with the shape of the enveloping developable GBTBézier surface, see Fig. 12.
Some designing examples of spine curve developable GBTBézier surfaces
To demonstrate the designing feature of a spine curve developable GBTBézier surface constructed from the tangent lines of the spine curve Sx, some design examples are given in this portion. For constructing a cubic spine curve developable GBTBézier surface, the central points of the control planes of a cubic spine curve developable GBTBézier surface are taken in four different quadrants I, II, III, IV with nonidentical distance from the origin, and the particular coordinates of control planes \(Q_{0}\), \(Q_{1}\), \(Q_{2}\), \(Q_{3}\) of a cubic spine curve developable GBTBézier surface are described as follows:
With different values of shape parameters μ and ν, a class of cubic spine curve developable GBTBézier surfaces can be designed on the requirement of provided control planes \(Q_{0}\), \(Q_{1}\), \(Q_{2}\), \(Q_{3}\) in Fig. 13 and Fig. 14 respectively, with distinct shapes.
Continuity requirements among developable GBTBézier surfaces
The designing of freeform 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 GBTBézier surfaces are \(G^{1}\) and \(G^{2}\) (Farin–Boehm and beta) continuity, etc. [15, 17, 21, 25, 30]. Now, we take an interpretation of GBTBézier curve having weight coefficients in 4D homogeneous space as
where \(g_{i,k}(z,\mu,\nu )\ (i=0,1,\ldots,k)\) are \(k^{th}\) order GBTB basis functions; \(\hat{R}=[\hat{R}_{0}, \hat{R}_{1},\ldots,\hat{R}_{k}]\), here \(\hat{R}_{i}=(\omega _{i} R_{i}, \omega _{i})\) is the weighted control point of \(R_{i}\ (i=0,1,\ldots,k)\) having weight factor \(\omega _{i}\). It is understood that the weighted GBTBézier curve share the basic features with GBTBézier curves. When the weighted control points \(\hat{R}_{i}=(\omega _{i} R_{i}, \omega _{i})\ (i=0,1,\ldots,k)\) in a 4D homogeneous space are considered as the control planes \(Q_{i}\ (i=0,1,\ldots,k)\) in a 3D Euclidean space, the planes \(\{\Pi _{x}\}\) constructed from control planes \(Q_{i}\ (i=0,1,\ldots,k)\) are dual to the weighted GBTBézier curves constructed from \(\hat{R}_{i}=(\omega _{i} R_{i}, \omega _{i})(i =0,1,\ldots,k)\). Hence some geometric design methods of developable GBTBézier surfaces (like continuity requirements, tangent planes, and terminal properties) in a 4D homogeneous space are identical to those of GBTBézier curves. Therefore, for continuity requirements of developable GBTBézier surfaces, it is imagined that the single parameter families of planes of two developable GBTBézier surfaces \(H_{1}(x;\mu _{1},\nu _{1})\) and \(H_{2}(x;\mu _{2},\nu _{2})\) of order k and l, respectively, that require to be joined together are defined as follows:
where \(\mu _{i}, \nu _{i}\ (i=1, 2)\) are shape parameters and the control planes of \(\{\Pi _{x,1}\}\) and \(\{\Pi _{x,2}\}\) are \(Q_{i,1}\ (i=0,1,\ldots,k)\) and \(Q_{j,2}\ (j=0,1,\ldots,l)\), respectively.
\(G^{1}\) continuity among developable GBTBézier surfaces
Here, we want to establish the first order geometric continuity or \(G^{1}\) continuity among two or more weighted GBTBézier curves in a 4D homogeneous space. Now suppose that the singleparameter families of planes of the two contiguous weighted GBTBézier curves, which need to be spliced together, are as follows:
where \(\mu _{i}, \nu _{i}\ (i=1,2)\) are shape parameters of two weighted GBTBézier curves \(\hat{G}_{i}(x,\mu _{i}, \nu _{i})\) \((i=1, 2)\). \(\hat{R}_{i,1}=(\omega _{i,1} R_{i,1}, \omega _{i,1})\ (i=0,1,\ldots,k)\) and \(\hat{R}_{j,2}=(\omega _{j,2} Q_{j,2}, \omega _{j,2})\ (j=0,1,\ldots,l)\) are control points of \(\hat{G}_{i}(x,\mu _{i}, \nu _{i})\ (i=1, 2)\), and \(\omega _{i,1}\ (i=0,1,\ldots,k)\) and \(\omega _{j,2}\ (j=0,1,\ldots,l)\) are weight factors of \(\hat{G}_{i}(x,\mu _{i}, \nu _{i})\ (i=1, 2)\) respectively. Thus, from the definition of \(G^{1}\) continuity among two parametric curves, the adequate and essential requirements for \(G^{1}\) continuous connection among two or more contiguous weighted GBTBézier curves \(\hat{G}_{1}(x)\) and \(\hat{G}_{2}(x)\) are
where \(\gamma > 0\) is a constant. For control points \(R_{i,j}\ (i=0,1,\ldots,k; j=1, 2)\) of two weighted GBTBézier curves \(\hat{G}_{i}(x,\mu _{i}, \nu _{i})\ (i=1, 2)\) in a 4D homogeneous space understood as the control planes in 3D Euclidean space, the geometric continuity among two curves in a 4D homogeneous space certifies the geometric continuity among two conforming developable surfaces developed by using the duality principle in a 3D Euclidean space. Hence, from expression (5.4), the subsequent result can be acquired.
Theorem 2
Adequate and essential requirements for \(G^{1}\) smooth connection among two contiguous developable GBTBézier surfaces \(H_{1}(x,\mu _{1},\nu _{1})\) of order k and \(H_{2}(x,\mu _{2},\nu _{2})\) of order l at the connection are
Proof
If \(H_{1}(x,\mu _{1},\nu _{1})\) and \(H_{2}(x,\mu _{2},\nu _{2})\) want to reach \(G^{1}\) continuity, it is compulsory that \(H_{1}(x,\mu _{1},\nu _{1})\) and \(H_{2}(x,\mu _{2},\nu _{2})\) reach \(G^{0}\) continuity at the common first, which implies
In addition, \(H_{1}(x,\mu _{1},\nu _{1})\) and \(H_{2}(x,\mu _{2},\nu _{2})\) want to share a common tangent plane at connection, that is to say,
According to (3.11) and (3.12), we have
Using (5.8) into (5.7), we get
Ultimately, on the ground of (5.6), we can achieve
Hence equations (5.6) and (5.10) comprise the adequate and essential requirements of \(G^{1}\) smooth connection betwixt two developable GBTBézier surfaces of order k and l sequentially. □
Farin–Boehm \(G^{2}\) continuity requirements among developable GBTBézier surfaces
Now, we will derive the Farin–Boehm \(G^{2}\) continuity requirements at the joints for the construction of piecewise developable GBTBézier surfaces. Similar to the \(G^{1}\) continuity, when the control points of the two GBTBézier curves \(\hat{G}_{1}(x,\mu _{1},\nu _{1})\), \(\hat{G}_{2}(x,\mu _{2},\nu _{2})\) in a 4D homogeneous space are taken as control planes \(Q_{i,j}\ (i=1,2,\ldots,k; j=1,2)\) in a 3D Euclidean space, then the Farin–Boehm \(G^{2}\) continuity among two GBTBézier curves in a 4D homogeneous space ensures the Farin–Boehm \(G^{2}\) continuity among two developable GBTBézier surfaces determined by \(Q_{i,j}\ (i=1,2,\ldots,k; j=1,2)\) in a 3D Euclidean space [22]. Therefore, when two contagious developable GBTBézier surfaces in (5.2) want to reach Farin–Boehm \(G^{2}\) continuity, they must satisfy the following requirements:
Equation (5.2) represents that the developable GBTBézier surfaces defined by \(H_{1}(x, \mu _{1},\nu _{1})\), \(H_{2}(x,\mu _{2},\nu _{2})\) need to have the same tangent plane, characteristic point, and generator at the joint.
Theorem 3
The essential and satisfactory requirements for achieving Farin–Boehm \(G^{2}\) continuity betwixt two contiguous developable GBTBézier surfaces \(H_{1}(x,\mu _{1},\nu _{1})\) of order k and \(H_{2}(x,\mu _{2},\nu _{2})\) of order l at the connection are
where \({c= \frac{2(k2)+\pi (1+\nu _{1})}{2(l2)+\pi (1+\mu _{2})},}\) \({c_{k}=4(k2)(k3+\pi (1+\nu _{1}))}\), \({c_{l}=4(l2)(l3+\pi (1+\mu _{2})).}\)
Proof
If \(H_{1}(x,\mu _{1},\nu _{1})\) and \(H_{2}(x,\mu _{2},\nu _{2})\) want to reach Farin–Boehm \(G^{2}\) continuity, it is essential that \(H_{1}(x,\mu _{1},\nu _{1})\) and \(H_{2}(x,\mu _{2},\nu _{2})\) fulfil \(G^{1}\) continuity at the joint boundary first, which implies
also
According to (3.11) and (3.12), we have
Simply by using the values of \(Q_{0,2},Q_{1,2}\) from expression (5.6) into expression (5.8), we can get the required Farin–Boehm \(G^{2}\) continuity requirements (5.5). □
\(G^{2}\) beta continuity among developable GBTBézier surfaces
Theorem 4
For a smooth continuous connection among two adjoining developable GBTBézier surfaces \(H_{1}(x,\mu _{1},\nu _{1})\) and \(H_{2}(x,\mu _{2},\nu _{2})\) of order k and l, respectively, the necessary and sufficient \(G^{2}\) beta continuity requirements are
where \({c= \frac{2(k2)+\pi (1+\nu _{1})}{2(l2)+\pi (1+\mu _{2})},}\) \({c_{k}=4(k2)(k3+\pi (1+\nu _{1}))}\), \({c_{l}=4(l2)(l3+\pi (1+\mu _{2})).}\)
Proof
To reach \(G^{2}\) beta continuity among \(H_{1}(x,\mu _{1},\nu _{1})\) and \(H_{2}(x,\mu _{2},\nu _{2})\), there are some subsequent requirements which must be satisfied at the common boundary
where \(\gamma >0\) and λ are constants. From the terminal properties of developable GBTBézier surfaces and substituting expressions (3.11) and (3.12) into expression (5.17), we can achieve \(G^{2}\) beta continuity expression (5.16). Hence the three equations of expression (5.16) establish the adequate and essential requirements of \(G^{2}\) beta continuity requirements among two or more developable GBTBézier surfaces. □
Steps for smooth continuous connection between two adjacent developable GBTBézier surfaces
Using smooth continuity requirements among two contiguous developable surfaces, a huge amount of complicated surfaces can be made easily. To show \(G^{2}\) smooth continuity (Farin–Boehm and beta) steps betwixt two adjoining developable GBTBézier surfaces, an example is used here. For establishing a \(G^{2}\) beta continuous connection among two or more contiguous cubic enveloping developable GBTBézier surfaces, the following steps are performed.
Step 1. Preliminary developable GBTBézier surface \(H_{1}(x; \mu _{1}, \nu _{1})\), its control planes \(Q_{i,1}\ (i=0,1,2,3)\), and shape parameters \(\mu _{1}, \nu _{1}\) can be given openly.
Step 2. Let \(Q_{k,1}=Q_{0,2}\), or in other words, the developable GBTBézier surfaces expressed by \(H_{1}(x; \mu _{1}, \nu _{1})\) and \(H_{2}(x; \mu _{2}, \nu _{2})\)) have a combined control plane to fulfil the \(G^{0}\) continuity condition.
Step 3. For any provided values of \(\mu _{2}, \nu _{2}\) and constant \(\gamma >0\), the 2nd control plane \(Q_{1,2}\) of the developable GBTBézier surface \(H_{2}(x; \mu _{2}, \nu _{2})\) can be calculated from the second equation of (5.16).
Step 4. From the \(3rd\) equation of (5.16), the control plane \(Q_{2,2}\) of the developable GBTBézier surface \(H_{2}(x; \mu _{2},\nu _{2})\) 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 \(H_{2}(x; \mu _{2}, \nu _{2})\) can be selected openly for establishing \(G^{2}\) beta smooth continuous connection among two or more adjoining developable GBTBézier surfaces.
From iteration of the above steps among two developable GBTBézier surfaces, \(G^{2}\) beta smooth continuity can be attained among multiple developable GBTBézier surfaces, and it can also be applied on other continuity conditions in the same manners.
Some modeling examples of smooth developable GBTBézier surfaces
This portion will give some designing examples of \(G^{2}\) smooth connection (beta and Farin–Boehm) among two adjoining cubic developable GBTBézier surfaces sequentially. Additionally, the effect of shape parameters on combined surfaces is also examined.
Example 5.1
Figures 15–16 exhibit the graphs of Farin–Boehm \(G^{2}\) continuity among two adjoining cubic enveloping developable GBTBézier surfaces. In these figures, the blue surfaces are primary developable GBTBézier surfaces with subsequent control planes
The green surfaces are the secondary enveloping developable GBTBézier 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 GBTBézier surfaces with fixed value of ν. These four graphs indicate that without disturbing control planes, the shape of the combined developable GBTBézier surface can be simply modified by amending the shape parameters. Figure 16 represents the impact of shape parameter ν on the combined developable GBTBézier surface for a fixed value of μ.
Example 5.2
Geometric design of \(G^{2}\) beta continuity among two contiguous cubic enveloping developable GBTBézier surfaces is illustrated in Fig. 17. In Fig. 17, the green surface represents first cubic enveloping developable GBTBézier surface \(H1(x;\mu _{1}, \nu _{1})\) with control planes described in (5.18), whereas red surface represents the 2nd cubic enveloping developable GBTBézier surface \(H2(x;\mu _{2}, \nu _{2})\) which fulfills the \(G^{2}\) beta continuity requirements with \(H1(x;\mu 1, \nu 1)\). By setting all shape parameters \(\mu _{1}, \nu _{1}, \mu _{2}, \nu _{2}\) of the corresponding cubic enveloping developable GBTBézier surfaces equal to 1, the coordinates of the control planes of cubic enveloping developable GBTBézier surface \(H2(x;\mu _{2}, \nu _{2})\) 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) (\(\gamma =1, \lambda =0\)). Figure 17 represents the combined enveloping developable GBTBézier surface having \(G^{2}\) beta connection among them after amending the values of its shape parameters, while Fig. 18 indicates the combined enveloping developable GBTBézier surface for multiple values of scale factor γ. Figure 18 indicates that a unified merged developable GBTBézier surface can be achieved by setting scaling factor \(\gamma =1\), unconcerned of modifying the shape parameter, and at connection a gap will be achieved on merged developable GBTBézier surfaces when \(\gamma \neq 1\). We can use these characteristics to design a complex surface according to our needs.
Conclusions
In this research work, developable GBTBézier 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 GBTBézier surfaces have been examined. Additionally, the geometric continuity requirements (\(G^{1}\), \(G^{2}\) Farin–Boehm and \(G^{2}\) beta) among two contiguous developable GBTBézier surfaces have been derived for the construction of complicated developable GBTBézier surfaces. These proposed developable GBTBézier surfaces have been shown to reduce the flaws of line and plane geometric description in modeling developable GBTBézier 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.
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Acknowledgements
We thank Dr. Muhammad Amin for his assistance in proofreading of the manuscript. This research was supported by the Department of Mechanical Engineering, Shizuoka University, Hamamatsu, 4328561 Shizuoka, Japan.
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Maqsood, S., Abbas, M., Miura, K.T. et al. Shapeadjustable developable generalized blended trigonometric Bézier surfaces and their applications. Adv Differ Equ 2021, 459 (2021). https://doi.org/10.1186/s13662021036143
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DOI: https://doi.org/10.1186/s13662021036143
Keywords
 GBTB basis functions
 Shape control of developable GBTBézier curve
 Developable GBTBézier surfaces
 Duality
 Enveloping developable GBTBézier surfaces
 Spine curve developable GBTBézier surfaces
 Properties
 Continuity conditions
 Modeling examples