3.1 GBTBézier cylinder with shape parameters
Let \({F_{m}(z, \mu , \nu )=\sum^{m}_{k=0}Q_{k}f_{k,m}(z), z \in [0,1]}\) be a GBTBézier curve of degree m and V be a 3D unit vector, the representation of the GBTBézier cylinder \(T(z,z_{1},\mu , \nu )\) is obtained by sweeping \({F_{m}(z, \mu , \nu )}\) a distance α (\(\alpha >0\)) along V [25]. For sweep direction representing the parameter by \(z_{1}\), clearly the GBTBézier cylinder should meet the following requirements:

(i)
The curve \(T(\overline{z},z_{1}, \mu , \nu )\) is a straight line segment from point \(F_{m}(\overline{z}, \mu , \nu )\) to \(F_{m}(\overline{z}, \mu , \nu )+ \alpha \mathbf{V}\) for fixed z̅ (\(0\leq \overline{z} \leq 1\)).

(ii)
For fixed \(\overline{z}_{1}\) (\(0\leq \overline{z}_{1} \leq 1\)), we have
$$\begin{aligned} T(z, \overline{z_{1}}, \mu , \nu ) =&F_{m}(z, \mu , \nu )+ \overline{z}_{1}\alpha \mathbf{V} \\ =& \sum^{m}_{k=0}(Q_{k}+ \overline{z}_{1}\alpha \mathbf{V})f_{k,m}(z). \end{aligned}$$
(8)
Proposition 3.1
The translational invariance property is used to define GBTBézier curves in (8). Thus, in the perspective of the aforementioned conditions, the expression of the GBTBézier cylinder can be described as follows:
$$ T_{\mathrm{cylinder}}(z, z_{1}, \mu , \nu )=\sum ^{m}_{k=0}(Q_{k}+z_{1}\alpha \mathbf{V})f_{k,m}(z),\quad z,z_{1}\in [0,1], $$
(9)
where \(\alpha > 0\) is a certain real constant. Taking \({F_{m}(z, \mu , \nu )}\) as directrix of the GBTBézier cylinder, expression (9) is called the GBTBézier cylinder.
Moreover, the GBTBézier cylinder \({T_{\mathrm{cylinder}}}\) is a surface which is generated by translating a small portion of the straight line from \({F_{m}(0)}\) to \({F_{m}(0)+ \alpha \mathbf{V}}\) along an assigned curve \({F_{m}(z, \mu , \nu )}\). Anyhow, the cylinder in (9) leads to some complications in designing GBTBézier curves and surfaces as it is not a tensor product GBTBézier surface. To overcome this problem, equation (9) is transformed into the tensor product GBTBézier surface described in (7). It is observed that the mthdegree GBTB basis functions defined by (2) are only described for parameter m (\(m\geq 2\)), but they are not given for \(m=1\). Thus, extending the definition of GBTB basis functions, we describe the firstdegree GBTB basis functions as
$$ {f_{k,1}(z)=(1z)^{1k}z^{k}}\quad (k=0, 1). $$
(10)
Expression (10) shows that the GBTB basis functions of order one are the traditional Bernstein basis functions of order one.
Theorem 3
A GBTBézier cylinder in (9) can also be written in its tensor product form as follows:
$$ T_{\mathrm{cylinder}}(z, z_{1}, \mu , \nu , \mu _{1}, \nu _{1})=\sum^{m}_{k=0} \sum^{1}_{l=0} Q_{k,l}f_{k,m}(z) f_{l,n}(z_{1}),\quad z,z_{1} \in [0,1], $$
(11)
where \({Q_{k,0}=Q_{k}, Q_{k,1}= Q_{k}+d \mathbf{V}, k=0,1,\ldots,m}\), are control points, \(f_{k,m}(z)\), \({f_{l,1}(z_{1})}\), \(l=0,1\), are GBTB basis functions described by (2) and (10), respectively, and μ, ν are shape control parameters.
Proof
By taking control points \({Q_{k,0}=Q_{k}}\) and \({Q_{k,1}= Q_{k}+\alpha \mathbf{V}= Q_{k,0}+\alpha \mathbf{V}}\) and basis function given in (10), we have
$$\begin{aligned} T_{\mathrm{cylinder}}(z, z_{1}, \mu , \nu ) =&\sum ^{m}_{k=0}(Q_{k}+z_{1}d \mathbf{V})f_{k,m}(z) \\ =&\sum^{m}_{k=0} \bigl[Q_{k,0}+z_{1}(Q_{k,1}Q_{k,0}) \bigr]f_{k,m}(z) \\ =&\sum^{m}_{k=0} \bigl[(1z_{1})Q_{k,0}+z_{1}Q_{k,1} \bigr]f_{k,m}(z) \\ =&\sum^{m}_{k=0}\sum ^{1}_{l=0} Q_{k,l}f_{k,m}(z) f_{l,1}(z_{1}). \end{aligned}$$
(12)
Therefore, GBTBézier cylinder in (9) can be exactly expressed in terms of a tensor product GBTBézier surface of degree \((m,1)\). Hence the proof of the result. □
Since a GBTBézier curve is the directrix of a GBTBézier cylinder, it carries some remarkable benefits of the general Bézier cylinder. Furthermore, the GBTBézier cylinder (11) has an edge that by keeping the control mesh unchanged, the GBTBézier cylinder can be transformed into any shape merely by changing the shape parameters.
Here, some detailed steps are described to design a GBTBézier cylinder.
Algorithm 1

1.
Choose an arbitrary 3D unit vector V and a GBTBézier curve as the directrix of GBTBézier cylinder.

2.
Describe a GBTBézier cylinder in the manner as given in equation (11).

3.
Take control points \({Q}_{k,l}\) as \({Q}_{k,0}= {Q}_{k}\), \({Q}_{k,1}= {Q}_{k}+\alpha \mathbf{V}\) (\(k=0,1,2,\ldots,m\)), where α is the distance along a normal vector of the plane.

4.
Select the distance α and draw the GBTBézier cylinder with different shape parameters via Wolfram Mathematica software.

5.
GBTBézier cylinder can also be drawn with varying values of α and keeping the values of shape parameters unchanged.
Example 3.1
A model of GBTBézier cylinder with GBTBézier curve as directrix is given in this example. Suppose that a GBTBézier curve \(F_{4}(z;\mu ,\nu )\) is constructed by five control points \(Q_{0}=(5, 2, 0)\), \(Q_{1}=(0,12,0)\), \(Q_{2}=(10,20,0)\), \(Q_{3}=(20,12,0)\), \(Q_{4}=(15, 2, 0)\). Now we generate a GBTBézier cylinder by interpreting \(F_{4}(z;\mu ,\nu )\) a distance α with the normal vector of the plane (here \(\alpha =20\), \(\mathbf{V}=(0, 0, 1)\)) and taking the curve \(F_{4}(z;\mu ,\nu )\) as the directrix. By Theorem 3, the equation of the GBTBézier cylinder can be represented as follows:
$$ T_{\mathrm{cylinder}}(z, z_{1}, \mu , \nu )=\sum ^{4}_{k=0}\sum^{1}_{l=0} Q_{k,l}f_{k,4}(z) f_{l,1}(z_{1}), \quad z,z_{1}\in [0,1], $$
(13)
where \(Q_{k,0}=Q_{k}\), \(Q_{k,1}=Q_{k}+(0, 0, 20)\) (\(k=0,1,\ldots,4\)) are control net points. Figure 4 represents the GBTBézier cylinders with 10 mesh points for fixed values of \(\alpha =20\) and different shape parameters μ, ν which clearly make its confined convex hull.
A closed GBTBézier cylinder is illustrated in Fig. 5 to show the impact of α and shape control parameters μ, ν on GBTBézier cylinder, whereas the \(C^{2}\) continuity of two GBTBézier cylinders is illustrated in Fig. 6 with shape control parameters μ, \(\mu _{1}\), ν, \(\nu _{1}\). Figures 4, 5, and 6 present that the effect of shape parameters on GBTBézier cylinders (open or close) is the same as the effect on GBTBézier curves and surfaces.
3.2 GBTBézier bilinear surface with shape parameters
The authors in [25] proposed a nonuniform rational Bspline (NURBS) representation of the surface (nonrational) called bilinear surface by bilinearly interpolating among four line segments, \({Q_{0,0}Q_{1,0}}\), \({Q_{0,1}Q_{1,1}}\), \({Q_{0,0}Q_{0,1}}\), and \({Q_{1,0}Q_{1,1}}\) using four control points \({Q_{0,0},Q_{1,0}}\), \({Q_{0,1}, Q_{1,1}}\) in a 3D space which is given by
$$ T_{\mathrm{bi}\text{}\mathrm{linear}}(z, z_{1}, \mu , \nu , \mu _{1}, \nu _{1})=\sum^{1}_{k=0} \sum^{1}_{l=0} Q_{k,l}N_{k,1}(z) N_{l,1}(z_{1}),\quad z,z_{1} \in [0,1], $$
(14)
where \(\mathbf{Z}=\mathbf{Z_{1}}=\{0, 0, 1, 1\}\) is the node vector and \(Q_{k,l}\), \(k,l\in {0,1}\) are taken as control points. Among opposite boundary lines, the bilinear surface (14) performs a clear linear interpolation in both direction [25].
Proposition 3.2
The characteristics of the bilinear surface (14) are as follows: \((a)\) given angular points are interpolated by the surface; \((b)\) line segments are taken as boundary curves; \((c)\) the curves \({T(\overline{z}, z_{1})}\) or \({T(z, \overline{z}_{1})}\) are the line segments for fixed z̅ or \(\overline{z}_{1}\). The transformation of a bilinear surface in terms of tensor product GBTBézier surface is given by
$$ T_{\mathrm{bi}\textit{}\mathrm{linear}}(z, z_{1}, \mu , \nu , \mu _{1}, \nu _{1})=\sum^{1}_{k=0} \sum^{1}_{l=0} Q_{k,l}f_{k,1}(z) f_{l,1}(z_{1}), \quad z,z_{1} \in [0,1], $$
(15)
where \(f_{k,1}(z)\), \(f_{l,1}(z_{1})\) (\(k,l=0,1\)) are basis functions defined in (10).
However, the bilinear surfaces described in (14) and (15) cannot be modified. This inconvenience of shape adjustment is due to the given four angular points. So, we need to generalize the classical bilinear surface (14) into a GBTBézier bilinear surface which is defined by taking a GBTBézier curve. Some properties of such surfaces are given as follows:

1.
Angular points are interpolated by the surface;

2.
Line segments are taken as the boundary curves of the surface;

3.
\(T_{m,n}(\overline{z}, z_{1})\) or \(T_{m,n}(z, \overline{z}_{1})\) is a GBTBézier curve for fixed z̅ or \(\overline{z}_{1}\);

4.
The model of GBTBézier surface is used by the bilinear surface.
From the characteristics of GBTBézier curves and surfaces, the actual procedure for developing a GBTBézier bilinear surface is described. The steps of the algorithm are as follows.
Algorithm 2

1.
In a 3D space, take \({Q_{0,0}, Q_{m,0}, Q_{0,n}, Q_{m,n}}\) to be the four control points. These points are used as the angular points of the GBTBézier surface.

2.
Split the line segments \({Q_{0,0}Q_{m,0}}\), \({Q_{0,n}Q_{m,n}}\) and \({Q_{0,0}Q_{0,n}}\), \({Q_{m,0}Q_{m,n}}\) into m and n same parts (an irregular division can also be made) accordingly and indicate the central partition points as \({Q_{1,0}}, {Q_{2,0}},\ldots, {Q_{m1,0}}\), \({Q_{1,n}}, {Q_{2,n}},\ldots, {Q_{m1,n}}\) and \({Q_{0,1}}, {Q_{0,2}},\ldots, {Q_{0,n1}}\), \({Q_{m,1}}, {Q_{m,2}},\ldots, {Q_{m,n1}}\). Moreover all the central partition points and \(Q_{0,0}\), \(Q_{m,0}\), \(Q_{0,n}\), \(Q_{m,n}\) defined above are used as the exterior control mesh points of the GBTBézier surface.

3.
Connect the control points achieved in step 1 and 2 with the remaining mesh points \(Q_{k,l}\) (\(k=1,2,\ldots,m  1\); \(l=1,2,\ldots,n1\)) to form the following GBTBézier bilinear:
$$ T_{\mathrm{bi}\textit{}\mathrm{linear}}(z, z_{1}, \mu , \nu , \mu _{1}, \nu _{1})=\sum^{m}_{k=0} \sum^{n}_{l=0} Q_{k,l}f_{k,m}(z) f_{l,n}(z_{1}), \quad z,z_{1}\in [0,1]. $$
(16)
Modeling example of a third degree GBTBézier bilinear surface is given below.
Example 3.2
Let \({Q_{0,0}=(0, 0, 0)}\), \({Q_{3,0}=(3, 0, 0)}\), \({Q_{0,3}=(0, 3, 0)}\), \({Q_{3,3}=(3, 3, 0)}\) be the four angular points of the bilinear surface and
$$ \textstyle\begin{cases} Q_{1,0}=(1, 0, 0),\quad\quad Q_{2,0}=(2, 0, 0),\quad\quad Q_{1,3}=(1, 3, 0), \quad\quad Q_{2,3}=(2, 3, 0), \\ Q_{0,1}=(0, 1, 0), \quad\quad Q_{0,2}=(0, 2, 0), \quad\quad Q_{3,1}=(3, 2, 0), \quad\quad Q_{3,2}=(3, 2, 0) \end{cases} $$
be the eight central partition points produced by dividing the line segments \({Q_{0,0}Q_{3,0}}\), \({Q_{0,3}Q_{3,3}}\), \({Q_{0,0}Q_{0,3}}\), and \({Q_{3,0}Q_{3,3}}\) into three equal parts. Decisively, the remaining control points \({Q_{1,1}=(1, 1, 3)}\), \({Q_{2,1}=(2, 1, 3)}\), \({Q_{1,2}=(1, 2, 3)}\), \({Q_{2,2}=(2, 2, 3)}\) can be chosen freely.
By using control points and equation (15), the GBTBézier bilinear surface can be designed as shown in Fig. 7 and Fig. 8. Bilinear surfaces have sixteen control points, in which the exterior twelve control mesh points are coplanar. Furthermore, Fig. 8 exhibits the graphs of the GBTBézier bilinear surfaces that have four noncoplanar boundary curves.
3.3 GBTBézier ruled surface along shape parameters
Let \(b_{k}(z)\) (\(k = 1, 2\)) be any two space curves (Bspline curves, Bézier curves, or NURBS curves), then in \(z_{1}\) ruled direction, a ruled surface \(T(z, z_{1})\) [25] is created if a linear interpolation is implemented among isoparametric points of the two curves \(b_{1}(z)\) and \(b_{2}(z)\). Also, a ruled surface has an astonishing property that, for fixed z, \(T(\overline{z}, z_{1})\) is a line segment joining points \(b_{1}(\overline{z})\) and \(b_{2}(\overline{z})\) (line segments are called straight generatrices, whereas the curves \(b_{1}(z)\) and \(b_{2}(z)\) are named as guide lines, for short GLs). This portion particularly demonstrates how the ruled surface with shape parameters is expressed using tensor product GBTBézier surface design. The necessary condition to express the ruled surface into tensor product form is that both should have the same degree. Let
$$ \textstyle\begin{cases} F_{m,0}(z, \mu , \nu )=\sum^{m}_{k=0} f^{0}_{k,m}(z)Q_{0,k},\quad 0\leq z \leq 1, \\ F_{m,1}(z, \mu _{1}, \nu _{1})=\sum^{m}_{k=0} f^{1}_{k,m}(z)Q_{1,k}, \quad 0 \leq z\leq 1, \end{cases} $$
(17)
be the two GLs of the ruled surface of degree m, where \(f^{l}_{k,m}(z)\), \(k=0,1,\ldots,m\), \(l=0,1\), are the GBTB functions described by (2). The GBTBézier ruled surfaces \({T_{\mathrm{ruled}}(z, z_{1})}\) should fulfil the following two conditions according to its definition [25]:

(1)
For fixed value of z̅, the curve \({T_{\mathrm{ruled}}(\overline{z}, z_{1})}\) is a line segment.

(2)
The guild lines fulfil the \({T_{\mathrm{ruled}}(z, 0)=F_{m,0}(z, \mu , \nu )}\) and \({T_{\mathrm{ruled}}(z, 1)=F_{m,1}(z, \mu _{1}, \nu _{1})}\).
In accordance with the definition of GBTBézier surfaces, the equation of the GBTBézier ruled surfaces \(T_{\mathrm{ruled}}(z, z_{1})\) that meet the aforementioned requirement is
$$ T_{\mathrm{ruled}}(z, z_{1}, \mu , \nu , \mu _{1}, \nu _{1})=\sum^{1}_{l=0} \Biggl( f_{l,1}(z_{1})\sum^{m}_{k=0} f_{k,m}(z)Q_{l,k} \Biggr), \quad 0\leq z, z_{1}\leq 1, $$
(18)
where μ, \(\mu _{l}\), ν, \(\nu _{l}\) are the shape parameters; \(Q_{l,k}\) (\(l=0,1\); \(k=0,1,\ldots,m\)) are the mesh points defined by (17) and \({f_{k,m}(z)}\) and \({f_{l,1}(z_{1})}\) are the GBTB basis functions given in (2) and (10), respectively. Equation (18) represents the tensor product GBTBézier surface in the form of a GBTBézier modified ruled surface. The GBTBézier ruled surfaces defined in (18) take over all characteristics and benefits of GBTBézier surfaces because these surfaces are expressed in the form of tensor product GBTBézier surfaces and the GLs are two GBTBézier curves that have the same degree. The NURBS modeled ruled surface defined in [25] has some limitations, such like complex mathematics, ambiguous weight elements, the annoyance of finding the surface order and the node vector [26]. On the other hand, the required surface has some benefits including straightforward computation, obvious implication of the shape parameters, and the free choice of vertexes.
Proposition 3.3
It is worth recalling that the straight generatrices of a GBTBézier ruled surfaces are the straight lines that are obtained by joining isoparametric points, instead of joining the same length of an arc of the two GLs. Generally, joining the same length of an arc of any two GLs gives us a different ruled surface.
Proposition 3.4
The technique of splicing GBTBézier surfaces can be utilized to generate all types of complicated combined GBTBézier ruled surfaces for the reason that the GBTBézier ruled surface is a special case of tensor product GBTBézier surface. In this technique, the two adjacent GBTBézier ruled surfaces satisfy \(C^{1}\), \(C^{2}\), and \(C^{3}\) continuity conditions because both GLs of the ruled surfaces satisfy these conditions (see Fig. 10and Fig. 11).
The algorithm for designing a GBTBézier ruled surface is as follows.
Algorithm 3

1.
Consider any two GLs in the form of GBTBézier curves.

2.
Use tensor product form of the GBTBézier ruled surface as given in (18).

3.
The guild lines must satisfy the conditions \({T_{\mathrm{ruled}}(z, 0)=F_{m,0}(z, \mu , \nu )}\) and \({T_{\mathrm{ruled}}(z,1)=F_{m,1}(z, \mu _{1}, \nu _{1})}\).

4.
Take the control points \(Q_{i,j}\) in a 3D plane according to designer choice.

5.
Draw the GBTBézier ruled surface in Wolfram Mathematica software with suitable values of shape parameters.

6.
Two adjacent GBTBézier ruled surfaces can be connected by \(C^{1}\), \(C^{2}\), and \(C^{3}\) continuity conditions given in Theorem 2.
Example 3.3
Let \(F_{4,l}(z, \mu _{1}, \nu _{1})\) (\(l=0, 1\)) be two GBTBézier curves of degree 4 in a 3dimensional space, then from equation (18) the equation of the GBTBézier ruled surfaces formed by these GBTBézier curves as the GLs is given by
$$ T_{\mathrm{ruled}}(z, z_{1}, \mu _{0}, \nu _{0}, \mu _{1}, \nu _{1})=\sum ^{1}_{l=0} \Biggl( f_{l,1}(z_{1}) \sum^{4}_{k=0} f^{l}_{k,m}(z)Q_{l,k} \Biggr), \quad 0\leq z, z_{1}\leq 1, $$
(19)
where
$$ \textstyle\begin{cases} Q_{1,0}=(1, 0, 4), \quad\quad Q_{0,1}=(2, 0, 8), \quad\quad Q_{0,2}=(4, 0, 8), \\ Q_{0,3}=(5, 0, 4),\quad\quad Q_{0,4}=(6, 0, 8), \\ Q_{0,1}=(1, 6, 4),\quad\quad Q_{1,1}=(2, 6, 0),\quad\quad Q_{1,2}=(4, 6, 0),\\ Q_{1,3}=(6, 6, 4),\quad\quad Q_{1,4}=(7, 6, 2), \end{cases} $$
are the control points of GLs. Figure 9 illustrates the GBTBézier ruled surfaces determined by (19) with various shape parameters.
Two adjacent GBTBézier ruled surfaces satisfying the \(C^{1}\) and \(C^{2}\) continuity in z direction can be presented in Figs. 10 and 11 with third and fourth degree GBTBézier curves taken as GLs of the right and left GBTBézier ruled surface, respectively. Figure 12 shows the \(C^{3}\) continuity of two adjacent GBTBézier ruled surfaces that have forth degree GBTBézier curves as GLs in z direction.
3.4 GBTBézier swung surface along shape parameters
A swung surface can be generated from the surface of revolution [25]. Using a tensor product GBTBézier surface design with shape parameters, a swung surface is described in this portion. Let
$$ \textstyle\begin{cases} F_{m}(z, \mu , \nu )=\sum^{m}_{k=0} f_{k,m}(z)Q_{k,0}, \quad 0 \leq z\leq 1, \\ F_{n}(z_{1}, \mu _{1}, \nu _{1})=\sum^{n}_{l=0} f_{l,n}(z_{1})Q_{1,l}, \quad 0\leq z_{1}\leq 1, \end{cases} $$
(20)
be the two GBTBézier curves described by (3), where \(Q_{k,0}=(Q^{X}_{k,0}, 0, Q^{Z}_{k,0})\) and \(Q_{l,1}=(Q^{X}_{l,1}, Q^{Y}_{l,1}, 0)\) are their control points, and \(F_{m}(z, \mu , \nu )\) and \(F_{n}(z_{1}, \mu _{1}, \nu _{1})\) are known as profile curves (PCs) and trajectory curves (TCs), respectively. They are described in the \(XOZ\) plane and \(XOY\) plane, respectively. Vector form of the these curves and their corresponding nonzero coordinate functions \({F^{X}_{m}(z), F^{Z}_{m}(z)}\) and \({F^{X}_{n}(z_{1}), F^{Y}_{n}(z_{1})}\) are used to construct a swung surface given by
$$ \textstyle\begin{cases} F_{m}(z, \mu , \nu )=\{F^{X}_{m}(z), 0, F^{Z}_{m}(z)\}, \\ F_{n}(z_{1}, \mu _{1}, \nu _{1})=\{F^{X}_{n}(z_{1}), F^{Y}_{n}(z_{1}), 0\}, \end{cases} $$
(21)
and the GBTBézier swung surface is defined by [25, 26]
$$ T_{\mathrm{swung}}(z, z_{1})= \bigl( \lambda F^{X}_{m}(z) F^{X}_{n}(z_{1}), \lambda F^{X}_{m}(z) F^{Y}_{n}(z_{1}), F^{Z}_{m}(z) \bigr), $$
(22)
where λ (\(\lambda > 0\)) is an arbitrary scaling factor.
Proposition 3.5
The GBTBézier swung surface expressed by (22) has the following geometric characteristics: if the profile curve \({F_{m}(z)}\) is swung about zaxis and rescale in the X and Y directions, the swung surface \({T_{\mathrm{swung}}(z, z_{1})}\) is achieved in Fig. 13with different values of scaling factor λ.
Proposition 3.6
Here, some properties of z and \(z_{1}\) lines of swung surfaces are as follows:

1.
If the z line is fixed, the curves \({T_{\mathrm{swung}}(\overline{z}, z_{1})}\) which have identical trajectory curve \(F_{n}(z_{1})\) but are scaled as \(\lambda F^{X}_{m}(z)\) in X and Y directions are produced.

2.
The fixed \(\overline{z}_{1}\) line produces curves \({Z_{\mathrm{swung}}(z, \overline{z}_{1})}\) with the same profile curve \(F_{m}(z)\), and the connection between the curves \({T_{\mathrm{swung}}(z, \overline{z}_{1})}\) and the PCs \(F_{m}(z)\) is: the resulting curves \({T_{\mathrm{swung}}(z, \overline{z}_{1})}\) are achieved by rotating the PCs \(F_{m}(z)\) into the plane with the vector \({(F^{X}_{n}(\overline{z}_{1}), F^{Y}_{n}(\overline{z}_{1}), 0)}\), and use the scale factor \({\lambda \vert F_{n}(z_{1}) \vert }\) to scale the rotated curve along X and Y coordinates while Z coordinate stays unscaled.
Theorem 4
The GBTBézier swung surface (22) can be converted into its tensor product form:
$$ T_{\mathrm{swung}}(z, z_{1}, \mu , \nu , \mu _{1}, \nu _{1})=\sum^{m}_{k=0} \sum^{n}_{l=0} Q_{k,l}f_{k,m}(z) f_{l,n}(z_{1}), \quad z,z_{1} \in [0,1], $$
(23)
where \({Q_{k,l}=(\lambda Q^{X}_{k,0}Q^{X}_{l,1}, \lambda Q^{X}_{k,0}Q^{Y}_{l,1}, Q^{Z}_{k,0}), k=0,1,\ldots,m; l=0,1,\ldots,n}\), are the control net points of the GBTBézier swung surface, \({Q^{X}_{k,0}, Q^{Z}_{k,0}}\) and \({Q^{X}_{l,1}, Q^{Y}_{k,1}}\) are elements of the control points \({Q_{k,0}}\) and \({Q_{l,1}}\). The shapes of profile and trajectory curve are controlled by the shape parameters μ, ν and \(\mu _{1}\), \(\nu _{1}\), respectively.
Proof
From equations (20) and (21) of swung surface, we have
$$ \textstyle\begin{cases} F^{X}_{m}(z)=\sum^{m}_{k=0} Q^{X}_{k,0}f_{k,m}(z), F^{Z}_{m}(z)= \sum^{m}_{k=0} Q^{Z}_{k,0}f_{k,m}(z), \\ F^{X}_{n}(z_{1})=\sum^{n}_{l=0} Q^{X}_{l,1}f_{l,n}(z_{1}), F^{Y}_{n}(z_{1})= \sum^{n}_{l=0} Q^{Y}_{l,1}f_{l,n}(z_{1}). \end{cases} $$
(24)
Now, substituting equation (24) into equation (22) and using the partition of unity property \({\sum^{m}_{k=0} f_{k,m}(z)=1}\), we have
$$\begin{aligned}& T_{\mathrm{swung}}(z, z_{1}) \\& \quad = \bigl( \lambda F^{X}_{m}(z) F^{X}_{n}(z_{1}), \lambda F^{X}_{m}(z) F^{Y}_{n}(z_{1}), F^{Z}_{m}(z) \bigr) \\& \quad = \Biggl( \lambda \sum^{m}_{k=0} Q^{X}_{k,0}f_{k,m}(z)\sum ^{n}_{l=0} Q^{X}_{l,0}f_{l,n}(z_{1}), \lambda \sum^{m}_{k=0} Q^{X}_{k,0}f_{k,m}(z) \sum^{n}_{l=0} Q^{Y}_{l,0}f_{l,n}(z_{1}), \sum^{m}_{k=0} Q^{Z}_{k,0}f_{k,m}(z) \Biggr) \\& \quad = \Biggl( \sum^{m}_{k=0} \sum^{n}_{l=0} f_{k,m}(z) f_{l,n}(z_{1}) \bigl( \lambda Q^{X}_{k,0}Q^{X}_{l,0} \bigr), \sum^{m}_{k=0} \sum ^{n}_{l=0} f_{k,m}(z) f_{l,n}(z_{1}) \bigl(\lambda Q^{X}_{k,0}Q^{Y}_{l,0} \bigr), \\& \quad\quad{} \sum^{m}_{k=0} \sum ^{n}_{l=0} f_{k,m}(z) f_{l,n}(z_{1}) Q^{z}_{k,0} \Biggr) \\& \quad = \sum^{m}_{k=0} \sum ^{n}_{l=0} f_{k,m}(z) f_{l,n}(z_{1}) \bigl(\lambda Q^{X}_{k,0}Q^{X}_{l,0}, \lambda Q^{X}_{k,0}Q^{Y}_{l,0}, Q^{Z}_{k,0}\bigr). \end{aligned}$$
(25)
Thus, a tensor product GBTBézier surface design of degree \((n, m)\) is used to express the swung surfaces (22). □
The swung surface described by Theorem 4 is called GBTBézier swung surface. It appears that the GBTBézier swung surface carried all the characteristics and benefits of GBTBézier surface. The shape parameters are used to modify GBTBézier swung surface by keeping control net points fixed. For smooth connection among adjacent GBTBézier swung surfaces like previous surfaces, if the PCs or the TCs are combined by \(C^{1}\) or \(C^{2}\) continuity for GBTBézier curve, then the resulting GBTBézier swung surface is also a \(C^{1}\) or \(C^{2}\) continuous surface in the direction of profile or trajectory curve. It is worth noting that if the swung surface \(T_{\mathrm{swung}}(z, z_{1})\) is either open or close in z and \(z_{1}\) directions, then the corresponding profile curve \(F_{m}(z)\) and the trajectory curve \(F_{n}(z_{1})\) both will be either open or closed. Additionally, if the scale factor \(\lambda =1\) and the trajectory curve \(F_{n}(z_{1})\) circle with a unit radius with a center at the origin, then the GBTBézier swung surface is a surface of revolution. However, the surface of revolution no longer is a tensor product GBTBézier swung surface which is described by Theorem 4.
The detailed algorithm for designing a GBTBézier swung surface is described as follows.
Algorithm 4

1.
Take a profile curve in \(XOZ\) and a trajectory curve in the \(XOY\) plane in the form of GBTBézier curves.

2.
Define a GBTBézier swung surface by using equation (23).

3.
Take the control net points of the GBTBézier swung surface in the pattern \({Q_{k,l}=(\lambda Q^{X}_{k,0}Q^{X}_{l,1}} , {\lambda Q^{X}_{k,0}Q^{Y}_{l,1}} , Q^{Z}_{k,0})\), \(k=0,1,\ldots,m\); \(l=0,1,\ldots,n\), where \({Q^{X}_{k,0}, Q^{Z}_{k,0}}\) and \({Q^{X}_{l,1}, Q^{Y}_{k,1}}\) are elements of the control points \({Q_{k,0}}\) and \({Q_{l,1}}\) described in expression (20).

4.
Construct the GBTBézier swung surfaces in Wolfram Mathematica software using suitable values of scaling factor λ and shape control parameters μ, ν, \(\mu _{1}\), and \(\nu _{1}\)

5.
Two adjacent GBTBézier swung surfaces can be connected by \(C^{1}\) and \(C^{2}\) continuity conditions given in Theorem 2.
Example 3.4
Figures 13–15 show the GBTBézier swung surfaces for the different values of scale factor and shape parameters. In these figures, the PCs \(F_{m}(z)\) and the TCs \(F_{n}(z_{1})\) are both GBTBézier curves of degree three and the mesh points are
$$ \textstyle\begin{cases} Q_{0,0}=(3, 0, 5), \quad\quad Q_{1,0}=(1, 0, 3), \\ Q_{2,0}=(1, 0, 2),\quad\quad Q_{3,0}=(4, 0, 1), \\ Q_{0,1}=(4, 1, 0), \quad\quad Q_{1,1}=(1, 2, 0), \\ Q_{2,1}=(1, 4, 0),\quad\quad Q_{3,1}=(5, 5, 0). \end{cases} $$
(26)
It can be observed from Fig. 13 that the scaling of swung surfaces in both directions is handled by scale factor λ keeping shape parameters μ and ν same and fixed. Figures 14 and 15 exhibit the graphs of GBTBézier swung surfaces along the same control points of the PCs and TCs as taken in Fig. 13, but the values of shape parameters are different and scaling factor \(\lambda =0.15\) is fixed. Figures 16, 17, and 18 depict the \(C^{1}\) and \(C^{2}\) continuity connection of two adjacent GBTBézier swung surfaces for various values of shape control parameters and scaling factor λ.
3.5 GBTBézier swept surfaces with shape parameters
In this section, we especially tackle the problem of constructing a surface by sweeping a section curve (SC) around an arbitrary TC [11]. The GBTBézier curves of order m and n are taken as SCs and TCs in a 3D space, respectively. These two curves are given by
$$ \textstyle\begin{cases} F_{m}(z; \mu , \nu )=\sum^{m}_{k=0} f_{k,m}(z)Q_{k,m}, \quad 0 \leq z\leq 1, \\ F_{n}(z_{1}; \mu _{1}, \nu _{1})=\sum^{n}_{l=0} f_{l,n}(z_{1})Q_{l,n}, \quad 0\leq z_{1}\leq 1. \end{cases} $$
(27)
Generally, a swept surface is given by [25]
$$ T_{\mathrm{swept}}(z, z_{1};\mu , \nu , \mu _{1}, \nu _{1})=F_{n}(z_{1};\mu _{1}, \nu _{1})+\mathbf{S}(z_{1})F_{m}(z; \mu , \nu ), $$
(28)
where \(\mathbf{S}(z_{1})\) is a \(3\times 3\) matrix integrating scaling and rotation of the SC \(F_{m}(z)\). In this study, the SC \(F_{m}(z)\) and the TCs \(F_{n}(z_{1})\) both are GBTBézier curves, but the type and shape might be random (single or composite, planar or nonplanar, open or closed). Normally, equation (28) can generate an unnecessary surface with degeneracy and without continuity. Moreover, in many situations the swept surface \(T_{\mathrm{swept}}(z, z_{1})\) is not exactly expressible as a surface model existing in available literature such as Bspline, Bézier, and NURBS surfaces. Practically, swept surfaces can be generated by the following two categories:

\(\mathbf{M}(z_{1})\) is an identity matrix, which for all \(z_{1}\) and \(F_{m}(z)\) is just translated by \(F_{n}(z_{1})\).

\(\mathbf{M}(z_{1})\) is not an identity matrix.
The second type is very complicated and debatable, which can be referred to literature [25]. Particularly, we consider the first case of swept surfaces which can be defined by
$$ T_{\mathrm{swept}}(z, z_{1}; \mu , \nu , \mu _{1}, \nu _{1})=F_{n}(z_{1})+F_{m}(z). $$
(29)
The curves \(F_{m}(z)\) and \(F_{n}(z_{1})\) are GBTBézier curves described by (27). The surface (29) is termed as the GBTBézier swept surface.
Theorem 5
A GBTBézier swept surface (29) can be expressed in its tensor product form:
$$ T_{\mathrm{swept}}(z, z_{1}, \mu , \nu , \mu _{1}, \nu _{1})=\sum^{m}_{k=0} \sum^{n}_{l=0}f_{k,m}(z) f_{l,n}(z_{1}) \widehat{Q}_{k,l}, \quad z,z_{1}\in [0,1], $$
(30)
where \(\widehat{Q}_{k,l}=Q_{k,m}+Q_{l,n}\) (\(k=0,1,\ldots,m\); \(l=0,1,\ldots,n\)) are the control net points.
Proof
From the definition of GBTBézier swept surface, and substituting equation (27) into equation (29), we have
$$\begin{aligned} T_{\mathrm{swept}}(z, z_{1}; \mu , \nu , \mu _{1}, \nu _{1}) =&F_{n}(z_{1})+F_{m}(z) \\ =&\sum^{n}_{l=0}Q_{l,n}f_{l,n}(z_{1})+ \sum^{m}_{k=0}Q_{k,m}f_{k,m}(z) \\ =&\sum^{m}_{k=0}f_{k,m}(z) \sum^{n}_{l=0}Q_{l,n}f_{l,n}(z_{1})+ \sum^{n}_{l=0}f_{l,n}(z_{1}) \sum^{m}_{k=0}Q_{k,m}f_{k,m}(z) \\ =&\sum^{m}_{k=0}\sum ^{n}_{l=0}f_{k,m}(z) f_{l,n}(z_{1})Q_{l,n}+ \sum^{m}_{k=0}\sum ^{n}_{l=0}f_{k,m}(z)f_{l,n}(z_{1})Q_{k,m} \\ =&\sum^{m}_{k=0}\sum ^{n}_{l=0}f_{k,m}(z) f_{l,n}(z_{1}) ( Q_{l,n}+Q_{k,m} ) \\ =&\sum^{m}_{k=0}\sum ^{n}_{l=0}f_{k,m}(z) f_{l,n}(z_{1}) \widehat{Q}_{k,l}. \end{aligned}$$
(31)
Thus, the GBTBézier swept surface described by (29) can be exactly expressed as tensor product GBTBézier surface design. □
Proposition 3.7
Equation (30) can be demonstrated:

1.
As the tensor product GBTBézier surface is used to express the GBTBézier swept surface, so it holds all the characteristics and benefits of the GBTBézier surface.

2.
By keeping control mesh fixed and varying the values of shape control parameters, the shape of the GBTBézier swept surface can be modified. Furthermore, the composite GBTBézier swept surfaces can also be constructed like the previous surfaces.
A GBTBézier swept surface can be constructed as follows:
Algorithm 5

1.
Consider a SC which sweeps to the trajectory curve (both curves in the form of GBTBézier curves).

2.
Construct the GBTBézier swept surface by using equation (30).

3.
Follow the pattern \(\widehat{Q}_{k,l}=Q_{k,m}+Q_{l,n}\) (\(k=0,1,\ldots,m\); \(l=0,1,\ldots,n\)) and take the control points of GBTBézier swept surface where \(Q_{k,m}\), \(Q_{l,n}\) are the control points of section and TCs, respectively.

4.
Draw GBTBézier swept surfaces in Wolfram Mathematica software by taking different values of shape control parameters μ, ν in their respective value range \([1,1]\).

5.
Two adjacent GBTBézier swept surfaces can be connected by \(C^{1}\) and \(C^{2}\) continuity conditions given in Theorem 2.
Example 3.5
The shape control parameters of the two contiguous curves segments are used μ, \(\mu _{1}\), ν, \(\nu _{1}\) and \(\mu _{2}\), \(\mu _{3}\), \(\nu _{2}\), \(\nu _{3}\), respectively, and these values are identical for two SCs as well as are similar for two TCs case. The mesh points are taken as \((2, 0, 2)\), \((1, 0, 3)\), \((5, 0, 4)\), \((6, 0, 3)\) and \((6, 0, 3)\), \((7, 0, 2)\), \((3, 0, 1)\), \((2, 0, 2)\). The control points of the two TCs are taken as \((4, 0, 0)\), \((0, 1, 0)\), \((0, 2, 0)\), \((4, 3, 0)\) and \((4, 3, 0)\), \((8, 4, 0)\), \((8, 5, 0)\), \((4, 6, 0)\). Figure 19 and Fig. 20 depict the composite curves of two contiguous GBTBézier swept surfaces with different shape parameters which can be connected by \(C^{1}\) and \(C^{2}\) continuity constraints, respectively.
3.6 GBTBézier rotation surfaces with shape parameters
Geometric shapes of many objects are rotation surfaces in nature. Rotation surface is a type of special and common geometric profile in the field of product modeling design and has a broad demand in many fields such as aeronautic, architectural engineering, industrial designing, machining, and computer graphics. In CAD/CAM technology, how we can obtain 3D mathematical designs of rotation surfaces efficiently and swiftly is an important research issue. This portion is especially focused on how to develop a \(C^{1}\) or \(C^{2}\) continuous 3D mathematical model of GBTBézier rotation surface by taking a combined GBTBézier curve. Rotation surfaces are created by revolving a curve along a straight line in space with a constant angle. When the GBTBézier curves \(F_{m} (z; \mu , \nu )\) (\(0 < z \leq 1\)) are considered as the generating line and ϕ (\(0 < \phi \leq 2\pi \)) as a fixed angle around the rotation axis, then a rotation surface \(R_{\mathrm{rotation}}\) can be constructed in the \(XOZ\) plane. Let us imagine that the generating line is a GBTBézier curve \(F_{m} (z; \mu , \nu )\) (\(0 < z \leq 1\)) in the \(XOZ\) plane and has Zaxis as the rotation axis, then the expression for rotation surface \(R_{\mathrm{rotation}}\) is described as follows [25]:
$$\begin{aligned} R_{\mathrm{rotation}} (z, \phi ) =& \begin{pmatrix} \cos \phi &\sin \phi & 0 \\ \sin \phi &\cos z\phi & 0 \\ 0 &0&1 \end{pmatrix} F _{m}(z) \\ =&\bigl[X_{m}(z)\cos \phi , X_{m}(z)\sin \phi , Z_{m}(z)\bigr]^{T}, \end{aligned}$$
(32)
where \(0 < \phi \leq 2\pi \), the generating line \(F _{m}(z)=[X_{m}(z), 0, Z_{m}(z)]^{T}\), \(z \in [0, 1]\). However, the rotation surface expressed by equation (32) comprises trigonometric functions that will enhance the mathematical complication and rounding error in developing a rotation surface. Practically, architects often desire to modify the shape of a rotating surface automatically and rapidly. Therefore, from the technique in [26], we define a class of GBTBézier rotation surfaces by taking the GBTBézier curves with their generatrices.
Theorem 6
Suppose that a GBTBézier curve \(F _{m}(z)\) can be expressed by a variety of control points \(Q_{k}\) (\(k=0,1,\ldots,m\)) whose coordinates are taken as \(\{X_{k}, 0, Z_{k}\}\) (\(k=0,1,\ldots,m\)) in the \(XOZ\) plane, then the expression for entire rotation surfaces generated by rotating the generating line \(F _{m}(z)\) around Zaxis in one rotation is described as
$$ \textstyle\begin{cases} {T^{Z}_{\mathrm{rotation}}(z, z_{1}, \mu , \nu )=\{ \frac{12z_{1}}{2z^{2}_{1}2z_{1}+1}X_{m}(z), \frac{2z_{1}2z^{2}_{1}}{2z^{2}_{1}2z_{1}+1}X_{m}(z), Z_{m}(z) \}}, \\ {\widetilde{T}^{Z}_{\mathrm{rotation}}(z, z_{1}, \mu , \nu )=\{ \frac{12z_{1}}{2z^{2}_{1}2z_{1}+1}X_{m}(z),  \frac{2z_{1}2z^{2}_{1}}{2z^{2}_{1}2z_{1}+1}X_{m}(z), Z_{m}(z) \}}, \end{cases} $$
(33)
where \({T^{Z}_{\mathrm{rotation}}}\) and \({\widetilde{T}^{Z}_{\mathrm{rotation}}}\) are symmetric about the \({XOZ}\) plane, and by combining these two terms, we can get a complete GBTBézier rotation surface. The \(X_{m}(z)\) and \(Z_{m}(z)\) components of generating line \(F _{m}(z)\) can be derived from the following expression:
$$ \textstyle\begin{cases} X_{m}(z)=\sum^{m}_{k=0} f_{k,m}(z)X_{k}, \quad z \in [0, 1], \\ Z_{m}(z)=\sum^{m}_{l=0} f_{l,m}(z)Z_{l}, \quad z \in [0, 1], \end{cases} $$
(34)
where \(f_{k,m}(z)\) are GBTB degree m given in equation (2).
Proof
The proof of the above results given in Theorem 6 is as described in [21]. □
Conclusively, using the same results, we can develop a rotation surface if the generating line and the rotation axis are \({XOY}\) or \({YOZ}\) and Xaxis or Yaxis, respectively.
Proposition 3.8
If the generating line \(F _{m}(z)\) is revolved around the Zaxis with a fixed angle ϕ (\(0 < \phi \leq 2\pi \)), then the resulting surface is partisan GBTBézier rotation surface and is defined as follows:

(a)
\({T^{Z}_{\mathrm{rotation}}}\) in (33) is used to create the partially rotation surface if \(0 < \phi \leq \pi \), where ϕ is the rotation angle and range of the parameter z becomes \(X_{m}(z)\).

(b)
If \(0 < \phi \leq 2\pi \), where ϕ is the rotation angle, we can utilize \({T^{Z}_{\mathrm{rotation}}}\) to construct the first half and then utilize \({\widetilde{T}^{Z}_{\mathrm{rotation}}}\) to construct the remaining according to the technique in \((a)\), with \((2\pi \phi )/ \pi \leq z \leq 1\) value range of parameter z.
Proposition 3.9
From the reality of GBTBézier rotation surfaces which have various independent shape parameters, we can generate the desired shape of a rotation surface both locally and globally, by modifying shape control parameters. Moreover, translational and rotation transformation allows us to transfer the GBTBézier rotation surfaces in any direction to a particular position.
For constructing a GBTBézier rotation surface in the \(XOZ\) plane, the detailed algorithm is as follows.
Algorithm 6

1.
Take GBTB functions with shape parameters \(\mu , \nu \in [1,1]\).

2.
Consider the control points \(\{X_{k}, 0, Z_{k}\}\) (\(k=0,1,\ldots,m\)) in the 3D plane.

3.
Select the appropriate values of shape control parameters μ, ν from the interval \([1,1]\).

4.
Calculate the functions \(X_{m}(z)\) and \(Z_{m}(z)\) by using expression (34).

5.
Substitute the functions \(X_{m}(z)\) and \(Z_{m}(z)\) into expression (33) to achieve \({{T}^{Z}_{\mathrm{rotation}}(z, z_{1}, \mu , \nu )}\) and \({\widetilde{T}^{Z}_{\mathrm{rotation}}(z, z_{1}, \mu , \nu )}\) rotation surfaces.

6.
Plot these two functions in Wolfram Mathematica software and join them to obtain GBTBézier rotation surfaces in the \(XOZ\) plane.
Example 3.6
Provided a variety of control points \(Q_{l}\) (\(1= 0,1,2,3,4\)) in the \({XOZ}\) plane have the coordinates
$$ \textstyle\begin{cases} Q_{0}=(4, 0, 15), \\ Q_{1}=(0, 0, 13), \\ Q_{2}=(0, 0, 10), \\ Q_{3}=(10, 0, 5), \\ Q_{4}=(4, 0, 0). \end{cases} $$
(35)
From equations (34) and (35), the parametric equation of GBTBézier curve \(F_{4}(z, \mu , \nu )\) described by control points \(Q_{i}\) (\(i=0,1,2,3,4\)) can be expressed as
$$ F_{4}(z, \mu , \nu )=\bigl\{ X_{4}(z), 0, Z_{4}(z)\bigr\} \quad (0\leq z \leq 1). $$
Hence, by using equations (33) and (35) and rotating the generating line \(F_{4}(z, \mu , \nu )\) in one revolution around zaxis, the GBTBézier rotation surfaces are constructed as follows:
$$ \textstyle\begin{cases} {T^{Z}_{\mathrm{rotation}}(z, z_{1}, \mu , \nu )=\{ \frac{12z_{1}}{2z^{2}_{1}2z_{1}+1}X_{4}(z), \frac{2z_{1}2z^{2}_{1}}{2z^{2}_{1}2z_{1}+1}X_{4}(z), Z_{4}(z) \}}, \\ {\widetilde{T}^{Z}_{\mathrm{rotation}}(z, z_{1}, \mu , \nu )=\{ \frac{12z_{1}}{2z^{2}_{1}2z_{1}+1}X_{4}(z),  \frac{2z_{1}2z^{2}_{1}}{2z^{2}_{1}2z_{1}+1}X_{4}(z), Z_{4}(z) \}}. \end{cases} $$
(36)
The graphs of GBTBézier rotation surfaces using control points (35) are presented in Fig. 21 with different shape parameters μ, ν and with similar control polygon of generating lines.
Figures 22–27 also represent some GBTBézier rotation surfaces with different values of shape parameters and control polygon of generating lines.