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 take an interpretation of GBT-Bézier curve having weight coefficients in 4*D* homogeneous space as

$$\begin{aligned} \hat{G}(x,\mu,\nu )=\sum^{k}_{i=0} g_{i,k}(x,\mu,\nu )\hat{R}_{i},\quad 0 \leq x\leq 1, \end{aligned}$$

(5.1)

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 GBT-Bézier curve share the basic features with GBT-Bézier curves. When the weighted control points \(\hat{R}_{i}=(\omega _{i} R_{i}, \omega _{i})\ (i=0,1,\ldots,k)\) in a 4*D* homogeneous space are considered as the control planes \(Q_{i}\ (i=0,1,\ldots,k)\) in a 3*D* Euclidean space, the planes \(\{\Pi _{x}\}\) constructed from control planes \(Q_{i}\ (i=0,1,\ldots,k)\) are dual to the weighted GBT-Bézier curves constructed from \(\hat{R}_{i}=(\omega _{i} R_{i}, \omega _{i})(i =0,1,\ldots,k)\). Hence some geometric design methods of developable GBT-Bézier surfaces (like continuity requirements, tangent planes, and terminal properties) in a 4*D* 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 \(H_{1}(x;\mu _{1},\nu _{1})\) and \(H_{2}(x;\mu _{2},\nu _{2})\) of order *k* and *l*, respectively, that require to be joined together are defined as follows:

$$\begin{aligned} \textstyle\begin{cases} \{\Pi _{x,1}\}: H_{1}(x;\mu _{1},\nu _{1})={a_{0,1}(x), a_{1,1}(x), a_{2,1}(x), a_{3,1}(x)},\quad 0\leq x\leq 1, \\ \{\Pi _{x,2}\}: H_{2}(x;\mu _{2},\nu _{2})={a_{0,2}(x), a_{1,2}(x), a_{2,2}(x), a_{3,2}(x)}, \quad 0\leq x\leq 1, \end{cases}\displaystyle \end{aligned}$$

(5.2)

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.

### 5.1
\(G^{1}\) continuity among developable GBT-Bézier surfaces

Here, we want to establish the first order geometric continuity or \(G^{1}\) continuity among two or more weighted GBT-Bézier curves in a 4*D* homogeneous space. Now suppose that the single-parameter families of planes of the two contiguous weighted GBT-Bézier curves, which need to be spliced together, are as follows:

$$\begin{aligned} \textstyle\begin{cases} \hat{G}_{1}(x,\mu _{1},\nu _{1})=\sum^{k}_{i=0} g_{i,k}(x,\mu _{1}, \nu _{1})\hat{R}_{i,1},\quad 0\leq x\leq 1, \\ \hat{G}_{2}(x,\mu _{2},\nu _{2})=\sum^{l}_{j=0} g_{j,l}(x,\mu _{2}, \nu _{2})\hat{R}_{j,2}, \quad 0\leq x\leq 1, \end{cases}\displaystyle \end{aligned}$$

(5.3)

where \(\mu _{i}, \nu _{i}\ (i=1,2)\) are shape parameters of two weighted GBT-Bézier 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 GBT-Bézier curves \(\hat{G}_{1}(x)\) and \(\hat{G}_{2}(x)\) are

$$\begin{aligned} \textstyle\begin{cases} \hat{G}_{2}(0,\mu _{2},\nu _{2})=\hat{G}_{1}(1,\mu _{1},\nu _{1}), \\ \hat{G}'_{2}(0,\mu _{2},\nu _{2})=\gamma \hat{G}'_{1}(1,\mu _{1},\nu _{1}), \end{cases}\displaystyle \end{aligned}$$

(5.4)

where \(\gamma > 0\) is a constant. For control points \(R_{i,j}\ (i=0,1,\ldots,k; j=1, 2)\) of two weighted GBT-Bézier curves \(\hat{G}_{i}(x,\mu _{i}, \nu _{i})\ (i=1, 2)\) in a 4*D* homogeneous space understood as the control planes in 3*D* Euclidean space, the geometric continuity among two curves in a 4*D* homogeneous space certifies the geometric continuity among two conforming developable surfaces developed by using the duality principle in a 3*D* 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 GBT*-*Bézier 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*

$$\begin{aligned} \textstyle\begin{cases} {Q_{0,2}=Q_{k,1}}, \\ {Q_{1,2}=Q_{k,1}+\gamma \frac{k-2+\frac{\pi }{2}(1+\nu _{1})}{l-2+\frac{\pi }{2}(1+\mu _{2})}(Q_{k,1}-Q_{k-1,1}).} \end{cases}\displaystyle \end{aligned}$$

(5.5)

### 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

$$\begin{aligned} Q_{0,2}=Q_{k,1}. \end{aligned}$$

(5.6)

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,

$$\begin{aligned} H'_{2}(0,\mu _{2},\nu _{2})=\gamma H'_{1}(1,\mu _{1},\nu _{1}). \end{aligned}$$

(5.7)

According to (3.11) and (3.12), we have

$$\begin{aligned} \textstyle\begin{cases} {H'_{1}(1,\mu _{1},\nu _{1})=(k-2+\frac{\pi }{2}(1+\nu _{1}))(Q_{k,1}-Q_{k-1,1}),} \\ {H'_{2}(0,\mu _{2},\nu _{2})=(l-2+\frac{\pi }{2}(1+\mu _{2}))(Q_{1,2}-Q_{0,2}).} \end{cases}\displaystyle \end{aligned}$$

(5.8)

Using (5.8) into (5.7), we get

$$\begin{aligned} \gamma \biggl(k-2+\frac{\pi }{2}(1+\nu _{1})\biggr) (Q_{k,1}-Q_{k-1,1})=\biggl(l-2+ \frac{\pi }{2} (1+\mu _{2})\biggr) (Q_{1,2}-Q_{0,2}). \end{aligned}$$

(5.9)

Ultimately, on the ground of (5.6), we can achieve

$$\begin{aligned} Q_{1,2}=Q_{k,1}+\gamma \frac{k-2+\frac{\pi }{2}(1+\nu _{1})}{l-2+\frac{\pi }{2}(1+\mu _{2})}(Q_{k,1}-Q_{k-1,1}). \end{aligned}$$

(5.10)

Hence equations (5.6) and (5.10) comprise the adequate and essential requirements of \(G^{1}\) smooth connection betwixt two developable GBT-Bézier surfaces of order *k* and *l* sequentially. □

### 5.2 Farin–Boehm \(G^{2}\) continuity requirements among developable GBT-Bézier surfaces

Now, we will derive the Farin–Boehm \(G^{2}\) continuity requirements at the joints for the construction of piecewise developable GBT-Bézier surfaces. Similar to the \(G^{1}\) continuity, when the control points of the two GBT-Bézier curves \(\hat{G}_{1}(x,\mu _{1},\nu _{1})\), \(\hat{G}_{2}(x,\mu _{2},\nu _{2})\) in a 4*D* 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 GBT-Bézier curves in a 4*D* homogeneous space ensures the Farin–Boehm \(G^{2}\) continuity among two developable GBT-Bézier surfaces determined by \(Q_{i,j}\ (i=1,2,\ldots,k; j=1,2)\) in a 3*D* Euclidean space [22]. Therefore, when two contagious developable GBT-Bézier surfaces in (5.2) want to reach Farin–Boehm \(G^{2}\) continuity, they must satisfy the following requirements:

$$\begin{aligned} \textstyle\begin{cases} H_{2}(0,\mu _{2},\nu _{2})=H_{1}(1,\mu _{1},\nu _{1}), \\ H'_{2}(0,\mu _{2},\nu _{2})=H'_{1}(1,\mu _{1},\nu _{1}), \\ H''_{2}(0,\mu _{2},\nu _{2})=H''_{1}(1,\mu _{1},\nu _{1}). \end{cases}\displaystyle \end{aligned}$$

(5.11)

Equation (5.2) represents that the developable GBT-Bézier 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 GBT*-*Bézier 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*

$$\begin{aligned} \textstyle\begin{cases} Q_{0,2}=Q_{k,1}, \\ Q_{1,2}= Q_{k,1}+c(Q_{k,1}-Q_{k-1,1}), \\ Q_{2,2}=Q_{k,1}+c(Q_{k,1}-Q_{k-1,1})\\ \phantom{Q_{2,2}=}{} + \frac{1}{c_{l}+\pi ^{2}(1-\nu _{2})}[c_{k}(Q_{k,1}-2Q_{k-1,1}+Q_{k-2,1})- \pi ^{2}(1-\mu _{1}) \\ \phantom{Q_{2,2}=} {}\times (Q_{k-1,1}-Q_{k-2,1})+(2\pi ^{2}\nu _{1}+c(c_{l}+2 \pi ^{2}\mu _{2}))(Q_{k,1}-Q_{k-1,1})], \end{cases}\displaystyle \end{aligned}$$

(5.12)

*where* \({c= \frac{2(k-2)+\pi (1+\nu _{1})}{2(l-2)+\pi (1+\mu _{2})},}\) \({c_{k}=4(k-2)(k-3+\pi (1+\nu _{1}))}\), \({c_{l}=4(l-2)(l-3+\pi (1+\mu _{2})).}\)

### 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

$$\begin{aligned} \textstyle\begin{cases} {Q_{0,2}=Q_{k,1}}, \\ {Q_{1,2}=Q_{k,1}+ \frac{2(k-2)+\pi (1+\nu _{1})}{2(l-2)+\pi (1+\mu _{2})}(Q_{k,1}-Q_{k-1,1}),} \end{cases}\displaystyle \end{aligned}$$

(5.13)

also

$$\begin{aligned} {H''_{2}(0,\mu _{2},\nu _{2})=H''_{1}(1, \mu _{1},\nu _{1}).} \end{aligned}$$

(5.14)

According to (3.11) and (3.12), we have

$$\begin{aligned} &4(l-2) \bigl(l-3+\pi (1+\mu _{2})\bigr) (Q_{0,2}-2Q_{1,2}+Q_{2,2})+2\pi ^{2}\mu _{2}(Q_{0,2}-Q_{1,2}) \\ &\quad{}- \pi ^{2}(1-\nu _{2}) (Q_{1,2}-Q_{2,2})=4(k-2) \\ &\quad{}\times \bigl(k-3+\pi (1+\nu _{1})\bigr) (Q_{k-2,1}-2Q_{k-1,1}+Q_{k,1}) \\ &\quad{}+ \pi ^{2}(1-\mu _{1}) (Q_{k-2,1}-Q_{k-1,1}) -2 \pi ^{2}\nu _{1}(Q_{k-1,1}-Q_{k,1}). \end{aligned}$$

(5.15)

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). □

### 5.3
\(G^{2}\) beta continuity among developable GBT-Bézier surfaces

### Theorem 4

*For a smooth continuous connection among two adjoining developable GBT*-*Bézier 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*

$$\begin{aligned} \textstyle\begin{cases} Q_{0,2}=Q_{k,1}, \\ Q_{1,2}=Q_{k,1}+c\gamma (Q_{k,1}-Q_{k-1,1}), \\ Q_{2,2}=Q_{k,1}+c\gamma (Q_{k,1}-Q_{k-1,1})\\ \phantom{Q_{2,2}=}{} + \frac{1}{c_{l}+\pi ^{2}(1-\nu _{2})}[c_{k}\gamma ^{2}(Q_{k,1}-2Q_{k-1,1}+Q_{k-2,1})- \pi ^{2}\gamma ^{2}(1-\mu _{1}) \\ \phantom{Q_{2,2}=}{}\times (Q_{k-1,1}-Q_{k-2,1})+(\gamma (2\pi ^{2} \nu _{1}\gamma +\mathit{cc}_{l}+2c\pi ^{2}\mu _{2})\\ \phantom{Q_{2,2}=}{}+\lambda (k-2+\frac{\pi }{2}(1+ \nu _{1}))) (Q_{k,1}-Q_{k-1,1})], \end{cases}\displaystyle \end{aligned}$$

(5.16)

*where* \({c= \frac{2(k-2)+\pi (1+\nu _{1})}{2(l-2)+\pi (1+\mu _{2})},}\) \({c_{k}=4(k-2)(k-3+\pi (1+\nu _{1}))}\), \({c_{l}=4(l-2)(l-3+\pi (1+\mu _{2})).}\)

### 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

$$\begin{aligned} \textstyle\begin{cases} {Q_{0,2}= H_{2}(0,\mu _{2},\nu _{2})=H_{1}(1,\mu _{1}, \nu _{1})=Q_{k,1},} \\ H'_{2}(0,\mu _{2},\nu _{2})=\gamma H'_{1}(1,\mu _{1}, \nu _{1}), \quad \gamma >0, \\ {H''_{2}(0,\mu _{2},\nu _{2})= \gamma ^{2} H''_{1}(1, \mu _{1},\nu _{1})+\lambda H'_{1}(1,\mu _{1},\nu _{1}),} \end{cases}\displaystyle \end{aligned}$$

(5.17)

where \(\gamma >0\) and *λ* are constants. From the terminal properties of developable GBT-Bézier 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 GBT-Bézier surfaces. □

### 5.4 Steps for smooth continuous connection between two adjacent developable GBT-Bézier 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 GBT-Bézier surfaces, an example is used here. For establishing a \(G^{2}\) beta continuous connection among two or more contiguous cubic enveloping developable GBT-Bézier surfaces, the following steps are performed.

*Step 1*. Preliminary developable GBT-Bézier 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 GBT-Bézier 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 GBT-Bézier 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 GBT-Bézier 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 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.

### 5.5 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.

### Example 5.1

Figures 15–16 exhibit the graphs of Farin–Boehm \(G^{2}\) continuity among two adjoining cubic enveloping developable GBT-Bézier surfaces. In these figures, the blue surfaces are primary developable GBT-Bézier surfaces with subsequent control planes

$$\begin{aligned} \textstyle\begin{cases} {Q_{0,1}=(0, -20, 10, 400)}, \\ {Q_{1,1}=(0, -10, 20, 400)}, \\ {Q_{2,1}=(0, 10, 20, 400)}, \\ {Q_{3,1}=(0, 20, 10, 400)}. \end{cases}\displaystyle \end{aligned}$$

(5.18)

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, 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 \(H1(x;\mu _{1}, \nu _{1})\) with control planes described in (5.18), whereas red surface represents the 2nd cubic enveloping developable GBT-Bézier 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 GBT-Bézier surfaces equal to 1, the coordinates of the control planes of cubic enveloping developable GBT-Bézier surface \(H2(x;\mu _{2}, \nu _{2})\) are calculated as follows:

$$\begin{aligned} \textstyle\begin{cases} {Q_{0,2}=(0, 20, 10, 400)}, \\ {Q_{1,2}=(0, 30, 0, 400)}, \\ {Q_{2,2}=(0, 45.708, -35.708, 400)}, \end{cases}\displaystyle \end{aligned}$$

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 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 \(\gamma =1\), unconcerned of modifying the shape parameter, and at connection a gap will be achieved on merged developable GBT-Bézier surfaces when \(\gamma \neq 1\). We can use these characteristics to design a complex surface according to our needs.