# Modeling of Free-Form Complex Curves Using SG-Bézier Curves with Constraints of Geometric Continuities

^{1}

^{2}

^{*}

## Abstract

**:**

^{1}and G

^{2}continuity between two adjacent SG-Bézier curves. Furthermore, the detailed steps of smooth continuity for two SG-Bézier curves, and the influence rules of shape parameters on the composite curves, are studied. We also give some important applications of SG-Bézier curves. The modeling examples show that our methods in this paper are very effective, can easily be performed, and can provide an alternative powerful strategy for the design of complex curves.

## 1. Introduction

^{n}continuity); and ② geometric continuity (namely, G

^{n}continuity). Compared with the traditional Bézier curves, the SG-Bézier curves in [23] have more advantages and play a key role in representing complex curves. Nonetheless, the expressions of SG-Bézier curves are polynomials, and one will also be facing the issue of smooth joining to construct composite curves. According to the basis functions and terminal properties of SG-Bézier curves, we give the geometric continuity conditions between two SG-Bézier curves in this paper. The modeling examples show that the shape adjustability of the composite curves are more flexible, which lends itself to the wide application of the SG-Bézier curves.

^{1}and G

^{2}continuity conditions for SG-Bézier curves. In Section 4, we then investigate the steps of smooth joining for SG-Bézier curves and give some examples. In Section 5, we analyze the shape adjustment of piecewise SG-Bézier curves. Finally, some practical applications are shown in Section 6. At the end of the paper, we give a short conclusion in Section 7.

## 2. The family of SG-Bézier Curves

#### 2.1. SG-Bernstein Basis Functions of Degree n

**Definition**

**1.**

**Remark**

**1.**

#### 2.2. SG-Bézier Curves of Degree n

**Definition**

**2.**

^{2}or R

^{3}, and M $\in $ R

^{k}

^{×k}(k = 2 or 3) is an arbitrary constant matrix.

**Theorem**

**1.**

**Proof.**

#### 2.3. The Influence of Shape Parameters on the Shapes of SG-Bézier Curves

## 3. Smooth Continuity Conditions of SG-Bézier Curves

#### 3.1. G^{1} Smooth Continuity Conditions of SG-Bézier Curves

**Theorem**

**2.**

^{1}smooth continuity at the common joint; then, they match the necessary and sufficient conditions as follows:

**Proof.**

^{1}continuity, they are required to achieve G

^{0}continuity at a common joint in the first place. This signifies that one needs to combine the end of ${\mathit{L}}_{n}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{j,1},{\mu}_{j,1},{\omega}_{1})$ with the beginning of ${\mathit{L}}_{m}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$, that is:

^{0}continuity at a common joint firstly.

^{1}smooth continuity at the common joint, and hence Theorem 2 is proved. □

^{1}continuity conditions will degenerate into the C

^{1}continuity conditions when we set $\alpha =1$, and the signs in Equation (18) are the same as that in Equation (10).

#### 3.2. G^{2} Smooth Continuity Conditions of SG-Bézier Curves

**Theorem**

**3.**

^{2}smooth continuity if and only if:

**Proof.**

^{2}continuity, they are required to achieve G

^{1}continuity at a common joint in the first place, which satisfies:

^{2}continuity requires that the vice-normal vector ${\mathit{L}}_{n}$ has the same direction at the joint as ${\mathit{L}}_{m}$. According to Equations (16) and (20), we can obtain the four vectors ${{L}^{\prime}}_{n}(1\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{j,1},{\mu}_{j,1},{\omega}_{1})$, ${{L}^{\u2033}}_{n}(1\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{j,1},{\mu}_{j,1},{\omega}_{1})$, ${{L}^{\prime}}_{m}(0\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$, and ${{L}^{\u2033}}_{m}(0\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$, which are coplanar. Thus, using Equation (20), we have:

^{2}continuity requires that the value of the curvatures ${\kappa}_{1}(1)$ is equal to the value of ${\kappa}_{2}(0)$, i.e., ${\kappa}_{1}(1)={\kappa}_{2}(0)$. Combining with Equations (23) and (24), we see that $\beta ={\zeta}^{2}$. Substituting $\beta $ into Equation (22), we have:

^{2}smooth continuity at a common joint if they meet Equations (20) and (26) simultaneously, thus proving Theorem 3. □

^{2}continuity conditions in Equation (19) will degenerate to the corresponding C

^{2}continuity conditions in Equation (27).

## 4. Steps and Examples of Smooth Continuity for SG-Bézier Curves

#### 4.1. The Concrete Steps of Smooth Joining between Two SG-Bézier Curves

^{2}smooth joining between SG-Bézier curves. Meanwhile, other smooth continuity conditions can be carried out similarly.

^{2}smooth continuity between two splicing SG-Bézier curves can be given as follows in terms of Theorem 2:

**Step 1.**For any degree n, we give the initial curve ${\mathit{L}}_{n}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{j,1},{\mu}_{j,1},{\omega}_{1})$ with the shape parameters ${\omega}_{1},{\lambda}_{j,1},{\mu}_{j,1}(j=1,2,\cdots ,[n/2]+1)$ and the control points ${\mathit{P}}_{j}^{1}(j=0,1,\cdots ,n)$.

**Step 2.**Set ${\mathit{P}}_{0}^{2}={\mathit{P}}_{n}^{1}$, so that ${\mathit{L}}_{n}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{j,1},{\mu}_{j,1},{\omega}_{1})$ and ${\mathit{L}}_{m}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$ achieve the G

^{0}continuity at their common joint.

**Step 3.**For any given degree m, shape parameters ${\omega}_{2}$, ${\lambda}_{1\uff0c2}$ of ${\mathit{L}}_{m}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$ and constant $\zeta >0$, they should satisfy the equation $m+{\omega}_{2}-{\omega}_{2}{\lambda}_{1,2}\ne 0$. Based on the second equation in Equation (19), we can compute the second control point ${\mathit{P}}_{1}^{2}$ of ${\mathit{L}}_{m}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$.

**Step 4.**Based on steps (2) and (3), we can compute the third control point ${\mathit{P}}_{2}^{2}$ of ${\mathit{L}}_{m}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$ by using the third equation in Equation (19), once we give an arbitrary constant $\psi $ and shape parameter ${\lambda}_{2\uff0c2}$.

**Step 5.**Finally, the G

^{2}smooth continuity between two splicing SG-Bézier curves can be achieved if we freely give the remaining m-2 shape parameters and control points of ${\mathit{L}}_{m}(t\text{\hspace{0.17em}};\text{\hspace{0.17em}}{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2},{\omega}_{2})$.

^{2}smooth continuity can be obtained by repeating the steps above.

#### 4.2. Examples of C^{1}, G^{1} Smooth Continuity between SG-Bézier Curves

^{1}smooth continuity. Given the initial blue SG-Bézier curve of degree 3, the red SG-Bézier curve of degree 3 is constructed by the continuity conditions in Equation (18), with the blue curve reaching C

**smooth continuity. The shape parameters for these two SG-Bézier curves of degree 3 are ${\lambda}_{1,1}={\mu}_{1,1}={\lambda}_{1,2}={\mu}_{1,2}=2$, ${\lambda}_{2,1}={\lambda}_{2,2}=3$ and ${\omega}_{1}={\omega}_{2}=0.5$, with the scale factor $\zeta =1$. Note that the control polygons and control points of each SG-Bézier curve are denoted as broken lines and circular points, respectively; and similarly hereinafter.**

^{1}^{1}smooth continuity. Given the initial blue SG-Bézier curve of degree 3, the red SG-Bézier curve of degree 4 is constructed by the G

^{1}continuity conditions in Equation (10), with the blue curves reaching G

^{1}smooth continuity. The local and global shape parameters for the cubic and quartic SG-Bézier curves are ${\lambda}_{1,2}={\mu}_{1,2}=5$, ${\lambda}_{1,1}={\mu}_{1,1}={\lambda}_{2,1}=4$, ${\lambda}_{2,2}={\mu}_{2,2}=8$, and ${\omega}_{1}={\omega}_{2}=0.5$, with the scale factor $\zeta =1/2$. We can see from Figure 5 that the overall modeling of the ‘fish’ are smooth and natural.

#### 4.3. Examples of C^{2}, G^{2} Smooth Continuity between SG-Bézier Curves

^{2}continuity conditions in Equation (27) with the blue curves reaching C

^{2}smooth continuity. The shape parameters for these two SG-Bézier curves are ${\omega}_{1}={\omega}_{2}=0.5$, ${\lambda}_{1,1}={\mu}_{1,1}=3$, ${\lambda}_{1,2}={\mu}_{1,2}=4$, and ${\lambda}_{2,1}=5,{\lambda}_{2,2}={\mu}_{2,2}=8$, with the scale factors $\zeta =1,\psi =0$.

^{2}continuity conditions in Equation (19), with the red curve reaching G

^{2}smooth continuity. The shape parameters for these two cubic SG-Bézier curves are ${\lambda}_{1,1}={\lambda}_{1,2}={\mu}_{1,1}={\mu}_{1,2}=3$, ${\lambda}_{2,1}={\lambda}_{2,2}=5$, and ${\omega}_{1}={\omega}_{2}=0.5$, with the scale factors $\zeta =1$, $\psi =1/2$. In addition, the line of curvature corresponding to Figure 7 plotted in Figure A1 visually supports the claim on G

^{2}continuity (see Appendix A).

^{2}smooth continuity of the splicing curve remaining unchanged, the value of the scale factor $\zeta $ will change the position of the second and third control points. Similarly, Figure 8b shows the change of the third control point of the second curve in the splicing curve by altering the scale factor $\psi $. Analogously, the broken lines and circular points in Figure 8 represent the control polygons and control points of each SG-Bézier curve, respectively. Notice that the asterisks labeled on the control polygon denote the altered control points.

## 5. Shape Adjustment of the Composite SG-Bézier Curves with Smooth Continuity

^{1}, G

^{2}and C

^{1}, C

^{2}smooth continuity of the composite curves remain unchanged. In what follows, we shall discuss the shape adjustability of the composite SG-Bézier curves with G

^{1}, G

^{2}and C

^{1}, C

^{2}continuity, which were constructed by two SG-Bézier curves. Moreover, a similar discussion can be carried out on the multiple curves case.

#### 5.1. Shape Adjustment of the Composite SG-Bézier Curves with C^{1} or G^{1} Smooth Continuity

**Proposition**

**1.**

^{1}smooth continuity, some significant conclusions can be obtained when keeping its control points and C

^{1}smooth continuity unchanged.

**Proof.**

^{1}continuity conditions of Equation (18) just involve the shape parameters ${\omega}_{1},{\mu}_{1,1}$ and ${\omega}_{2},{\lambda}_{1,2}$, then we can modify the local shape by altering the shape parameters that are not contained in Equation (18). Thus, conclusion (1) is proved. On the other hand, if we change at least one of the shape parameters ${\omega}_{1},{\mu}_{1,1}$, then at least one of the shape parameters ${\omega}_{2},{\lambda}_{1,2}$ should be changed to ensure that the C

^{1}smooth continuity is maintained unchanged for the splicing curve (to modify the shape parameters in terms of the second equation in Equation (18)). Therefore, the global shape adjustment of the splicing curve can be achieved by changing the values of the shape parameters:${\mu}_{j,1}(j=2,3,\dots ,[n/2]+1)$, ${\lambda}_{j,1}(j=1,2,\dots ,[n/2]+1),$ and ${\lambda}_{\tilde{j},2}(\tilde{j}=2,3,\dots ,[m/2]+1),$ ${\mu}_{\tilde{j},2}(\tilde{j}=1,2,\cdots ,[m/2]+1)$. This completes the proof of Proposition 1. □

**Proposition**

**2.**

^{1}smooth continuity between splicing curves unchanged, we have:

**Proof.**

^{1}continuity merely requires that the tangent directions at the common joint of the two splicing SG-Bézier curves are the same, that is:

^{1}smooth continuity the same. Thus, Proposition 2 is proven. □

#### 5.2. Modeling Examples of Shape Adjustment for the Composite SG-Bézier Curves with C^{1} or G^{1} Smooth Continuity

^{1}smooth continuity of two SG-Bézier curves. In Figure 9, the blue curve is the pre-given SG-Bézier curve, and the red SG-Bézier curve is now constructed by the continuity conditions in Equation (18). The shape parameters are ${\lambda}_{1,1}={\mu}_{1,1}={\lambda}_{1,2}={\mu}_{1,2}=2$, ${\lambda}_{2,1}={\lambda}_{2,2}=3$ and ${\omega}_{1}={\omega}_{2}=1$. Figure 9b shows the local shape adjustment of the composite curve just by modifying one shape parameter ${\lambda}_{1,1}$. The upper part of the splicing curve is now modified, where the shape parameters of solid lines, dashed lines, and dotted lines are ${\lambda}_{1,1}=3,\text{\hspace{0.17em}}1.5,\text{\hspace{0.17em}}0$, respectively. Figure 9c shows the local shape adjustment by altering the shape parameters $\text{\hspace{0.17em}}{\mu}_{1,2}$, and now, the lower part of the splicing curve is modified, with the shape parameters ${\mu}_{1,2}=3$ (solid lines), $\text{\hspace{0.17em}}{\mu}_{1,2}=1.5$ (dashed lines), and ${\mu}_{1,2}=0$ (dotted lines). We see that the global shape adjustment can be achieved in Figure 9d by changing the shape parameters ${\omega}_{1},{\omega}_{2}$ and ${\mu}_{1,1},{\mu}_{1,2}$, with the shape parameters ${\omega}_{1}={\omega}_{2}=1,{\mu}_{1,1},{\mu}_{1,2}=3$ (solid lines), ${\omega}_{1}={\omega}_{2}=0.8,{\mu}_{1,1},{\mu}_{1,2}=2$ (dashed lines), and ${\omega}_{1}={\omega}_{2}=0.1,{\mu}_{1,1},{\mu}_{1,2}=2$ (dotted lines). The rest of the shape parameters are equal to those in Figure 9a, expect that they are modified in the other three figures.

^{1}smooth continuity. In Figure 10a, the G

^{1}smooth continuity and the shape parameters are the same as that in Figure 5b. Figure 10b shows the global shape adjustment by altering the shape parameters ${\omega}_{1},{\lambda}_{1,1},{\mu}_{1,1},{\lambda}_{2,1}$. Now, we only adjust the left part of the splicing curve with the shape parameters ${\omega}_{1}=0.5,\text{\hspace{0.17em}}{\lambda}_{1,1}=4,\text{\hspace{0.17em}}{\mu}_{1,1}=4,{\lambda}_{2,1}=4$ (solid lines), ${\omega}_{1}=0.2,\text{\hspace{0.17em}}{\lambda}_{1,1}=2,\text{\hspace{0.17em}}{\mu}_{1,1}=2,{\lambda}_{2,1}=2$ (dashed lines), and ${\omega}_{1}=1,\text{\hspace{0.17em}}{\lambda}_{1,1}=1,\text{\hspace{0.17em}}{\mu}_{1,1}=0,{\lambda}_{2,1}=0$ (dotted lines). Figure 10c shows the graph of the local adjustment for the splicing curve by modifying shape parameters ${\omega}_{2},{\lambda}_{1,2},{\mu}_{1,2},{\lambda}_{2,2},{\mu}_{2,2}$, where we only make a adjustment for the right part of the splicing curve with the shape parameters ${\mu}_{1,2}=5,{\lambda}_{2,2}=8,{\mu}_{2,2}=8$ (solid lines), ${\omega}_{2}=1,{\lambda}_{1,2}=4.5,{\mu}_{1,2}=5,{\lambda}_{2,2}=10,{\mu}_{2,2}=10$ (dashed lines), and ${\omega}_{2}=0,{\lambda}_{1,2}=0.5,{\mu}_{1,2}=0.5,{\lambda}_{2,2}=0.5,{\mu}_{2,2}=0.5$ (dotted lines). Figure 10d shows the graph of the global adjustment by changing the shape parameters ${\omega}_{1},{\lambda}_{1,1},{\mu}_{1,1},{\lambda}_{2,1},{\omega}_{2},{\lambda}_{1,2},{\mu}_{1,2},{\lambda}_{2,2},{\mu}_{2,2}$, with the shape parameters ${\omega}_{1}=0.5,\text{\hspace{0.17em}}{\lambda}_{1,1}=4,{\mu}_{1,1}=4,{\lambda}_{2,1}=4,{\omega}_{2}=0.5,{\lambda}_{1,2}=5,{\mu}_{1,2}=5,{\lambda}_{2,2}=8,$ ${\mu}_{2,2}=8$ (solid lines), ${\omega}_{1}=1,\text{\hspace{0.17em}}{\lambda}_{1,1}=4,\text{\hspace{0.17em}}{\mu}_{1,1}=4,{\lambda}_{2,1}=6,{\omega}_{2}=1,{\lambda}_{1,2}=4.5,{\mu}_{2,2}=10,{\mu}_{1,2}=5,{\lambda}_{2,2}=10$ (dashed lines), and ${\omega}_{1}=0,\text{\hspace{0.17em}}{\lambda}_{1,1}=1,\text{\hspace{0.17em}}{\mu}_{1,1}=1,{\lambda}_{2,1}=0,{\omega}_{2}=0,{\lambda}_{1,2}=1,{\mu}_{1,2}=1,{\lambda}_{2,2}=1,{\mu}_{2,2}=1$ (dotted lines). The rest of the shape parameters are the same as those in Figure 10a, expect for those that are modified in Figure 10.

#### 5.3. Shape Adjustment of the Composite SG-Bézier Curves with C^{2} or G^{2} Smooth Continuity

**Proposition**

**3.**

^{2}smooth continuity for the splicing curve remain unchanged, we have the following conclusions:

^{2}smooth continuity is kept unchanged, the global shape of the composite curve can also be adjusted by modifying the following shape parameters and control points:

**Proof.**

^{2}continuity conditions in Equation (27), if the control points and C

^{2}smooth continuity remain unchanged, we can alter the local shape of the composite curve by changing these shape parameters, which are not included in the conditions in Equation (27). Thus, conclusion (1) is proved. □

^{2}smooth continuity the same. Now, we only adjust the local shape of the composite curve. So, we cannot modify the global shape only by altering the shape parameters. As ${\omega}_{1}$ and ${\omega}_{2}$ are the global shape parameters, we might alter the two control points ${P}_{1}^{2}$ and ${P}_{2}^{2}$ simultaneously to ensure that C

^{2}smooth continuity is unchanged once we adjust the global shape by changing ${\omega}_{1}$ and ${\omega}_{2}$. Similarly, as the C

^{2}smooth continuity conditions involve the parameters ${\omega}_{1}$, ${\omega}_{2}$, ${\mu}_{1,1}$, ${\mu}_{2,1}$, ${\lambda}_{1,2}$ and ${\lambda}_{2,2}$, we can adjust the global shape of the two parts that go beyond the splicing curve by modifying the shape parameters ${\omega}_{1}$, ${\mu}_{1,1}$, ${\mu}_{2,1}$ and ${\omega}_{2}$, ${\lambda}_{1,2}$, ${\lambda}_{2,2}$, as well as the two control points ${P}_{1}^{2}$ and ${P}_{2}^{2}$ to ensure that the C

^{2}smooth continuity is unchanged. Obviously, the global shape adjustment can be achieved by modifying the shape parameters ${\omega}_{1},{\lambda}_{j,1},{\mu}_{j,1}(j=1,2,\cdots ,[n/2]+1)$, ${\omega}_{2},{\lambda}_{\tilde{j},2},{\mu}_{\tilde{j},2}(\tilde{j}=1,2,\cdots ,[m/2]+1)$, and the control points ${P}_{1}^{2}$ and ${P}_{2}^{2}$. Thus, conclusion (2) is proved. This completes the proof of Proposition 3.

**Proposition**

**4.**

^{2}smooth continuity, some significant conclusions can be obtained when keeping its control points and G

^{2}smooth continuity unchanged.

^{2}smooth continuity remains unchanged, we can modify the global shape of the composite curve by altering the following shape parameters and control points:

**Proof.**

#### 5.4. Examples of Shape Adjustment between SG-Bézier Curves with C^{2} and G^{2} Smooth Continuity

^{2}smooth continuity between two SG-Bézier curves of degree 3. In this figure, the blue curve is the given SG-Bézier curve, and the red curve is constructed by the continuity conditions in Equation (27), which together with the blue curve reaches C

^{2}smooth continuity. The solid lines denote the curves before adjustment, while the dashed lines and dotted lines denote the curves after the adjustment of the shape parameters. The circular points denote the points before adjustment, and the asterisks indicate the modified control points. The shape parameters in Figure 11a are ${\omega}_{1}=1,\text{\hspace{0.17em}}{\lambda}_{1,1}=2,\text{\hspace{0.17em}}{\mu}_{1,1}=2,{\lambda}_{2,1}=3$ and ${\omega}_{2}=0.5,\text{\hspace{0.17em}}{\lambda}_{1,2}=4,\text{\hspace{0.17em}}{\mu}_{1,2}=4,{\lambda}_{2,2}=3$. Figure 11b displays the global shape adjustment of the composite curve by altering the shape parameters ${\lambda}_{1,1},{\lambda}_{2,1}$, where the shape parameters for the solid lines, dashed lines, and dotted lines are ${\lambda}_{1,1}=2,{\lambda}_{2,1}=3$, ${\lambda}_{1,1}=0,{\lambda}_{2,1}=0$, and ${\lambda}_{1,1}=4,{\lambda}_{2,1}=6$, respectively. Figure 11c displays the local shape adjustment by using a single shape parameter ${\mu}_{1,2}$, where the shape parameters are ${\mu}_{1,2}=4$ (solid lines), ${\mu}_{1,2}=2$ (dashed lines), and ${\mu}_{1,2}=0$ (dotted lines), respectively. Based on Figure 11a, Figure 11d illustrates the global shape adjustment by modifying the shape parameters ${\omega}_{1},{\omega}_{2},{\lambda}_{1,2}$. From the figure, we realize the global shape adjustment for the dashed lines merely using the shape parameters ${\omega}_{1},{\omega}_{2}$ (with ${\omega}_{1}={\omega}_{2}=0.1$), and the dotted lines by changing shape parameters ${\omega}_{1},{\omega}_{2},{\lambda}_{1,2}$ (with ${\omega}_{1}=0.5,{\omega}_{2}=0.8,{\lambda}_{1,2}=3$) . The rest of the shape parameters are equal to those in Figure 11a expect for the ones that are modified in the other three figures.

^{2}smooth continuity between the two SG-Bézier curves of degree 3. The shape parameters in Figure 12a are the same as those in Figure 7b. As argued in the paragraph above, the solid line means the curves before adjustment, while the dotted lines and dashed lines are the curves after the shape parameters are changed. The circular points in Figure 12 denote the control points before adjustment, and the asterisks marked on the control polygon are the modified control points. Figure 12b displays the local shape adjustment by changing the shape parameters ${\lambda}_{1,1},{\lambda}_{2,1}$, with the shape parameters ${\lambda}_{1,1}=3,{\lambda}_{2,1}=5$ (solid lines), ${\lambda}_{1,1}=2,{\lambda}_{2,1}=2$ (dashed lines), and ${\lambda}_{1,1}=0,{\lambda}_{2,1}=0$ (dotted lines). Figure 12c illustrates the local shape adjustment by using a single shape parameter ${\mu}_{1,2}$, with the shape parameter ${\mu}_{1,2}=3$ (solid lines), ${\mu}_{1,2}=1.5$ (dashed lines) and ${\mu}_{1,2}=0$ (dotted lines). Figure 12d gives an example showing the global shape adjustment by modifying the shape parameters ${\omega}_{1},{\omega}_{2},{\mu}_{1,1}$. The dashed lines achieve the global shape adjustment merely by using the shape parameters ${\omega}_{1},{\omega}_{2}$ (with ${\omega}_{1}=0.8,{\omega}_{2}=0.1$). The dotted lines realize the global shape adjustment in Figure 12d by altering the shape parameters ${\omega}_{1},{\omega}_{2},{\mu}_{1,1}$ (with ${\omega}_{1}=0.2,$ ${\omega}_{2}=0.8,{\mu}_{1,1}=1$). The rest of the shape parameters are all equal to those in Figure 12a expect for the ones that are modified in the other three figures.

## 6. Applications

## 7. Conclusions

^{1}, C

^{2}and G

^{1}, G

^{2}smooth continuity conditions for two adjacent nth-degree SG-Bézier curves, and study the effect of shape parameters on the shapes of the composite curves. Moreover, we give the concrete steps of G

^{2}smooth continuity between SG-Bézier curves. Furthermore, we utilize some practical examples to verify the validity of the proposed continuity conditions. The theoretical analysis and modeling examples show that our proposed continuity conditions between two SG-Bézier curves are not only easy to carry out, they also make the shapes of the curves adjusted more conveniently under these conditions. This paper provides an effective scheme for the shape design and construction of complex curves in the engineering that encode the value in the future applications. Notice that the introduction of shape parameters not only brings an advantage to the shape modification of composite SG-Bézier curves, it also provides optimized parameters for the shape optimization design of the SG-Bézier ones. Therefore, the research on how to utilize the genetic algorithm [26] to solve the model of curve shape optimization, which takes the shape parameters as the optimization variables, will be addressed in our future work.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Curvature of Composite SG-Bézier Curves in Figure 7

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## Share and Cite

**MDPI and ACS Style**

Hu, G.; Bo, C.; Wu, J.; Wei, G.; Hou, F.
Modeling of Free-Form Complex Curves Using SG-Bézier Curves with Constraints of Geometric Continuities. *Symmetry* **2018**, *10*, 545.
https://doi.org/10.3390/sym10110545

**AMA Style**

Hu G, Bo C, Wu J, Wei G, Hou F.
Modeling of Free-Form Complex Curves Using SG-Bézier Curves with Constraints of Geometric Continuities. *Symmetry*. 2018; 10(11):545.
https://doi.org/10.3390/sym10110545

**Chicago/Turabian Style**

Hu, Gang, Cuicui Bo, Junli Wu, Guo Wei, and Fei Hou.
2018. "Modeling of Free-Form Complex Curves Using SG-Bézier Curves with Constraints of Geometric Continuities" *Symmetry* 10, no. 11: 545.
https://doi.org/10.3390/sym10110545