# Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{1}and G

^{2}between two adjacent SG-Bézier curves, and studied the smooth joining steps of SG-Bézier curves and the influence of shape parameters on complex curves [21]. At present, there is no report on the degree reduction of SG-Bézier curves.

^{k}

^{,k}continuity at the endpoints. Lee et al. [27] gave a multi-degree reduction algorithm by using the transformation between Bernstein basis and Legendre basis. In 2006, Rabah et al. [28] gave a multi-degree reduction algorithm for Bézier curves with endpoints constraints by using the transformation matrix between Bernstein basis and Chebyshev basis and the elevation matrix of Chebyshev polynomial. One method is to transform the problem of degree reduction into solving the optimization of objective function by using intelligent optimization algorithm; Ahn et al. [29] showed that the constrained polynomial degree reduction in the L

_{2}-norm equals best weighted Euclidean approximation of Bézier coefficients. In 2016, Ait-Haddou and Bartoň [30] illustrated that a weighted least squares approximation of Bézier coefficients with factored Hahn weights provides the best constrained polynomial degree reduction with respect to the Jacobi L

_{2}-norm. Lu and Qin [31] proposed a method for the degree reduction of S-λ curves using a GSA algorithm. Based on the theory of grey wolf optimizer (GWO) algorithm [32,33,34,35], this paper considers the degree reduction of SG-Bézier curves under unrestricted condition and constraint condition of C

^{0}and C

^{1}. The numerical examples show that the algorithm can find the global optimal SG-Bézier curve more quickly and accurately.

## 2. The Definition of SG-Bézier Curves

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Definition**

**2.**

## 3. Grey Wolf Optimizer (GWO) Algorithm

#### 3.1. The Basic Principles of Grey Wolf Optimizer

#### 3.1.1. Social Hierarchy

#### 3.1.2. Encircling Prey

**A**and

**C**represent coefficient vectors; r

_{1}and r

_{2}are random numbers between [0, 1]; and a linearly decrements from 2 to 0 as the number of iterations increases during the iteration.

#### 3.1.3. Attack Prey

**D**

_{α},

**D**

_{β}, and

**D**

_{δ}respectively represent the distance between the search individual (ω wolf) and the α, β and δ wolves;

**X**

_{1},

**X**

_{2}, and

**X**

_{3}indicate, respectively, that α, β, and δ wolves guide the direction of individual (ω wolf) movement;

**X**

_{ω}(t + 1) indicates the position of the grey wolf individual after updating.

#### 3.2. Algorithmic Flow of Grey Wolf Optimizer

**a**,

**C**, and

**A**, and generate the S initial wolf group:

**X**

_{i}(i = 0, 1, …, m).

**X**

_{α},

**X**

_{β}, and

**X**

_{δ}.

**A**,

**C**according to Equations (7)–(9).

**X**

_{α},

**X**

_{β}, and

**X**

_{δ}.

**X**

_{α}, and the algorithm ends.

## 4. Degree Reduction of SG-Bézier Curves with the Grey Wolf Optimizer

#### 4.1. The Basic Idea of SG-Bézier Curve Degree Reduction

**P**,

**Q**are the control point vectors of the curve before and after the degree reduction, ω is the global shape parameter, and λ and μ are the local shape control parameters.

#### 4.2. Initialization of the Grey Wolf Population

_{j}∈ [0, 1] (j = 0, 1, …, m), and then use the following equation to find the control points on the interval [

**P**

_{min},

**P**

_{max}]:

**Q**

_{j}=

**P**α

_{min}+_{j}(

**P**−

_{max}**P**), j = 0, 1, ..., m,

_{min}**Q**

_{j}is the initial control points of the degree-reduced SG-Bézier curves

**Q**(t),

**P**

_{min}= (x

_{min}, y

_{min}),

**P**

_{max}= (x

_{max}, y

_{max}). When the endpoints satisfy the C

^{0}constraint, taking

**Q**

_{0}=

**P**

_{0}and

**Q**

_{m}=

**P**

_{n}to ensure the endpoints interpolation; when the endpoints satisfy the C

^{1}constraint, we have:

#### 4.3. Selection of Fitness Function

**P**(t

_{j}),

**Q**(t

_{j})) is Euclidean distance,

**P**(t

_{j}) and

**Q**(t

_{j})(j = 0, 1, …, s) are the different points on the SG-Bézier curves before and after degree reduction, respectively.

#### 4.4. The Algorithm Description for Degree Reduction of SG-Bézier Curves

Algorithm 1 Grey wolf optimizer algorithm: Steps to the degree reduction of SG-Bézier curves. | |

Step 1 | Input the degree and control points sequence of SG-Bézier curves before degree reduction {P_{0}, P_{1}, …, P_{n}}. |

Step 2 | Draw SG-Bézier curve before degree reduction and its control polygon. |

Step 3 | Initialize the parameters, set the population size and the maximum number of iterations MaxIter, determine the number of points on the curve selected in the error analysis before and after the reduction, and give the upper and lower bounds of the search individual information dimension and the search individual information dimension. |

Step 4 | According to Equation (17), generate the initial population Q_{j}(j = 0, 1, …, m). |

Step 5 | According to Equation (19), calculate the fitness value of the grey wolf individual, rank all the fitness values, and record the position Q_{α}, Q_{β}, Q_{δ}of the optimal grey wolf α, the sub-optimal grey wolf β and the third optimal grey wolf ω. |

Step 6 | According to Equation (12), update the position of the current grey wolf. |

Step 7 | Update parameters a, A and C by Equations (7)–(9). |

Step 8 | Calculate the fitness values of all grey wolves after being updated, and determine the new Q_{α}, Q_{β} and Q_{δ}. |

Step 9 | To determine whether the maximum number of iterations is reached, yes, execute Step 10; No, repeat Step 5–Step 9. |

Step 10 | Output optimization parameter result Q_{α}. |

Step 11 | Draw SG-Bézier curve and its control polygon after degree reduction. |

Algorithm 2 The pseudo-code of the GWO algorithm: Degree reduction of SG-Bézier curve | |

Input Parameters: s: Population size; d: Dimensions of searching individuals; lb: The lower bound of searching individual dimension; ub: The upper bound of searching individual dimension; N: The number of points on the curve selected during error analysis before and after the reduction; m: Maximum number of iterations; | |

01 | wolf population ${\mathbf{Q}}_{j}\left(j=1,2,\dots ,m\right)\leftarrow $ initialization(wolf_size); |

02 | fitness $\leftarrow $ evaluate(wolf population); /*Calculating individual fitness of grey wolf*/ |

03 | while (evaluation_number<max_evaluation_number) do/*Termination condition*/ |

04 | ${\mathbf{Q}}_{\alpha},{\mathbf{Q}}_{\beta},{\mathbf{Q}}_{\delta}\leftarrow $ select the first best three wolves(wolf population); /*Select the optimal, sub-optimal, and second optimal wolf*/ |

05 | while (i < m) do |

06 | new position of the wolf $\leftarrow $ update(the current wolf); /*Update the location of the current search individuals*/ |

07 | evaluation_number + 1; |

08 | end |

09 | update(a,A,C) |

10 | fitness $\leftarrow $ evaluate(new position of the wolf); |

11 | evaluation_number + 1; |

12 | end |

13 | ${\mathbf{Q}}_{\alpha}\leftarrow $ select the best wolf(the entire wolves); |

14 | output ${\mathbf{Q}}_{\alpha}$ |

## 5. Examples of Degree Reduction Approximation Curves

**Example**

**1.**

^{0}and C

^{1}. Under each kind of constraints, two different situations are given: changing global shape parameters (Figure 4, Figure 5 and Figure 6) and changing local shape parameters (Figure 7, Figure 8 and Figure 9). The errors of the curves after the degree reduction are shown in Table 2 and Table 3, respectively.

**Example**

**2.**

^{0}and C

^{1}. Under each kind of constraints, two different situations are given: changing global shape parameters (Figure 10, Figure 11 and Figure 12) and changing local shape parameters (Figure 13, Figure 14 and Figure 15). The errors of the curves after the degree reduction are shown in Table 4 and Table 5, respectively.

**Example**

**3.**

## 6. Conclusions

^{0}and C

^{1}. These examples fully demonstrate the characteristics of the global optimization of the grey wolf optimizer algorithm in the process of degree reduction. That is to say this method is applicable for CAD/CAM modeling systems. The research on how to utilize the GWO algorithm [32,33,34,35] or genetic algorithm [36] 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

## References

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**Figure 4.**Degree reduction of SG-Bézier curves of degree 6 under unrestricted condition in Example 5.1; changing global shape parameter. (

**a**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0}; (

**b**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.2}; (

**c**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.3}; (

**d**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.5}.

**Figure 5.**Degree reduction of SG-Bézier curve of degree 6 under C

^{0}constraint condition in Example 5.1; changing global shape parameter. (

**a**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0}; (

**b**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.2}; (

**c**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.3}; (

**d**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.5}.

**Figure 6.**Degree reduction of SG-Bézier curve of degree 6 under C

^{1}constraint condition in Example 5.1; changing global shape parameter. (

**a**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0}; (

**b**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.2}; (

**c**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.3}; (

**d**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, µ

_{3}= 3; ω = 0.5}.

**Figure 7.**Degree reduction of SG-Bézier curve of degree 6 under unrestricted condition in Example 5.1; changing local shape parameters. (

**a**) {ω = 0, λ

_{2}= 3, µ

_{1}= 3, µ

_{2}= 3; λ

_{1}= 2}; (

**b**) {ω = 0, λ

_{1}= 3, µ

_{1}= 3, µ

_{2}= 3; λ

_{2}= 4}; (

**c**) {ω = 0, λ

_{1}= 3, λ

_{2}= 3, µ

_{2}= 3; µ

_{1}= 2}; (

**d**) {ω = 0, λ

_{2}= 3, µ

_{1}= 3; λ

_{1}= 2, µ

_{2}= 2}.

**Figure 8.**Degree reduction of SG-Bézier curve of degree 6 under C

^{0}constraint condition in Example 5.1; changing local shape parameters. (

**a**) {ω = 0, λ

_{2}= 3, µ

_{1}= 3, µ

_{2}= 3; λ

_{1}= 2}; (

**b**) {ω = 0, λ

_{1}= 3, µ

_{1}= 3, µ

_{2}= 3; λ

_{2}= 4}; (

**c**) {ω = 0, λ

_{1}= 3, λ

_{2}= 3, µ

_{2}= 3; µ

_{1}= 2}; (

**d**) {ω = 0, λ

_{2}= 3, µ

_{1}= 3; λ

_{1}= 2, µ

_{2}= 2}.

**Figure 9.**Degree reduction of SG-Bézier curve of degree 6 under C

^{1}constraint condition in Example 5.1; changing local shape parameters. (

**a**) {ω = 0, λ

_{2}= 3, µ

_{1}= 3, µ

_{2}= 3; λ

_{1}= 2}; (

**b**) {ω = 0, λ

_{1}= 3, µ

_{1}= 3, µ

_{2}= 3; λ

_{2}= 4}; (

**c**) {ω = 0, λ

_{1}= 3, λ

_{2}= 3, µ

_{2}= 3; µ

_{1}= 2}; (

**d**) {ω = 0, λ

_{2}= 3, µ

_{1}= 3; λ

_{1}= 2, µ

_{2}= 2}.

**Figure 10.**Degree reduction of SG-Bézier curve of degree 8 under unrestricted condition in Example 5.2; changing global shape parameter. (

**a**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0}; (

**b**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.2};(

**c**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.3};(

**d**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.5}.

**Figure 11.**Degree reduction of SG-Bézier curve of degree 8 under C

^{0}constraint condition in Example 5.2; changing global shape parameter. (

**a**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0}; (

**b**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.2}; (

**c**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.3};(

**d**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.5}.

**Figure 12.**Degree reduction of SG-Bézier curve of degree 8 under C

^{1}constraint condition in Example 5.2; changing global shape parameter. (

**a**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0}; (

**b**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.2}; (

**c**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.3}; (

**d**) {λ

_{1}= 3, λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3; ω = 0.5}.

**Figure 13.**Degree reduction of SG-Bézier curve of degree 8 under unrestricted condition in Example 5.2; changing local shape parameters. (

**a**) {λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2}; (

**b**) {λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2, λ

_{2}= 4};(

**c**) {λ

_{3}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2, λ

_{2}= 4, µ

_{1}= 2};(

**d**) {λ

_{1}= 3, λ

_{2}= 3, µ

_{1}= 3, ω = 0; λ

_{3}= 2, µ

_{2}= 2}.

**Figure 14.**Degree reduction of SG-Bézier curve of degree eight under C

^{0}constraint condition in Example 5.2; changing local shape parameters. (

**a**) {λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2}; (

**b**) {λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2, λ

_{2}= 4}; (

**c**) {λ

_{3}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2, λ

_{2}= 4, µ

_{1}= 2}; (

**d**) {λ

_{1}= 3, λ

_{2}= 3, µ

_{1}= 3, ω = 0; λ

_{3}= 2, µ

_{2}= 2}.

**Figure 15.**Degree reduction of SG-Bézier curve of degree eight under C

^{1}constraint condition in Example 5.2; changing local shape parameters. (

**a**) {λ

_{2}= 3, λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2}; (

**b**) {λ

_{3}= 3, µ

_{1}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2, λ

_{2}= 4}; (

**c**) {λ

_{3}= 3, µ

_{2}= 3, ω = 0; λ

_{1}= 2, λ

_{2}= 4, µ

_{1}= 2}; (

**d**) {λ

_{1}= 3, λ

_{2}= 3, µ

_{1}= 3, ω = 0; λ

_{3}= 2, µ

_{2}= 2}.

**Figure 16.**Comparisons between the proposed method and GA method; under different constraints. (

**a**) {under unrestricted condition; grey wolf optimizer algorithm}; (

**b**) {under unrestricted condition; genetic algorithm}; (

**c**) {under C

^{0}constraint condition; grey wolf optimizer algorithm}; (

**d**) {under C

^{0}constraint condition; genetic algorithm}; (

**e**) {under C

^{1}constraint condition; grey wolf optimizer algorithm}; (

**f**) {under C

^{1}constraint condition; genetic algorithm}.

n | Parameters |
---|---|

even | $\omega $, ${\lambda}_{1},{\lambda}_{2},\cdots ,{\lambda}_{\left[n/2\right]}$, ${\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{\left[n/2\right]}$ |

odd | $\omega $, ${\lambda}_{1},{\lambda}_{2},\cdots ,{\lambda}_{\left[n/2\right]},{\lambda}_{\left[n/2\right]+1}$, ${\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{\left[n/2\right]}$ |

**Table 2.**Error for an approximate SG-Bézier curve of degree 6 to degree 4; changing global shape parameters.

Constraint Condition | ω = 0 | ω = 0.2 | ω = 0.3 | ω = 0.5 |
---|---|---|---|---|

Unrestricted | 4.9914 × 10^{−3} | 8.6669 × 10^{−3} | 7.4429 × 10^{−3} | 9.5617 × 10^{−3} |

C^{0} constraint | 4.1972 × 10^{−3} | 2.7182 × 10^{−3} | 3.1251 × 10^{−3} | 2.6399 × 10^{−3} |

C^{1} constraint | 9.1096 × 10^{−3} | 9.1890 × 10^{−3} | 9.1935 × 10^{−}^{3} | 9.1096 × 10^{−3} |

**Table 3.**Error for an approximate SG-Bézier curve of degree 6 to degree 4; changing local shape parameters.

Shape Parameters | Unrestricted | C^{0} Constraint | C^{1} Constraint |
---|---|---|---|

ω = 0, λ_{2} = 3, µ_{1} = 3, µ_{2} = 3; λ_{1} = 2 | 1.0108 × 10^{−2} | 2.6558 × 10^{−3} | 9.1097 × 10^{−3} |

ω = 0, λ_{1} = 3, µ_{1} = 3, µ_{2} = 3; λ_{2} = 4 | 4.4997 × 10^{−3} | 2.6803 × 10^{−3} | 9.1097 × 10^{−3} |

ω = 0, λ_{1} = 3, λ_{2} = 3, µ_{2} = 3; µ_{1} = 2 | 4.0165 × 10^{−3} | 1.0576 × 10^{−2} | 9.1097 × 10^{−3} |

ω = 0, λ_{2} = 3, µ_{1} = 3; λ_{1} = 2, µ_{2} = 2 | 7.3299 × 10^{−3} | 1.0956 × 10^{−2} | 9.1097 × 10^{−3} |

**Table 4.**Error for an approximate SG-Bézier curve of degree 8 to degree 5; changing global shape parameters.

Constraint Condition | ω = 0 | ω = 0.2 | ω = 0.3 | ω = 0.5 |
---|---|---|---|---|

Unrestricted | 2.5164 × 10^{−2} | 1.9874 × 10^{−2} | 9.4116 × 10^{−3} | 2.2389 × 10^{−2} |

C^{0} constraint | 4.9315 × 10^{−3} | 1.2441 × 10^{−2} | 8.8423 × 10^{−3} | 7.3303 × 10^{−3} |

C^{1} constraint | 2.9630 × 10^{−3} | 2.7537 × 10^{−3} | 3.9361 × 10^{−3} | 6.7623 × 10^{−3} |

**Table 5.**Error for an approximate SG-Bézier curve of degree 8 to degree 5; changing local shape parameters.

Shape Parameters | Unrestricted | C^{0} Constraint | C^{1} Constraint |
---|---|---|---|

ω = 0, λ_{2} = 3, λ_{3} = 3, µ_{1} = 3, µ_{2} = 3; λ_{1} = 2 | 3.3712 × 10^{−2} | 1.8863 × 10^{−2} | 2.7072 × 10^{−3} |

ω = 0, λ_{3} = 3, µ_{1} = 3, µ_{2} = 3; λ_{1} = 2, λ_{2} = 4 | 2.6413 × 10^{−2} | 1.7231 × 10^{−2} | 1.6661 × 10^{−3} |

ω = 0, λ_{3} = 3, µ_{2} = 3; λ_{1} = 2, λ_{2} = 4, µ_{1} = 2 | 3.0435 × 10^{−2} | 1.4649 × 10^{−2} | 2.0349 × 10^{−2} |

ω = 0, λ_{1} = 3, λ_{2} = 3, µ_{1} = 3; λ_{3} = 2, µ_{2} = 2 | 1.3253 × 10^{−2} | 5.2874 × 10^{−3} | 1.4618 × 10^{−3} |

Method | Unrestricted | C^{0} Constraint | C^{1} Constraint |
---|---|---|---|

GWO method | 6.1586 × 10^{−3} | 4.7146 × 10^{−3} | 7.4163 × 10^{−3} |

GA method | 1.3546 × 10^{−1} | 1.0971 × 10^{−1} | 2.4860 × 10^{−2} |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hu, G.; Qiao, Y.; Qin, X.; Wei, G.
Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm. *Symmetry* **2019**, *11*, 1242.
https://doi.org/10.3390/sym11101242

**AMA Style**

Hu G, Qiao Y, Qin X, Wei G.
Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm. *Symmetry*. 2019; 11(10):1242.
https://doi.org/10.3390/sym11101242

**Chicago/Turabian Style**

Hu, Gang, Yu Qiao, Xinqiang Qin, and Guo Wei.
2019. "Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm" *Symmetry* 11, no. 10: 1242.
https://doi.org/10.3390/sym11101242