# Degree Reduction of S-λ Curves Using a Genetic Simulated Annealing Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definition of S-λ Curves

#### 2.1. S-λ Distributions and Basis Functions

_{1}may be equal to $+\infty $. $\lambda (x)$ is called the transformation factor (tf). For $x\in [0,{R}_{1})$, setting ${P}_{1,j}(x)=\frac{{A}_{j}{[\lambda (x)]}^{j}}{S[\lambda (x)]}$, and then ${P}_{1,j}(x)\in C[0,{R}_{1})$ and $\sum {}_{j=0}^{m}{P}_{1,j}(x)=1$. Now we give the following definition (Fan and Zeng 2012 [15]).

**Definition**

**1.**

**Example**

**1.**

**Example**

**2.**

#### 2.2. S-λ Curves

**Definition**

**2.**

**Example**

**3.**

**C**(t) with different generating functions S(t). From Figure 2, we can see clearly that if the j-th coefficient A

_{j}of the generating functions is even bigger, then the corresponding S-λ curve is even closer to the control point

**V**

_{j}[15].

## 3. Degree Reduction of S-λ Curves Using an Intelligent Algorithm

#### 3.1. The Degree Reduction Problem

#### 3.2. Basic Principles of the Degree Reduction for S-λ Curve

#### 3.2.1. Elementary Algorithms

#### 3.2.2. Initialization of the Control Parameters and the Group

#### 3.2.3. Selection of a Fitness Function

#### 3.2.4. Selection

#### 3.2.5. Crossover

#### 3.2.6. Mutation

#### 3.2.7. Termination Conditions

#### 3.2.8. Setting Parameters

#### 3.3. The Procedure of the Degree Reduction for S-λ Curve

#### 3.4. Results and Discussions

#### 3.4.1. Degree Reduction of S-λ Curves with Fixed Endpoints

#### 3.4.2. Degree Reduction of S-λ Curves with Unconstrained Endpoint

#### 3.4.3. Quantitative Evaluation of the Degree Reduction Algorithm

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Degree reduction of a third-order S-λ curve with fixed endpoints. The control points of the degree-reduced curve are {(1,1), (2.6958, 8.9393), (5.3362, 18.5222), (13.7203, 25.5980), (20.8985, 21.6798), (24.6082, 7.1808), (26,1)}.

**Figure 4.**Degree reduction of a fourth-order S-λ curve with fixed endpoints. The control points of the degree-reduced curve are {(1,1), (3.7973, 5.0336), (6.2139, 0.0000), (9,2)}.

**Figure 5.**Degree reduction of a sixth-order S-λ curve with fixed endpoints. The control points of the degree-reduced curve are {(1,1), (2.21112, 4.5028), (3.2254, 5.6096), (6.9537, 5.6776), (7.5792, 4.4878), (9,1)}.

**Figure 6.**Degree reduction of a third-order S-λ curve with unconstrained endpoints. The control points of the degree-reduced curve are {(0.9766, 0.8736), (2.3848, 10.1090), (6.2801, 13.9343), (12.4859, 34.1484), (21.7390, 16.0439), (24.2554, 8.6127), (25.9813, 1.0708)}.

**Figure 7.**Degree reduction of a fourth-order S-λ curve with unconstrained endpoints. The control points of the degree-reduced curve are {(1.9132, 0.9329), (0.8636, 2.5036), (1.0289, 3.6672), (3.5579, 4.4441), (3.9541, 4.5389), (5.7761, 1.8075), (3.8825, 1.0933)}.

**Figure 8.**Degree reduction of a sixth-order S-λ curve with unconstrained endpoints. The control points of the degree-reduced curve are {(1.0346, 0.8772), (1.8500, 4.3859), (3.8810, 6.6002), (6.4763, 4.1338), (7.6701, 5.4494), (9.0588, 0.7503)}.

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**MDPI and ACS Style**

Lu, J.; Qin, X.
Degree Reduction of *S-λ* Curves Using a Genetic Simulated Annealing Algorithm. *Symmetry* **2019**, *11*, 15.
https://doi.org/10.3390/sym11010015

**AMA Style**

Lu J, Qin X.
Degree Reduction of *S-λ* Curves Using a Genetic Simulated Annealing Algorithm. *Symmetry*. 2019; 11(1):15.
https://doi.org/10.3390/sym11010015

**Chicago/Turabian Style**

Lu, Jing, and Xinqiang Qin.
2019. "Degree Reduction of *S-λ* Curves Using a Genetic Simulated Annealing Algorithm" *Symmetry* 11, no. 1: 15.
https://doi.org/10.3390/sym11010015