# Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves

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## Abstract

**:**

## 1. Introduction

## 2. Convergence Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Numerical Experiments

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Remark**

**3.**

- (1)
- Divide a parameter interval $[a,b]$ of parametric curve $c\left(t\right)$ into M subintervals with equal length.
- (2)
- Randomly select an initial iterative parametric value in each subinterval.
- (3)
- Using the iterative method (2) and using each initial iterative parametric value, iterate, respectively. Suppose that the iterative parametric values are ${\alpha}_{1}$ ,${\alpha}_{2}$ ,…,${\alpha}_{M}$, respectively.
- (4)
- Calculate the local minimum distances ${d}_{1},{d}_{2},\cdots ,{d}_{M}$, where ${d}_{i}=\u2225p-c\left({\alpha}_{i}\right)\u2225$.
- (5)
- Calculate the global minimum distance $d=\u2225p-c\left(\alpha \right)\u2225=\left\{{d}_{1},{d}_{2},\cdots {d}_{M}\right\}$.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Table 1.**Step sizes $\Delta {t}_{1}$, $\Delta {t}_{2}$ in Example 1 for the Newton method and the iterative method (2).

$\mathit{p}=(2,2),{\mathit{t}}_{0}=-0.32$ | |||||||

iterative methods | Step | 1 | 2 | 3 | 4 | 5 | 6 |

Newton method | $\Delta {t}_{1}$ | NC | NC | NC | NC | NC | NC |

The iterative method (2) | $\Delta {t}_{2}$ | 1.82 | $2.8\times {10}^{-1}$ | $2.1\times {10}^{-3}$ | $8.3\times {10}^{-7}$ | $1.4\times {10}^{-13}$ | 0 |

$\mathit{p}=(\mathbf{2},\mathbf{2}),{\mathit{t}}_{\mathbf{0}}=\mathbf{4.2}$ | |||||||

iterative methods | Step | 2 | 3 | 4 | 5 | 6 | 7 |

Newton method | $\Delta {t}_{1}$ | NC | NC | NC | NC | NC | NC |

The iterative method (2) | $\Delta {t}_{2}$ | $-1.65\times {10}^{-1}$ | $-4.1\times {10}^{-2}$ | $3.5\times {10}^{-4}$ | $2.5\times {10}^{-8}$ | $2.2\times {10}^{-16}$ | 0 |

**Table 2.**Step sizes $\Delta {t}_{1}$, $\Delta {t}_{2}$ in Example 2 for the Newton method and the iterative method (2).

$\mathit{p}=(2,5)$, ${\mathit{t}}_{0}=1.7$ | ||||||||

iterative methods | Step | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Newton method | $\Delta {t}_{1}$ | NC | NC | NC | NC | NC | NC | NC |

The iterative method (2) | $\Delta {t}_{2}$ | −1.04 | $3.29\times {10}^{-1}$ | $2.94\times {10}^{-2}$ | $4.48\times {10}^{-4}$ | $1.05\times {10}^{-7}$ | $5.77\times {10}^{-15}$ | 0.0 |

$\mathit{p}=(\mathbf{2},\mathbf{5}),{\mathit{t}}_{\mathbf{0}}=-\mathbf{2.4}$ | ||||||||

iterative methods | Step | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

Newton method | $\Delta {t}_{1}$ | NC | NC | NC | NC | NC | NC | NC |

The iterative method (2) | $\Delta {t}_{2}$ | −1.34 | −0.461 | $9.08\times {10}^{-2}$ | $4.08\times {10}^{-3}$ | $8.69\times {10}^{-6}$ | $3.94\times {10}^{-11}$ | 0 |

**Table 3.**Comparison of the robustness and effectiveness for the Newton’s method and the iterative method (2).

${t}_{0}$ | −20 | −16 | −13 | −10 | −7 | −2 | 0 | 1 | 5 | 8 | 11 | 15 | 18 | 20 |

The iterative method (2) | 7 | 7 | 7 | 7 | 6 | 5 | 4 | 5 | 6 | 7 | 7 | 7 | 8 | 8 |

Newton’s method | NC | 11 | 11 | 10 | 10 | 7 | 5 | 6 | 9 | 10 | 11 | 11 | 12 | NC |

**Table 4.**Comparison of the robustness and effectiveness for the Newton’s method and the iterative method (2).

${t}_{0}$ | 5.75 | 5.76 | 5.77 | 5.78 | 5.79 | 5.80 | 5.82 | 5.84 | 5.85 | 5.87 |

The iterative method (2) | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 |

Newton method | NC | NC | NC | NC | NC | NC | NC | NC | NC | NC |

${t}_{0}$ | 5.88 | 5.89 | 5.90 | 5.92 | 5.93 | 5.95 | 5.96 | 5.97 | 5.99 | 6.00 |

The iterative method (2) | 7 | 6 | 6 | 8 | 10 | 25 | 23 | 24 | 21 | 22 |

Newton method | NC | NC | NC | NC | NC | NC | NC | NC | NC | NC |

**Table 5.**Comparison of the robustness and effectiveness for the Newton’s method and the iterative method (2).

${t}_{0}$ | −5.6 | −5.5 | −5.3 | −5.2 | −5.0 | −4.9 | −4.8 | −4.7 | −4.6 | −4.5 |

The iterative method (2) | 11 | 11 | 11 | 12 | 14 | 11 | 13 | 14 | 11 | 14 |

Newton method | NC | NC | NC | NC | NC | NC | NC | NC | NC | NC |

${t}_{0}$ | −4.3 | −4.0 | −3.8 | −3.4 | −3.0 | −2.3 | −2.0 | −1.0 | 0.4 | 1.0 |

The iterative method (2) | 13 | 11 | 16 | 7 | 10 | 6 | 11 | 11 | 12 | 14 |

Newton method | NC | NC | NC | NC | NC | NC | NC | NC | NC | NC |

**Table 6.**Comparison of the robustness and effectiveness for the Newton’s method and the iterative method (2).

${t}_{0}$ | −12.8 | −12.4 | −12.1 | −11.9 | −11.7 | −11.0 | −10.8 | −10.7 | −10.4 | −10.0 |

The iterative method (2) | 59 | 55 | 59 | 46 | 46 | 53 | 41 | 42 | 39 | 32 |

Newton method | NC | NC | NC | NC | NC | NC | NC | NC | NC | NC |

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**MDPI and ACS Style**

Li, X.; Wang, L.; Wu, Z.; Hou, L.; Liang, J.; Li, Q.
Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves. *Symmetry* **2017**, *9*, 210.
https://doi.org/10.3390/sym9100210

**AMA Style**

Li X, Wang L, Wu Z, Hou L, Liang J, Li Q.
Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves. *Symmetry*. 2017; 9(10):210.
https://doi.org/10.3390/sym9100210

**Chicago/Turabian Style**

Li, Xiaowu, Lin Wang, Zhinan Wu, Linke Hou, Juan Liang, and Qiaoyang Li.
2017. "Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves" *Symmetry* 9, no. 10: 210.
https://doi.org/10.3390/sym9100210