# An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes

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## Abstract

**:**

## 1. Introduction

## 2. Fundamental Concepts

#### 2.1. Non-Stationary Curves

#### 2.2. Non-Stationary Surfaces

## 3. Main Results and Numerical Applications

#### 3.1. Subdivision Depth for Non-Stationary Subdivision Curves

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

#### 3.2. Numerical Applications for Curve Models

**Experiment 1.**

**Remark 1.**

**Remark 2.**

**Experiment 2.**

#### 3.3. Subdivision Depth for Non-Stationary Subdivision Surfaces

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**Proof.**

#### 3.4. Numerical Application for Surface Models

**Experiment 3.**

**Experiment 4.**

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{Y}}_{{\mathit{\chi}}_{0}^{\mathit{c}}}\to $ | ${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ${\mathit{Y}}_{3}$ | ${\mathit{Y}}_{4}$ | ${\mathit{Y}}_{5}$ |
---|---|---|---|---|---|

1st level ($\kappa $ = 1) | 0.509795 | 0.254898 | 0.128345 | 0.064401 | 0.032343 |

2nd level ($\kappa $ = 2) | 0.502419 | 0.251211 | 0.125831 | 0.062972 | 0.031522 |

3rd level ($\kappa $ = 3) | 0.500603 | 0.250301 | 0.125207 | 0.062618 | 0.031318 |

4th level ($\kappa $ = 4) | 0.500151 | 0.250075 | 0.125052 | 0.062529 | 0.031267 |

5th level ($\kappa $ = 5) | 0.500038 | 0.250019 | 0.125013 | 0.062507 | 0.031254 |

$\u03f5$ | $2.52{e}^{-3}$ | $2.62{e}^{-6}$ | ||||||||

${Y}_{{\chi}_{0}^{c}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${Y}_{1}$ | 6 | 6 | 6 | 6 | 6 | 16 | 16 | 16 | 16 | 16 |

${Y}_{2}$ | 3 | 3 | 3 | 3 | 3 | 8 | 8 | 8 | 8 | 8 |

${Y}_{3}$ | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 5 |

${Y}_{4}$ | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |

${Y}_{5}$ | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |

$\u03f5$ | $2.67{e}^{-9}$ | $2.86{e}^{-12}$ | ||||||||

${Y}_{{\chi}_{0}^{c}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${Y}_{1}$ | 26 | 26 | 26 | 26 | 26 | 37 | 36 | 36 | 36 | 36 |

${Y}_{2}$ | 13 | 13 | 13 | 13 | 13 | 18 | 18 | 18 | 18 | 18 |

${Y}_{3}$ | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 |

${Y}_{4}$ | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 |

${Y}_{5}$ | 5 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 7 | 7 |

${\mathit{Y}}_{{\mathit{\chi}}_{0}^{\mathit{c}}}\to $ | ${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ${\mathit{Y}}_{3}$ | ${\mathit{Y}}_{4}$ | ${\mathit{Y}}_{5}$ |
---|---|---|---|---|---|

1st level ($\kappa $ = 1) | 0.564343 | 0.378685 | 0.229794 | 0.139262 | 0.084417 |

2nd level ($\kappa $ = 2) | 0.562954 | 0.375908 | 0.227357 | 0.137343 | 0.082985 |

3rd level ($\kappa $ = 3) | 0.562613 | 0.375226 | 0.22676 | 0.136874 | 0.082635 |

4th level ($\kappa $ = 4) | 0.562528 | 0.375056 | 0.226612 | 0.136757 | 0.082548 |

5th level ($\kappa $ = 5) | 0.562507 | 0.375014 | 0.226575 | 0.136728 | 0.082527 |

$\u03f5$ | $9.22{e}^{-3}$ | $6.57{e}^{-3}$ | ||||||||

${Y}_{{\chi}_{0}^{c}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${Y}_{1}$ | 6 | 6 | 6 | 6 | 6 | 14 | 14 | 14 | 14 | 14 |

${Y}_{2}$ | 3 | 3 | 3 | 3 | 3 | 8 | 8 | 8 | 8 | 8 |

${Y}_{3}$ | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 5 |

${Y}_{4}$ | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |

${Y}_{5}$ | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |

$\u03f5$ | $4.68{e}^{-7}$ | $3.33{e}^{-9}$ | ||||||||

${Y}_{{\chi}_{0}^{c}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${Y}_{1}$ | 23 | 23 | 23 | 23 | 23 | 32 | 31 | 31 | 31 | 31 |

${Y}_{2}$ | 13 | 13 | 13 | 13 | 13 | 18 | 18 | 18 | 18 | 18 |

${Y}_{3}$ | 9 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 |

${Y}_{4}$ | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 |

${Y}_{5}$ | 5 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 7 | 7 |

${\mathit{U}}_{{\mathit{\chi}}_{0}^{\mathit{s}}}{\mathit{V}}_{{\mathit{\chi}}_{0}^{\mathit{s}}}\to $ | ${\mathit{U}}_{1}{\mathit{V}}_{1}$ | ${\mathit{U}}_{2}{\mathit{V}}_{2}$ | ${\mathit{U}}_{3}{\mathit{V}}_{3}$ | ${\mathit{U}}_{4}{\mathit{V}}_{4}$ | ${\mathit{U}}_{5}{\mathit{V}}_{5}$ |
---|---|---|---|---|---|

1st level ($\kappa $ = 1) | 0.509795 | 0.254898 | 0.128345 | 0.064401 | 0.032368 |

2nd level ($\kappa $ = 2) | 0.502419 | 0.25121 | 0.12583 | 0.062972 | 0.031528 |

3rd level ($\kappa $ = 3) | 0.500603 | 0.250301 | 0.125207 | 0.062618 | 0.031319 |

4th level ($\kappa $ = 4) | 0.500151 | 0.250075 | 0.125052 | 0.062529 | 0.031267 |

5th level ($\kappa $ = 5) | 0.500038 | 0.250019 | 0.125013 | 0.062507 | 0.031254 |

$\u03f5$ | $6.51{e}^{-3}$ | $6.84{e}^{-6}$ | ||||||||

${U}_{{\chi}_{0}^{s}}{V}_{{\chi}_{0}^{s}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${U}_{1}{V}_{1}$ | 6 | 6 | 6 | 6 | 6 | 16 | 16 | 16 | 16 | 16 |

${U}_{2}{V}_{2}$ | 3 | 3 | 3 | 3 | 3 | 8 | 8 | 8 | 8 | 8 |

${U}_{3}{V}_{3}$ | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 5 |

${U}_{4}{V}_{4}$ | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |

${U}_{5}{V}_{5}$ | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |

$\u03f5$ | $7.17{e}^{-9}$ | $7.52{e}^{-12}$ | ||||||||

${U}_{{\chi}_{0}^{s}}{V}_{{\chi}_{0}^{s}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${U}_{1}{V}_{1}$ | 26 | 26 | 26 | 26 | 26 | 37 | 36 | 36 | 36 | 36 |

${U}_{2}{V}_{2}$ | 13 | 13 | 13 | 13 | 13 | 18 | 18 | 18 | 18 | 18 |

${U}_{3}{V}_{3}$ | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 |

${U}_{4}{V}_{4}$ | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 |

${U}_{5}{V}_{5}$ | 5 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 7 | 7 |

${\mathit{U}}_{{\mathit{\chi}}_{0}^{\mathit{s}}}{\mathit{V}}_{{\mathit{\chi}}_{0}^{\mathit{s}}}\to $ | ${\mathit{U}}_{1}{\mathit{V}}_{1}$ | ${\mathit{U}}_{2}{\mathit{V}}_{2}$ | ${\mathit{U}}_{3}{\mathit{V}}_{3}$ | ${\mathit{U}}_{4}{\mathit{V}}_{4}$ | ${\mathit{U}}_{5}{\mathit{V}}_{5}$ |
---|---|---|---|---|---|

1st level ($\kappa $ = 1) | 0.79049 | 0.402995 | 0.208033 | 0.105486 | 0.052898 |

2nd level ($\kappa $ = 2) | 0.783521 | 0.398725 | 0.205803 | 0.104247 | 0.052272 |

3rd level ($\kappa $ = 3) | 0.781815 | 0.397683 | 0.205258 | 0.103946 | 0.05212 |

4th level ($\kappa $ = 4) | 0.781391 | 0.397425 | 0.205123 | 0.10387 | 0.052082 |

5th level ($\kappa $ = 5) | 0.781285 | 0.39736 | 0.205089 | 0.103852 | 0.052073 |

$\u03f5$ | $1.67{e}^{-2}$ | $4.68{e}^{-5}$ | ||||||||

${U}_{{\chi}_{0}^{s}}{V}_{{\chi}_{0}^{s}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${U}_{1}{V}_{1}$ | 19 | 18 | 18 | 18 | 18 | 44 | 42 | 42 | 42 | 42 |

${U}_{2}{V}_{2}$ | 4 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 10 |

${U}_{3}{V}_{3}$ | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |

${U}_{4}{V}_{4}$ | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |

${U}_{5}{V}_{5}$ | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |

$\u03f5$ | $1.31{e}^{-7}$ | $3.67{e}^{-10}$ | ||||||||

${U}_{{\chi}_{0}^{s}}{V}_{{\chi}_{0}^{s}}$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ | $\kappa =1$ | $\kappa =2$ | $\kappa =3$ | $\kappa =4$ | $\kappa =5$ |

${U}_{1}{V}_{1}$ | 69 | 66 | 66 | 66 | 65 | 94 | 90 | 90 | 89 | 89 |

${U}_{2}{V}_{2}$ | 17 | 16 | 16 | 16 | 16 | 23 | 23 | 23 | 23 | 23 |

${U}_{3}{V}_{3}$ | 9 | 9 | 9 | 9 | 9 | 13 | 13 | 13 | 13 | 13 |

${U}_{4}{V}_{4}$ | 7 | 7 | 7 | 7 | 7 | 9 | 9 | 9 | 9 | 9 |

${U}_{5}{V}_{5}$ | 5 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 7 | 7 |

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**MDPI and ACS Style**

Abdul Karim, S.A.; Khan, F.; Mustafa, G.; Shahzad, A.; Asghar, M.
An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes. *Mathematics* **2023**, *11*, 2449.
https://doi.org/10.3390/math11112449

**AMA Style**

Abdul Karim SA, Khan F, Mustafa G, Shahzad A, Asghar M.
An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes. *Mathematics*. 2023; 11(11):2449.
https://doi.org/10.3390/math11112449

**Chicago/Turabian Style**

Abdul Karim, Samsul Ariffin, Faheem Khan, Ghulam Mustafa, Aamir Shahzad, and Muhammad Asghar.
2023. "An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes" *Mathematics* 11, no. 11: 2449.
https://doi.org/10.3390/math11112449