# A Computational Method for Subdivision Depth of Ternary Schemes

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results for Univariate Case

#### 2.1. Reformulation of Successive Convolutions

**Lemma**

**1.**

**Proof.**

**Case ${k}_{0}=1$:**From (5), we obtain a relation given in the following

**Case ${k}_{0}=2$:**From (6), we acquire

**General case:**By using the same technique, we acquire the reformulations for ${k}_{0}$-th convolutions, which is in the following

**Lemma**

**2.**

**Proof.**

**Case ${k}_{0}=1$:**

**Case ${k}_{0}=M+1$:**Consider

**Corollary**

**1.**

**Proof.**

#### 2.2. Subdivision Depth for Ternary Subdivision Curves

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 2.3. Application for Univariate Case

**Remark**

**1.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 3. Preliminary Results for Bivariate Case

**Lemma**

**3.**

**Proof.**

**Case ${k}_{0}=1$:**Consider an arbitrary sequence of vectors ${u}_{i,j}$. Then we have

**Case ${k}_{0}=2$:**Now, after applying two time convolution, we obtain

#### 3.1. Subdivision Depth for Ternary Subdivision Surfaces

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

#### 3.2. Application for Bivariate Case

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Ju, C.; Solomonik, E. Derivation and analysis of fast bilinear algorithms for convolution. arXiv
**2019**, arXiv:1910.13367v1. [Google Scholar] - Aslam, M.; Mustafa, G.; Ghaffar, A. (2n-1)-point ternary approximating and interpolating subdivision schemes. J. Appl. Math.
**2011**, 11, 1–12. [Google Scholar] [CrossRef][Green Version] - Faheem, K.; Mustafa, G. Ternary six-point interpolating subdivision scheme. Lobachevskii J. Math.
**2008**, 29, 153–163. [Google Scholar] [CrossRef] - Ghaffar, A.; Mustafa, G. The family of even-point ternary approximating schemes. ISRN Appl. Math.
**2012**, 12, 1–14. [Google Scholar] [CrossRef][Green Version] - Mustafa, G.; Ghaffar, A.; Khan, F. The odd-point ternary approximating schemes. Am. J. Comput. Math.
**2011**, 11, 111–118. [Google Scholar] [CrossRef][Green Version] - Hassan, M.F.; Ivrissimitzis, I.P.; Dodgson, N.A.; Sabin, M.A. An interpolating 4-point C
^{2}ternary stationary subdivision scheme. Comput. Aided Geom. Des.**2002**, 19, 1–18. [Google Scholar] [CrossRef] - Siddiqi, S.S.; Rehan, K. Symmetric ternary interpolating C
^{1}subdivision scheme. Int. Math. Forum**2012**, 7, 2269–2277. [Google Scholar] - Kwan, P.K.; Byung, G.L.; Gang, J.Y. A ternary 4-point approximating subdivision scheme. Appl. Math. Comput.
**2007**, 190, 1563–1573. [Google Scholar] - Mustafa, G.; Ashraf, P. A new 6-point ternary interpolating subdivision scheme and its differentiability. J. Inf. Comput. Sci.
**2010**, 5, 199–210. [Google Scholar] [CrossRef][Green Version] - Mustafa, G.; Irum, J.; Bari, M. A new 5-point ternary interpolating subdivision scheme and its differentiability. ISRN Comput. Math.
**2012**, 12, 1–10. [Google Scholar] [CrossRef][Green Version] - Siddiqi, S.S.; Ahmed, N.; Rehan, K. Ternary 4-point interpolating scheme for curve sketching. Res. J. Appl. Sci. Eng. Technol.
**2013**, 6, 1556–1561. [Google Scholar] [CrossRef] - Siddiqi, S.S.; Younis, M. Construction of ternary approximating subdivision schemes. UPB Sci. Bull. Ser. A Appl. Math. Phys.
**2014**, 76, 71–78. [Google Scholar] - Peng, K.; Tan, J.; Li, Z.; Zhang, L. Fractal behavior of a ternary 4-point rational interpolation subdivision scheme. Math. Comput. Appl.
**2018**, 23, 65. [Google Scholar] [CrossRef][Green Version] - Song, Q.; Zheng, H.C.; Peng, G.H. A nonlinear ternary circle-preserving interpolatory subdivision scheme. Appl. Mech. Mater.
**2013**, 427, 2170–2173. [Google Scholar] [CrossRef] - Aslam, M. A family of 5-point nonlinear ternary interpolating subdivision schemes with C
^{2}smoothness. Math. Comput. Appl.**2018**, 23, 18. [Google Scholar] [CrossRef][Green Version] - Beccari, C.; Casciola, G.; Romani, L. An interpolating 4-point C
^{2}ternary non-stationary subdivision scheme with tension control. Comput. Aided Geom. Des.**2007**, 24, 210–219. [Google Scholar] [CrossRef] - Cai, Z. Convergence, error estimation and some properties of four-point interpolation subdivision scheme. Comput. Aided Geom. Des.
**1995**, 12, 459–468. [Google Scholar] - Conti, C.; Jetter, K. Concerning order of convergence for subdivision. Numer. Algorithms
**2004**, 36, 345–363. [Google Scholar] [CrossRef] - Huning, S.; Wallner, J. Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature. Adv. Comput. Math.
**2019**, 45, 1689–1709. [Google Scholar] [CrossRef][Green Version] - Mustafa, G.; Chen, F.; Deng, J. Estimating error bounds for binary subdivision curves/surfaces. J. Comput. Appl. Math.
**2006**, 193, 596–613. [Google Scholar] [CrossRef][Green Version] - Mustafa, G.; Deng, J. Estimating error bounds for ternary subdivision curve/surfaces. J. Comput. Math.
**2007**, 25, 473–484. [Google Scholar] - Mustafa, G.; Hashmi, S.; Noshi, N.A. Estimating error bounds for tensor product binary subdivision volumetric model. Int. J. Comput. Math.
**2006**, 83, 879–903. [Google Scholar] [CrossRef] - Mustafa, G.; Hashmi, S.; Faheem, K. Estimating error bounds for non-stationary binary subdivision curves/surfaces. J. Inf. Comput. Sci.
**2007**, 2, 179–190. [Google Scholar] - Moncayo, M.; Amat, S. Error bounds for a class of subdivision schemes based on the two-scale refinement equation. J. Comput. Appl. Math.
**2011**, 236, 265–278. [Google Scholar] [CrossRef][Green Version] - Deng, C.; Jin, W.; Li, Y.; Xu, H. A formula for estimating the deviation of a binary interpolatory subdivision curve from its polygon. Appl. Math. Comput.
**2017**, 304, 10–19. [Google Scholar] [CrossRef] - Mustafa, G.; Hashmi, S. Subdivision depth computation for n-ary subdivision curves/surfaces. Vis. Comput.
**2010**, 26, 841–851. [Google Scholar] [CrossRef][Green Version] - Mustafa, G. Subdivision depth computation for tensor product ternary volumetric model. In Proceedings of the 2009 International Conference on Scientific Computing, CSC 2009, Las Vegas, NV, USA, 13–16 July 2009; pp. 151–158. [Google Scholar]
- Mustafa, G.; Hashmi, S.; Faheem, K. Subdivision depth for triangular surfaces. Alex. Eng. J.
**2016**, 55, 1647–1653. [Google Scholar] [CrossRef][Green Version] - Shahzad, A.; Faheem, K.; Ghaffar, A.; Mustafa, G.; Nisar, K.S.; Baleanu, D. A novel numerical algorithm to estimate the subdivision depth of binary subdivision schemes. Symmetry
**2020**, 12, 66. [Google Scholar] [CrossRef][Green Version] - Hassan, M.F.; Dodgson, N.A. Ternary and 3-Point Univariate Subdivision Scheme; Cohen, A., Marrien, J.L., Schumaker, L.L., Eds.; Curve and Surface fitting: Sant-Malo 2002; Nashboro Press: Brentwood, UK, 2003; pp. 199–208. [Google Scholar]

**Figure 2.**Comparison between first and fifth convolutions. This shows error decreases with the increase of convolution. Of course, it decreases with the increase of subdivision depth. (

**a**) 2-point tensor product scheme [12]; (

**b**) 3-point tensor product scheme [30]; (

**c**) 4-point tensor product scheme [8]; (

**d**) 4-point tensor product scheme [6].

Scheme/ ${\mathit{D}}_{{\mathit{k}}_{0}}$ | ${\mathit{D}}_{1}=\mathit{\delta}$ | ${\mathit{D}}_{2}$ | ${\mathit{D}}_{3}$ | ${\mathit{D}}_{4}$ | ${\mathit{D}}_{5}$ |
---|---|---|---|---|---|

2-point scheme [12] | 0.333333 | 0.111111 | 0.037037 | 0.012346 | 0.004115 |

3-point scheme [30] | 0.555467 | 0.253017 | 0.11107 | 0.047856 | 0.020353 |

4-point scheme [8] | 0.441358 | 0.176393 | 0.067478 | 0.027136 | 0.010728 |

4-point scheme [6] | 0.444444 | 0.171682 | 0.070918 | 0.029278 | 0.012089 |

${\mathit{D}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $6.88\times {10}^{-5}$ | $2.83\times {10}^{-7}$ | $1.16\times {10}^{-9}$ | $4.79\times {10}^{-12}$ | $1.97\times {10}^{-14}$ | $8.12\times {10}^{-17}$ | $3.34\times {10}^{-19}$ |
---|---|---|---|---|---|---|---|

${D}_{1}=\delta $ | 5 | 10 | 15 | 20 | 25 | 30 | 35 |

${D}_{2}$ | 3 | 5 | 8 | 10 | 13 | 15 | 18 |

${D}_{3}$ | 2 | 3 | 5 | 7 | 8 | 10 | 12 |

${D}_{4}$ | 1 | 2 | 4 | 5 | 6 | 7 | 9 |

${D}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

${\mathit{D}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $1.38\times {10}^{-3}$ | $2.81\times {10}^{-5}$ | $5.73\times {10}^{-7}$ | $1.16\times {10}^{-8}$ | $2.37\times {10}^{-10}$ | $4.83\times {10}^{-12}$ | $9.84\times {10}^{-14}$ |
---|---|---|---|---|---|---|---|

${D}_{1}=\delta $ | 8 | 15 | 21 | 28 | 34 | 41 | 48 |

${D}_{2}$ | 3 | 6 | 9 | 12 | 14 | 17 | 20 |

${D}_{3}$ | 2 | 4 | 5 | 7 | 9 | 11 | 12 |

${D}_{4}$ | 1 | 3 | 4 | 5 | 6 | 8 | 9 |

${D}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

${\mathit{D}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $1.26\times {10}^{-3}$ | $1.35\times {10}^{-5}$ | $1.45\times {10}^{-7}$ | $1.56\times {10}^{-9}$ | $1.67\times {10}^{-11}$ | $1.79\times {10}^{-13}$ | $1.92\times {10}^{-15}$ |
---|---|---|---|---|---|---|---|

${D}_{1}=\delta $ | 6 | 12 | 17 | 23 | 28 | 34 | 40 |

${D}_{2}$ | 3 | 5 | 8 | 11 | 13 | 16 | 18 |

${D}_{3}$ | 2 | 3 | 5 | 7 | 8 | 10 | 12 |

${D}_{4}$ | 1 | 3 | 4 | 5 | 6 | 8 | 9 |

${D}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

${\mathit{D}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $1.22\times {10}^{-3}$ | $1.47\times {10}^{-5}$ | $1.78\times {10}^{-7}$ | $2.16\times {10}^{-9}$ | $2.61\times {10}^{-11}$ | $3.15\times {10}^{-13}$ | $3.81\times {10}^{-15}$ |
---|---|---|---|---|---|---|---|

${D}_{1}=\delta $ | 6 | 12 | 17 | 22 | 28 | 33 | 39 |

${D}_{2}$ | 3 | 5 | 8 | 10 | 13 | 15 | 18 |

${D}_{3}$ | 2 | 3 | 5 | 7 | 8 | 10 | 12 |

${D}_{4}$ | 1 | 2 | 4 | 5 | 6 | 7 | 9 |

${D}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Scheme/${\mathit{F}}_{{\mathit{k}}_{0}}{\mathit{G}}_{{\mathit{k}}_{0}}$ | ${\mathit{F}}_{1}{\mathit{G}}_{1}$ | ${\mathit{F}}_{2}{\mathit{G}}_{2}$ | ${\mathit{F}}_{3}{\mathit{G}}_{3}$ | ${\mathit{F}}_{4}{\mathit{G}}_{4}$ | ${\mathit{F}}_{5}{\mathit{G}}_{5}$ |
---|---|---|---|---|---|

2-point scheme [12] | 0.333333 | 0.111111 | 0.037037 | 0.012346 | 0.004115 |

3-point scheme [30] | 0.61716 | 0.304535 | 0.145678 | 0.068303 | 0.031622 |

4-point scheme [8] | 0.505401 | 0.233098 | 0.096089 | 0.040774 | 0.016958 |

4-point scheme [6] | 0.54321 | 0.228729 | 0.10226 | 0.047435 | 0.021475 |

${\mathit{F}}_{{\mathit{k}}_{0}}{\mathit{G}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $2.06\times {10}^{-3}$ | $8.5\times {10}^{-7}$ | $3.49\times {10}^{-9}$ | $1.43\times {10}^{-11}$ | $5.92\times {10}^{-14}$ | $2.43\times {10}^{-16}$ | $1.003\times {10}^{-18}$ |
---|---|---|---|---|---|---|---|

${F}_{1}{G}_{1}$ | 5 | 10 | 15 | 20 | 25 | 30 | 35 |

${F}_{2}{G}_{2}$ | 3 | 5 | 8 | 10 | 13 | 15 | 18 |

${F}_{3}{G}_{3}$ | 1 | 3 | 5 | 6 | 8 | 10 | 11 |

${F}_{4}{G}_{4}$ | 1 | 2 | 4 | 5 | 6 | 7 | 9 |

${F}_{5}{G}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

${\mathit{F}}_{{\mathit{k}}_{0}}{\mathit{G}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $7.25\times {10}^{-3}$ | $2.29\times {10}^{-4}$ | $7.25\times {10}^{-6}$ | $2.29\times {10}^{-7}$ | $7.25\times {10}^{-9}$ | $2.29\times {10}^{-10}$ | $7.25\times {10}^{-12}$ |
---|---|---|---|---|---|---|---|

${F}_{1}{G}_{1}$ | 9 | 16 | 23 | 31 | 38 | 45 | 52 |

${F}_{2}{G}_{2}$ | 3 | 6 | 9 | 12 | 15 | 18 | 21 |

${F}_{3}{G}_{3}$ | 2 | 4 | 5 | 7 | 9 | 11 | 13 |

${F}_{4}{G}_{4}$ | 1 | 3 | 4 | 5 | 6 | 8 | 9 |

${F}_{5}{G}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

${\mathit{F}}_{{\mathit{k}}_{0}}{\mathit{G}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $6.59\times {10}^{-3}$ | $1.11\times {10}^{-4}$ | $1.18\times {10}^{-6}$ | $3.21\times {10}^{-8}$ | $5.45\times {10}^{-10}$ | $9.24\times {10}^{-12}$ | $1.56\times {10}^{-13}$ |
---|---|---|---|---|---|---|---|

${F}_{1}{G}_{1}$ | 7 | 13 | 19 | 25 | 31 | 37 | 43 |

${F}_{2}{G}_{2}$ | 3 | 6 | 9 | 11 | 14 | 17 | 20 |

${F}_{3}{G}_{3}$ | 2 | 4 | 5 | 7 | 9 | 10 | 12 |

${F}_{4}{G}_{4}$ | 1 | 3 | 4 | 5 | 6 | 8 | 9 |

${F}_{5}{G}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

${\mathit{F}}_{{\mathit{k}}_{0}}{\mathit{G}}_{{\mathit{k}}_{0}}$/$\mathit{\u03f5}$ | $7.31\times {10}^{-3}$ | $1.57\times {10}^{-4}$ | $3.37\times {10}^{-6}$ | $7.24\times {10}^{-8}$ | $1.55\times {10}^{-9}$ | $3.34\times {10}^{-11}$ | $7.17\times {10}^{-13}$ |
---|---|---|---|---|---|---|---|

${F}_{1}{G}_{1}$ | 8 | 14 | 20 | 26 | 33 | 39 | 45 |

${F}_{2}{G}_{2}$ | 3 | 5 | 8 | 11 | 13 | 16 | 18 |

${F}_{3}{G}_{3}$ | 2 | 3 | 5 | 7 | 8 | 10 | 12 |

${F}_{4}{G}_{4}$ | 1 | 3 | 4 | 5 | 6 | 8 | 9 |

${F}_{5}{G}_{5}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

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**MDPI and ACS Style**

Khan, F.; Mustafa, G.; Shahzad, A.; Baleanu, D.; M. Al-Qurashi, M. A Computational Method for Subdivision Depth of Ternary Schemes. *Mathematics* **2020**, *8*, 817.
https://doi.org/10.3390/math8050817

**AMA Style**

Khan F, Mustafa G, Shahzad A, Baleanu D, M. Al-Qurashi M. A Computational Method for Subdivision Depth of Ternary Schemes. *Mathematics*. 2020; 8(5):817.
https://doi.org/10.3390/math8050817

**Chicago/Turabian Style**

Khan, Faheem, Ghulam Mustafa, Aamir Shahzad, Dumitru Baleanu, and Maysaa M. Al-Qurashi. 2020. "A Computational Method for Subdivision Depth of Ternary Schemes" *Mathematics* 8, no. 5: 817.
https://doi.org/10.3390/math8050817