# Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Mathematical Model

## 4. Closed Form Complementary Solution

## 5. Continuity between Adjacent Surface Patches

#### 5.1. Continuity in Parametric Direction U

#### 5.2. Continuity in Parametric Direction V

## 6. Experiments and Application

#### 6.1. Creation of Single Surface

#### 6.2. Creation of Complicated Objects

#### 6.3. One Application of C2 Curve Network for Face Modeling

## 7. Discussion and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**(

**a**–

**c**) illustrate that our approach can connect surface patches with up to C2 continuities in both u and v parametric directions. (

**d**) shows the comparison between C1 and C2 continuity—our approach can achieve better continuity between patches.

**Figure 9.**Global comparison between the face model (red golden) created by the curve network with the proposed method and the original polygon face model (silver).

**Figure 10.**(

**a**) shows the patch surface generated by default shape control values, (

**b**) shows changing the values of shape control paremeters will generate different surfaces.

**Figure 12.**One face curve network template which can be used to represent and reconstruct different face shapes by our algorithm.

**Figure 13.**Reconstruct the surface of different Action Units from few curves by the proposed approach, and compare the realism between reconstructed surfaces and polygon surface.

**Figure 14.**GUI for showing the curve network structure and use default parameters to generate initialized surfaces.

Models | Male Face | Flower | Petal | Leaf | Dog | Plane |
---|---|---|---|---|---|---|

Polygon Verts | 4081 | 37,584 | 6594 | 1538 | 148,450 | 19,042 |

Curve Variables | 1467 | 6462 | 1132 | 298 | 24,066 | 4526 |

Data Size Comparison | 36% | 17% | 17% | 19% | 16% | 23% |

Proportion of Data Compression | 64% | 83% | 83% | 81% | 84% | 77% |

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## Share and Cite

**MDPI and ACS Style**

Bian, S.; Maguire, G.; Kokke, W.; You, L.; Zhang, J.J.
Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation. *Symmetry* **2020**, *12*, 38.
https://doi.org/10.3390/sym12010038

**AMA Style**

Bian S, Maguire G, Kokke W, You L, Zhang JJ.
Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation. *Symmetry*. 2020; 12(1):38.
https://doi.org/10.3390/sym12010038

**Chicago/Turabian Style**

Bian, Shaojun, Greg Maguire, Willem Kokke, Lihua You, and Jian J. Zhang.
2020. "Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation" *Symmetry* 12, no. 1: 38.
https://doi.org/10.3390/sym12010038