#
3D Modelling with C^{2} Continuous PDE Surface Patches

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}continuity between adjacent surface patches is well-maintained. Since surface creation of the model is transformed into the generation of cross sectional curves and few undetermined constants are required to describe cross sectional curves accurately, the proposed approach can save manual operations, reduce information storage, and generate 3D models quickly.

## 1. Introduction

## 2. Related Work

## 3. Algorithm Overview

#### 3.1. Curve Fitting

#### 3.2. Creation of ${C}^{2}$ Continuous PDE Surfaces

#### 3.2.1. PDE Surface Patch 1 Creation

#### 3.2.2. Creation of PDE Surface Patch 2

#### 3.2.3. Creation of PDE Surface Patch 3

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Determination of the Undetermined Functions A

## Appendix B. Determination of the Undetermined Functions B

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**Figure 3.**Curve fitting example. The ground truth curves are shown in blue and fitted generalized elliptic curves are in red.

**Figure 4.**Surface creation example. The six curves ${\mathit{C}}_{4}$ − ${\mathit{C}}_{9}$ (red) are used to construct the PDE surface patch 1.

**Figure 6.**The cross section curves of the human body and the created human body model in front and side views using ${C}^{2}$ continues PDE method.

**Figure 7.**Surface shape generation from cross section curves by using the method proposed in this paper. (

**a**) a smooth vase model, (

**b**) a horse belly model, (

**c**) front leg and nose models of an elephant.

n | 1 | 3 | 5 | 7 | 10 |
---|---|---|---|---|---|

ErM | 0.065371 | 0.040636 | 0.010601 | 0.005312 | 0.002534 |

ErA | 0.044169 | 0.012367 | 0.007068 | 0.001767 | 0.000883 |

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## Share and Cite

**MDPI and ACS Style**

Fu, H.; Bian, S.; Li, O.; Macey, J.; Iglesias, A.; Chaudhry, E.; You, L.; Zhang, J.J.
3D Modelling with *C*^{2} Continuous PDE Surface Patches. *Mathematics* **2022**, *10*, 319.
https://doi.org/10.3390/math10030319

**AMA Style**

Fu H, Bian S, Li O, Macey J, Iglesias A, Chaudhry E, You L, Zhang JJ.
3D Modelling with *C*^{2} Continuous PDE Surface Patches. *Mathematics*. 2022; 10(3):319.
https://doi.org/10.3390/math10030319

**Chicago/Turabian Style**

Fu, Haibin, Shaojun Bian, Ouwen Li, Jon Macey, Andres Iglesias, Ehtzaz Chaudhry, Lihua You, and Jian Jun Zhang.
2022. "3D Modelling with *C*^{2} Continuous PDE Surface Patches" *Mathematics* 10, no. 3: 319.
https://doi.org/10.3390/math10030319