# A Hermite Surface Triangle Modeling Method Considering High-Precision Fitting of 3D Printing Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{0}continuous surface by using a triangular rational quadratic Bezier surface to approximate a cubic linear interpolation function profile. The construction of Bezier surfaces requires control point information, which is harder to obtain directly when performing model fitting. NURBS and B-sample surfaces using surface approximation control meshes all have a similar problem to Bezier surfaces in that it is difficult to construct a direct mathematical relationship between the surface model and the original model.

## 2. Hermite Surface Characterization

#### 2.1. Definition of Hermite Curve in AMF

#### 2.2. Determine Feature Triangle and Mapping Relationship

## 3. Hermite Surface Triangle Model

## 4. Error Calculation and Evaluation Method

^{2}) of the sampled points can be described as:

## 5. Numerical Cases

**Case**

**1.**

**Case**

**2.**

## 6. Conclusions

**a**. Considering the different characteristics of the model, we will study an adaptive control method of

**a**that can satisfy high-precision fitting. In addition, based on the constructed Hermite surface model, we will study the adaptive layering technique of the surface model and surface path planning method considering the layered slicing and path planning errors. This will provide vital support to further enhance efficient and high-precision manufacturing of 3D printing.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Definition of AMF surface triangles: (

**a**) tangent vector of the vertex; (

**b**) normal vectors of the vertices.

**Figure 3.**Curved triangle parameters: (

**a**) parameters of each side; (

**b**) curved triangle; (

**c**) mapping of parameter fields.

**Figure 9.**Sampling point acquisition for basketball model: (

**a**) CAD model; (

**b**) grid division; (

**c**) sampling data points.

**Figure 11.**The vertical line intersects the curved triangle: (

**a**) the projection point (Z-direction); (

**b**) location of projection point.

**Figure 12.**Sampling point acquisition for rabbit model: (

**a**) solid model; (

**b**) grid division; (

**c**) sampling data points.

**Figure 14.**Sampling point acquisition for turbine model: (

**a**) CAD model; (

**b**) grid division; (

**c**) sampling data points.

Number | Models | Number of Facets | Z-Direction | Y-Direction | X-Direction | |||
---|---|---|---|---|---|---|---|---|

Mean Deviation (mm) | Variance (mm^{2}) | Mean Deviation (mm) | Variance (mm^{2}) | Mean Deviation (mm) | Variance (mm^{2}) | |||

1 | Hermite | 16,862 | 0.29 | 4.15 | 0.89 | 16.20 | 0.64 | 25.06 |

STL | 13.17 | 5561 | 3.47 | 59.78 | 2.31 | 102.10 | ||

2 | Hermite | 18,519 | 0.27 | 3.92 | 1.77 | 160.76 | 0.79 | 38.81 |

STL | 22.19 | 215,450 | 3.12 | 53.73 | 3.50 | 174.12 | ||

3 | Hermite | 22,820 | 0.50 | 16.88 | 0.92 | 23.84 | 0.43 | 9.66 |

STL | 2.06 | 57.23 | 3.16 | 57.51 | 2.10 | 112.78 | ||

4 | Hermite | 25,323 | 0.26 | 3.70 | 0.60 | 9.13 | 0.47 | 10.15 |

STL | 2.59 | 306.76 | 2.64 | 40.47 | 1.81 | 67.52 | ||

5 | Hermite | 27,239 | 0.25 | 3.67 | 0.84 | 21.53 | 0.38 | 8.80 |

STL | 2.76 | 259.87 | 2.69 | 45.57 | 2.84 | 461.31 |

Hermite Model | STL Model | |
---|---|---|

Z-direction | ||

X-direction | ||

Y-direction |

Hermite Model | STL Model | |
---|---|---|

Z-direction | ||

X-direction | ||

Y-direction |

Number | Models | Number of Facets | Z-Direction | Y-Direction | X-Direction | |||
---|---|---|---|---|---|---|---|---|

Mean Deviation (mm) | Variance (mm^{2}) | Mean Deviation (mm) | Variance (mm^{2}) | Mean Deviation (mm) | Variance (mm^{2}) | |||

1 | Hermite | 16,862 | 0.18 | 0.23 | 0.32 | 1.43 | 0.15 | 0.91 |

STL | 0.40 | 2.33 | 1.39 | 3.21 | 0.28 | 1.80 | ||

2 | Hermite | 18,519 | 0.16 | 0.22 | 0.33 | 1.32 | 0.13 | 0.74 |

STL | 0.38 | 2.15 | 1.20 | 2.86 | 0.27 | 1.87 | ||

3 | Hermite | 22,820 | 0.10 | 0.49 | 0.23 | 0.71 | 0.02 | 0.04 |

STL | 0.33 | 2.01 | 1.44 | 4.04 | 0.21 | 1.32 | ||

4 | Hermite | 25,323 | 0.09 | 0.31 | 0.24 | 0.65 | 0.02 | 0.17 |

STL | 0.29 | 1.74 | 1.32 | 3.81 | 0.17 | 1.09 | ||

5 | Hermite | 27,239 | 0.08 | 0.29 | 0.28 | 1.07 | 0.03 | 0.05 |

STL | 0.29 | 1.73 | 1.20 | 3.22 | 0.16 | 0.84 |

Number | Models | Number of Facets | Z-Direction | X-Direction | Y-Direction | |||
---|---|---|---|---|---|---|---|---|

Mean Deviation (mm) | Variance (mm^{2}) | Mean Deviation (mm) | Variance (mm^{2}) | Mean Deviation (mm) | Variance (mm^{2}) | |||

1 | Hermit | 18,272 | 0.25 | 0.08 | 0.18 | 0.07 | 0.24 | 0.07 |

STL | 0.35 | 0.13 | 0.30 | 0.19 | 0.33 | 0.16 | ||

2 | Hermit | 20,540 | 0.26 | 0.08 | 0.12 | 0.02 | 0.25 | 0.07 |

STL | 0.38 | 0.13 | 0.24 | 0.12 | 0.36 | 0.13 | ||

3 | Hermit | 22,752 | 0.23 | 0.05 | 0.13 | 0.06 | 0.24 | 0.07 |

STL | 0.37 | 0.14 | 0.26 | 0.18 | 0.33 | 0.16 | ||

4 | Hermit | 24,336 | 0.20 | 0.07 | 0.12 | 0.08 | 0.23 | 0.08 |

STL | 0.35 | 0.12 | 0.25 | 0.12 | 0.33 | 0.19 | ||

5 | Hermit | 28,592 | 0.21 | 0.07 | 0.11 | 0.10 | 0.20 | 0.04 |

STL | 0.34 | 0.12 | 0.25 | 0.12 | 0.32 | 0.08 |

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**MDPI and ACS Style**

Lian, R.; Jing, S.; Chen, Y.; Fan, J.
A Hermite Surface Triangle Modeling Method Considering High-Precision Fitting of 3D Printing Models. *Axioms* **2023**, *12*, 370.
https://doi.org/10.3390/axioms12040370

**AMA Style**

Lian R, Jing S, Chen Y, Fan J.
A Hermite Surface Triangle Modeling Method Considering High-Precision Fitting of 3D Printing Models. *Axioms*. 2023; 12(4):370.
https://doi.org/10.3390/axioms12040370

**Chicago/Turabian Style**

Lian, Ruichao, Shikai Jing, Yang Chen, and Jiangxin Fan.
2023. "A Hermite Surface Triangle Modeling Method Considering High-Precision Fitting of 3D Printing Models" *Axioms* 12, no. 4: 370.
https://doi.org/10.3390/axioms12040370