Fast High-Precision Bisection Feedback Search Algorithm and Its Application in Flattening the NURBS Curve
Abstract
:1. Introduction
1.1. Modeling and Deformation of the Ship Hull
1.2. Inversion Algorithms of NURBS Curves
1.3. Problems with the IR-BFS Inversion Algorithm
1.4. Research Objectives and Structure
2. Mathematical Background
2.1. NURBS Curve
2.2. IR-BFS Inversion Algorithm
2.3. Flattening Algorithm of the NURBS Curve
3. Framework of the Proposed Methodology
3.1. Overall Design of the FHP-BFS Algorithm
3.2. Flattening Algorithm Based on the FHP-BFS Algorithm
Algorithm 1: Flattening algorithm based on the fast high-precision bisection feedback search (FHP-BFS) algorithm. |
Input:—list of offsets table; —location points of flattening line segment ends; |
—degree of the interpolated NURBS curve. |
Output: — control point vector and knot vector of a flattened NURBS curve. |
1: function |
2: //interpolation operation of list |
3: // are parametric values of the endpoint of the flattening line segment |
4: //point inversion by the FHP-BFS algorithm |
5: |
6: // is the number of points projected successfully |
7: // is the knot refinement number |
8: |
9: while do |
10: |
11: //knot refinement algorithm |
12: Projection of the control point located on the same side of the flattening line segment |
13: update , s |
14: end while |
15: return |
16: end function |
4. Results
4.1. Comparison of Algorithms between FHP-BFS and IR-BFS
4.1.1. Validation of the Practical Effectiveness of the FHP-BFS Algorithm
4.1.2. Setting the Precision of the Threshold of the FHP-BFS Algorithm
4.2. Comparison with Other Algorithms
4.3. Evaluation of the Flattening Algorithm
5. Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
FHP-BFS | Fast high-precision bisection feedback search |
IR-BFS | Interval reformation and bisection feedback search |
NURBS | Non-uniform rational B-spline |
NR | Newton-Raphson |
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x (Station 4) | x (Station 14) | x (Station 32) | ||||||
---|---|---|---|---|---|---|---|---|
Index | y | z | Index | y | z | Index | y | z |
1 | 0.000 | 0.731 | 1 | 0.000 | 0.000 | 1 | 0.000 | 0.932 |
2 | 0.082 | 0.750 | 2 | 4.490 | 0.000 | 2 | 0.115 | 1.000 |
3 | 0.179 | 0.821 | 3 | 5.000 | 0.035 | 3 | 0.267 | 1.241 |
4 | 0.303 | 1.000 | 4 | 6.000 | 0.313 | … | … | … |
5 | 0.426 | 1.287 | 5 | 6.322 | 0.500 | 10 | 0.000 | 4.306 |
… | … | … | 6 | 6.541 | 0.710 | 11 | 0.000 | 7.306 |
23 | 6.191 | 6.953 | 7 | 6.732 | 1.000 | 12 | 0.164 | 7.371 |
24 | 6.359 | 7.610 | 8 | 6.850 | 1.319 | … | … | … |
25 | 6.531 | 8.630 | 9 | 6.900 | 1.719 | 14 | 2.000 | 8.877 |
26 | 6.638 | 9.520 | 10 | 6.900 | 15.00 | 15 | 2.505 | 9.284 |
27 | 6.638 | 15.00 | - | - | - | 16 | 2.505 | 15.00 |
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Zhu, K.; Shi, G.; Liu, J.; Shi, J. Fast High-Precision Bisection Feedback Search Algorithm and Its Application in Flattening the NURBS Curve. J. Mar. Sci. Eng. 2022, 10, 1851. https://doi.org/10.3390/jmse10121851
Zhu K, Shi G, Liu J, Shi J. Fast High-Precision Bisection Feedback Search Algorithm and Its Application in Flattening the NURBS Curve. Journal of Marine Science and Engineering. 2022; 10(12):1851. https://doi.org/10.3390/jmse10121851
Chicago/Turabian StyleZhu, Kaige, Guoyou Shi, Jiao Liu, and Jiahui Shi. 2022. "Fast High-Precision Bisection Feedback Search Algorithm and Its Application in Flattening the NURBS Curve" Journal of Marine Science and Engineering 10, no. 12: 1851. https://doi.org/10.3390/jmse10121851
APA StyleZhu, K., Shi, G., Liu, J., & Shi, J. (2022). Fast High-Precision Bisection Feedback Search Algorithm and Its Application in Flattening the NURBS Curve. Journal of Marine Science and Engineering, 10(12), 1851. https://doi.org/10.3390/jmse10121851