Fair curves are widely used in geometric design, modeling, and industrial manufacturing. This paper proposes a new curve fairing algorithm based on the smoothed target curvature normal. To accomplish this, the problem of constructing the target curve is converted into solving a second-order differential equation with boundary value constraints, which is then solved through the Rayleigh–Ritz method. The effectiveness of this approach is demonstrated through several illustrative examples.
| [1] |
Antoni J. The spectral kurtosis: a useful tool for characterising non-stationary signals. Mech. Syst. Signal Process., 2006, 20(2): 282-307.
|
| [2] |
Aszódi B, Czuczor S, Szirmay-Kalos L. NURBS fairing by knot vector optimization. J. WSCG, 2004, 12(1–3): 19-26
|
| [3] |
Dietz, U.: Fair surface reconstruction from point clouds. In: Proceedings of the International Conference on Mathematical Methods for Curves and Surfaces II Lillehammer, pp. 79–86 (1998)
|
| [4] |
Eck M, Hadenfeld J. Local Energy Fairing of B-spline Curves, 1995. Heidelberg, Springer.
|
| [5] |
Farin G. Curvature combs and curvature plots. Comput. Aided Des., 2016, 80: 6-8.
|
| [6] |
Farin G, Hoschek J, Kim M-S. Handbook of Computer Aided Geometric Design, 2002. Amsterdam, Elsevier
|
| [7] |
Forsgren A, Gill PE, Wright MH. Interior methods for nonlinear optimization. SIAM Rev., 2002, 44(4): 525-597.
|
| [8] |
Hadenfeld J. Local energy fairing of B-spline surfaces. Math. Methods Curves Surf., 1995, 10(129–147): 203-212
|
| [9] |
Hahmann, S., Konz, S.: Fairing bi-cubic B-spline surfaces using simulated annealing, 159–168 (1997)
|
| [10] |
Hahmann S, Konz S. Knot-removal surface fairing using search strategies. Comput. Aided Des., 1998, 30(2): 131-138.
|
| [11] |
Hua L, Huang N, Yi B, Zhao Y, Zhu L. Global toolpath smoothing for CNC machining based on B-spline approximation with tool tip position adjustment. Int. J. Adv. Manuf. Technol., 2023, 125(7): 3651-3665.
|
| [12] |
Jiang Y, Lin H, Huang W. Fairing-PIA: progressive-iterative approximation for fairing curve and surface generation. Vis. Comput., 2024, 40(3): 1467-1484.
|
| [13] |
Li W, Xu S, Zheng J, Zhao G. Target curvature driven fairing algorithm for planar cubic B-spline curves. Comput. Aided Geomet. Design, 2004, 21(5): 499-513.
|
| [14] |
Moreton, H.P., Séquin, C.H.: Minimum variation curves and surfaces for computer-aided geometric design. In: Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design, pp. 123–159. SIAM, Philadelphia, USA (1994)
|
| [15] |
Nishiyama Y, Nishimura Y, Sasaki T, Maekawa T. Surface faring using circular highlight lines. Comput.-Aided Design Appl., 2007, 4(1–4): 405-414.
|
| [16] |
Pigounakis KG, Sapidis NS, Kaklis PD. Fairing spatial B-spline curves. J. Ship Res., 1996, 40(04): 351-367.
|
| [17] |
Salvi, P., Suzuki, H., Várady, T.: Fast and local fairing of B-spline curves and surfaces. In: Advances in Geometric Modeling and Processing: 5th International Conference, GMP 2008, Hangzhou, China, Apr 23–25, 2008. Proceedings 5, pp. 155–163. Springer (2008)
|
| [18] |
Salvi, P., Várady, T.: Local fairing of freeform curves and surfaces. In: Proceedings of the 3rd Hungarian Graphics and Geometry Conference, Budapest, pp. 1493–1501 (2005)
|
| [19] |
Susanne C, Brenner L, Scott L. The Mathematical Theory of Finite Element Methods, 2008, 3, New York, Springer. 15
|
| [20] |
Tsuchie S, Yoshida N. $\alpha $-curves: extended log-aesthetic curves with variable shape parameter. Compute. Graph., 2024, 118: 60-70.
|
| [21] |
Yan Z, Schiller S, Wilensky G, Carr N, Schaefer S. $\kappa $-curves: Interpolation at local maximum curvature. ACM Trans. Graph. (TOG), 2017, 36(4): 1-7
|
| [22] |
Zhao, X.: A genetic algorithm-based method for smooth modeling of blade surfaces. In: New Materials. Machinery and Vehicle Engineering, vol. 37, pp. 160–165. IOS Press, Amsterdam (2023)
|
Funding
Open Research Fund Program of the State Key Laboratory of Hydroscience and Engineering(sklhse-KF-2025-D-01)
National Key R&D Plan of China(2024YFE0206600)
Fundamental Research Funds for the Central Universities(B240201091)
National Natural Science Foundation of China(12201292)
RIGHTS & PERMISSIONS
School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature
PDF
