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A Sufficient Convexity Condition for Parametric Bézier Surface over Rectangle
National Engineering Research Center of Die & Mold CAD, Shanghai Jiaotong University, Shanghai, China
National Engineering Research Center of Die & Mold CAD, Shanghai Jiaotong University, Shanghai, China
- 1 National Engineering Research Center of Die & Mold CAD, Shanghai Jiaotong University, Shanghai, China
- 2 National Engineering Research Center of Die & Mold CAD, Shanghai Jiaotong University, Shanghai, China
American Journal of Computational Mathematics·Volume 10 (2020)·Pages 252–265·Published 9 April 2020·DOI10.4236/ajcm.2020.102013
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Abstract
Surface convexity is a key issue in computer aided geometric design, which is widely applied in geometric modeling field, such as physical models, industrial design, automatic manufacturing, etc. In this paper, a sufficient convexity condition of the parametric Bézier surface over rectangles is proposed, which is firstly considered as a sufficient convexity condition for the Bézier control grid. The condition is proved by De Casteljau surface subdivision arithmetic, in which the recursive expressions elaborate that the control grid eventually converges to the surface. At last, two examples for the modeling of interpolation-type surface are discussed, one of which is a general surface and the other is a degenerate surface.
KeywordsConvexity ConditionBézier SurfaceDe Casteljau ArithmeticInterpolation-Type Surface
- Böhm, W., Farin, G. and Kahmann, J. (1984) A Survey of Curve and Surface Methods in CAGD. Computer Aided Geometric Design, 1, 1-60. https://doi.org/10.1016/0167-8396(84)90003-7
- Faux, I.D. and Pratt, M.J. (1979) Computational Geometry for Design and Manufacture. Halsted Press, Canberra, 220-222.
- Han, X.A., Ma, Y.C. and Huang, X.L. (2008) A Novel Generalization of Bézier Curve and Surface. Journal of Computational and Applied Mathematics, 217, 180-193. https://doi.org/10.1016/j.cam.2007.06.027
- Fang, K., Zhu, X., Luo, W., et al. (2009) Convexity Conditions of Planar Parametric Curves and its Properties. In: Fang, K., Zhu, X. and Luo, W., Eds., The 2009 International Symposium on Computer Science and Computational Technology (ISCSCI 2009), Academy Publisher, Huangshan, 258.
- Loop, C.T. and DeRose, T.D. (1989) A Multisided Generalization of Bézier Surfaces. ACM Transactions on Graphics, 8, 204-234. https://doi.org/10.1145/77055.77059
- Wu, X.H. and Sun, Y.H. (2017) Surface Design of Automobile AB Column and Ceiling Based on Alias. International Conference on Robots & Intelligent System (ICRIS), Huai’an, 15-16 October 2017, 321-324. https://doi.org/10.1109/ICRIS.2017.87
- Dubey, R.K., Thakur, S.S. and Rana, S.S. (2015) Positivity and Convexity Preserving Interpolation Using Quartic Trigonometric Spline. International Journal of Advanced Research, 3, 902-907.
- Zhou, S., Li, H., Wang, J., et al. (2014) Investigation Acoustic Effect of the Convexity-Preserving Axial Flow Fan Based on Bézier Function. Computers & Fluids, 102, 85-93. https://doi.org/10.1016/j.compfluid.2014.06.019
- Fang, K., Shen, L.M., Xu, X.Y., et al. (2012) Convexity Conditions for Parameterized Surfaces. JSW, 7, 1219-1226.
- Franchi, A., Genna, F. and Paterlini, F. (1990) Research Note on Quasi-Convexity of the Yield Function and Its Relation to Drucker’s Postulate. International Journal of Plasticity, 6, 369-375. https://doi.org/10.1016/0749-6419(90)90008-3
- Hussain, M., Majid, A.A. and Hussain, M.Z. (2014) Convexity-Preserving Bernstein-Bézier Quartic Scheme. Egyptian Informatics Journal, 15, 89-95. https://doi.org/10.1016/j.eij.2014.04.001
- Liu, R. and Qiao, X.J. (2011) Discussion on the Convexity of Cubic Triangular Bézier Surfaces. 2011 International Conference on Electrical and Control Engineering (ICECE), Yichang, 16-18 September 2011, 5235-5237. https://doi.org/10.1109/ICECENG.2011.6057025