Abstract
Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc´ıa-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.
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References
G Albrecht. A note on Farin points for rational triangular Bézier patches, Comput Aided Geom Design, 1995, 12(5): 507–512.
H J Cai, G J Wang. Minimizing the maximal ratio of weights of rational Bézier curves and surfaces, Comput Aided Geom Design, 2010, 27(9): 746–755.
G Farin. Curves and surfaces for computer-aided geometric design: a practical guide, 5th ed, Morgan Kaufmann, San Francisco, 2002.
L D García-Puente, F Sottile, C G Zhu. Toric degenerations of Bézier patches, ACM Trans Graphics, 2011, 30(5), no 110.
R Krasauskas. Toric surface patches, Adv Comput Math, 2002, 17(1-2): 89–113.
L Piegl. A geometric investigation of the rational Bézier scheme of computer aided design, Comput Ind, 1986, 7(5): 401–410.
L Piegl, W Tiller. The NURBS Book, Chapter 1, Curves and Surface Basics, Springer Verlag, Berlin, 1997, 1–46.
I Selimovic. New bounds on the magnitude of the derivative of rational Bézier curves and surfaces, Comput Aided Geom Design, 2005, 22(4): 321–326.
B Sturmfels. Gröbner Bases and Convex Polytopes, Amer Math Soc, Provindence, 1996.
H Theisel. Using Farin points for rational Bézier surfaces, Comput Aided Geom Design, 1999, 16(8): 817–835.
X Y Zhao, C G Zhu. Injectivity conditions of rational Bézier surfaces, Comput Graphics, 2015, 51: 17–25.
C G Zhu. Degenerations of toric ideals and toric varieties, J Math Anal Appl, 2012, 386(2): 613–618.
C G Zhu, L Yang, X Y Zhao, B Y Xia. Self-intersections of rational Bézier curves, J Comput Aided Design Comput Graphics, 2013, 25(5): 738–744.
C G Zhu, X Y Zhao. Self-intersections of rational Bézier curves, Graph Models, 2014, 76(5): 312–320.
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Supported by the National Natural Science Foundation of China (11671068, 11271060, 11601064, 11290143), Fundamental Research of Civil Aircraft (MJ-F-2012-04), and the Fundamental Research Funds for the Central Universities (DUT16LK38).
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Zhang, Y., Zhu, Cg. & Guo, Qj. Degenerations of rational Bézier surface with weights in the form of exponential function. Appl. Math. J. Chin. Univ. 32, 164–182 (2017). https://doi.org/10.1007/s11766-017-3457-9
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DOI: https://doi.org/10.1007/s11766-017-3457-9