Abstract
To tackle the problems in adjusting and controlling shapes of rotation surfaces, a new efficient method for quickly constructing generalized Bézier rotation surfaces with multiple shape parameters is proposed. Firstly, following the important idea of transfinite vectored rational interpolating function, the shape-adjustable generalized Bézier rotation surfaces are constructed using a generalized Bézier curve with multiple shape parameters. Secondly, the explicit function expression of the shape-adjustable generalized Bézier rotation surfaces is presented. The new rotation surfaces inherit the outstanding properties of the Bézier rotation surfaces, with a good performance on adjusting their local shapes by changing the shape parameters. Finally, some properties of the new rotation surfaces are discussed, and the influence rules of the shape parameters on the new rotation surfaces are studied. The modeling examples illustrate that the shape-adjustable generalized Bézier rotation surfaces provide a valuable way for the design of rotation surfaces.
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References
Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, New York (1997)
Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Academic Press, San Diego (2002)
Farin, G., Piper, B., Worsey, A.: The octant of a sphere as a non-degenerate triangular Bézier patch. Comput. Aided Geom. Des. 4(4), 329–332 (1987)
Wang, G.J.: A new method for representing the surface of revolution using rational B-splines in CAD. J. Softw. 1(4), 24–39 (1990)
Kang, B.S., Ma, X., Zhou, R.R.: On the representation of revolution surface and sphere by NURBS polynomial. J. Nanjing Univ. Aeronaut. Astronaut. 26(1), 80–87 (1994)
Zeng, T.J., Wang, W.M., Zhang, J.W.: Study on surfaces of revolution modeling with C-B-splines. J. Graph. 25(2), 104–108 (2004)
Ma, S.J., Liu, X.M.: Research of rotate surface of uniform T-B-spline. Comput. Eng. Des. 29(16), 4255–4256 (2008)
Ding, H., Zhu, L.M.: Geometric Theories and Methods for Digital Manufacturing of Complex Surfaces. Science Press, Beijing (2011)
Bourguignon, D., Cani, M.P., Drettakis, G.: Drawing for illustration and annotation in 3D. Comput. Graph. Forum 20(3), 114–122 (2001)
Dai, C.L., Ding, Y.L., Lu, X.: On generalized revolving surface based on metamorphose curve. Mech. Sci. Technol. 21(4), 537–539 (2002)
Rodrigues, A.B., Jorge, J.A.: Free form modeling with variational implicit surfaces. In: Proceedings of the 12th Encontro Português de Computacao Gráfica, pp.17–26. Porto, Portugal (2003)
Wong, K.Y.K., Mendonca, P.R.S., Cipolla, R.: Reconstruction of surfaces of revolution from single uncalibrated views. Image Vis. Comput. 22(10), 829–836 (2004)
Colombo, C., Bimbo, A.D., Pernici, F.: Metric 3D reconstruction and texture acquisition of surfaces of revolution from a single uncalibrated view. IEEE Trans. Pattern Anal. 27(1), 99–114 (2005)
Wu, Y.H., Wang, G.H., Wu, F.C., Hu, Z.Y.: Euclidean reconstruction of a circular truncated cone only from its uncalibrated contours. Image Vis. Comput. 24(8), 810–818 (2006)
Li, X.J., Liu, H., He, G., Liao, W.: Generation and shape adjustment of revolution surface based on stream curve. Mech. Sci. Technol. 27(3), 326–329 (2008)
Han, L., Raffaele, D.A.: Rotation surface modeling technique by cubic B-spline free drawing. J. Chin. Comput. Syst. 30(7), 1141–1144 (2009)
Gong, X.P., Li, M.Z., Lu, Q.P., Peng, Z.Q.: Continuous forming for rotary surface based on multi-point adjusting principle. Opt. Precis. Eng. 20(1), 117–123 (2012)
Wang, Y., Fang, M.: Non-homogeneous subdivision method of revolution surfaces based on generalized B-spline. J. Hangzhou Dianzi Univ. 33(2), 25–28 (2013)
Lionel, G., Hichem, B., Sebti, F.: Dupin cyclide blends between non-natural quadrics of revolution and concrete shape modeling applications. Comput. Graph. 42(1), 31–41 (2014)
Tan, J.Q.: Continued Fractions Theory and its Application. Science Press, Beijing (2007)
Gu, C.Q.: Bivariate Thiele-type matrix-valued rational interpolants. J. Comput. Appl. Math. 80(1), 71–82 (1997)
Tan, J.Q., Tang, S.: Bivariate composite vector valued interpolation. Math. Comput. 69(232), 1521–1532 (2000)
Zhu, X.L.: A new method for constructing circular arc. J. Hefei Univ. Technol. 25(2), 269–272 (2002)
Kim, S.H., Ahn, Y.J.: An approximation of circular arcs by quartic Bézier curves. Comput. Aided Des. 39(6), 490–493 (2007)
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Hu, G., Wei, G. & Wu, J. Shape-Adjustable Generalized Bézier Rotation Surfaces with Multiple Shape Parameters. Results Math 72, 1281–1313 (2017). https://doi.org/10.1007/s00025-017-0659-7
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DOI: https://doi.org/10.1007/s00025-017-0659-7