Abstract
A second-order algorithm is presented to calculate the parallel projection of a parametric curve onto a parametric surface in this chapter. The essence of our approach is to transform the problem of computing parallel projection curve on the parametric surface into that of computing parametric projection curve in the two-dimensional parametric domain of the surface. First- and second-order differential geometric characteristics of the parametric projection curve in the parametric domain of the surface are firstly analyzed. A marching method based on second-order Taylor Approximation is formulated to calculate the parametric projection curve. A first-order correction technique is developed to depress the error caused by the truncated higher order terms in the marching method. Several examples are finally implemented to demonstrate the effectiveness of the proposed scheme. Experimental results indicate that both the computational efficiency and accuracy of the presented method have dominant performance as compared with the first-order differential equation method.
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References
Sederberg, T. W., Finnigan, G. T., Li, X., Lin, H. W., & Ipson, H. (2008). Watertight trimmed NURBS. ACM Transaction on Graphics, 27(3), 79:1–79:8.
Chuang, J. H., Lin, C. H., & Hwang, W. C. (1995). Variable-radius blending of parametric surfaces. The Visual Computer, 11(10), 513–525.
Tam, H.-Y., Law, H. W., & Xu, H.-Y. (2004). A geometric approach to the offsetting of profiles on the three-dimensional surfaces. Computer-Aided Design, 36(10), 887–902.
Xu, H.-Y., Tam, H.-Y., Fang, X., & Hu, L. (2009). Quart-parametric interpolations for intersecting paths. Computer-Aided Design, 41(6), 432–440.
Pegna, J., & Wolter, F. E. (1996). Surface curve design by orthogonal projection of space curves onto free-form surface. Journal of Mechanical Design, 118(1), 45–52.
Wang, X., Wei, W., & Zhang, W.-Z. (2010). Projecting curves onto free-form surfaces. International Journal of Computer Applications in Technology, 37(2), 153–159.
Wang, X., An, L. L., Zhou L. S., & Zhang, L. (2010). Constructing G2 continuous curve on freeform surface with normal projection. Chinese Journal of Aeronautics, 23(1), 137–144.
Xu, H.-Y., Fang, X., Hu, L., Xiao F., & Li, D. (2010). An algorithm for curve orthogonal projections onto implicit surfaces. Journal of Computer-Aided Design and Computer Graphics, 22(12), 2103–2110.
Xu, H.-Y., Fang, X., Tam, H.-Y., Wu, X., & Hu, L. (2012). A second-order algorithm for curve orthogonal projection onto parametric surface. International Journal of Computer Mathematics, 89(1), 98–111.
Yang, Y.-J., Zeng, W., Yang, C.-L., Meng, X.-X., Yong J.-H., & Deng, B. (2012). G1 continuous approximate curves on NURBS surfaces. Computer-Aided Design, 44(9), 824–834.
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Fang, X., Xu, HY. (2014). A Second-Order Algorithm for Curve Parallel Projection on Parametric Surfaces. In: Wong, W.E., Zhu, T. (eds) Computer Engineering and Networking. Lecture Notes in Electrical Engineering, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-319-01766-2_2
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DOI: https://doi.org/10.1007/978-3-319-01766-2_2
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