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Journal of Zhejiang University SCIENCE A 2006 Vol.7 No.101 P.187~192

http://doi.org/10.1631/jzus.2006.AS0187


Numerical solution of geodesic through two given points on a simple surface


Author(s):  Wu Ming-Hua, Mo Guo-Liang, Yu Yi-Yue

Affiliation(s):  School of Computational Science, Zhejiang University City College, Hangzhou 310015, China; more

Corresponding email(s):   wmhua@zju.edu.cn, mogl@zucc.edu.cn

Key Words:  Geodesic, Filament winding, Functional of arc length


Wu Ming-Hua, Mo Guo-Liang, Yu Yi-Yue. Numerical solution of geodesic through two given points on a simple surface[J]. Journal of Zhejiang University Science A, 2006, 7(101): 187~192.

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PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2006.AS0187


Abstract: 
The algorithm for the approximate solution of a geodesic connecting two given points on a simple surface is discussed in this paper. It arises from practical demands of the filament winding technique. geodesic is the shortest path connecting two given points on a surface and it can also be regarded as the extremal curve of the arc length functional. The nonlinear equation system of the geodesic on some discrete points by means of the direct variation method is explored. By employing Newton’s iterative method, this nonlinear system is transformed into a linear one. And the approximate solution to the geodesic is obtained by solving the resultant linear system. This paper also proves that the iteration is convergent under certain circumstance. Moreover, the result is illustrated with three examples and an appropriate comparison between the analytical solution and the approximate solution to the geodesic is described on the cone surface.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

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[3] Hu, H.C., Hu, R.M., 1987. Variation. Architecture Publishing Company China, Beijing, p.7 (in Chinese).

[4] Leek, C., 1998. Filament winding breathes life into gas bottles. Reinforce Plastics, 9:52-53.

[5] Mazumdar, S.K., Hoa, S.V., 1995. Analytical models for low cost manufacturing of composite components by filament winding, Part I: Direct kinematics. Journal of Composite Materials, 29(11):1515-1541.

[6] Ghasemi Nejhad, M.N., Chandramouli, M.V., Yousefpour, A., 2001. Processing and Performance of continuous fiber ceramic composites by preceramic polymer pyrolysis: I—filament winding. Journal of Composite Materials, 35(24):2207-2237.

[7] Polini, W., Sorrentino, L., 2005. Winding trajectory and winding time in robotized filament winding of asymmetric shape parts. Journal of Composite Materials, 39(15): 1391-1411.

[8] Stoer, J., Bulirsch, R., 1980. Introduction to Numerical Analysis. Springer-Verlag, 247:542-544.

[9] Wu, M.H., Liang, Y.D., Yu, Y.Y., 2001. Stabilities of geodesics on torus. Appl. Math. J. Chinese Univ. Ser A, 16(4):481-485.

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