Numerical Treatment of Geodesic Differential Equations on Two Dimensional Surfaces

Author

Department of Mathematics, Mohaghegh Ardabili, Ardabil, P.O.Box 56199-11367, Iran

Abstract

This paper presents a brief instructions to nd geodesics equa-tions on two dimensional surfaces in R3. The resulting geodesic equations are solved numerically using Computer Program Matlab, the geodesics are dis-played through Figures.

Keywords


1. P.N. Azariadis and N.A. Aspragathos, Geodesic curvature preservation in surface flattening through constrained global optimization, Comput.-Aided Des., 33, 581-591 (2001).

2. P. Azariadis and N.A. Aspragathos, Design of plane developments of doubly curved surfaces,Comput. Aided Des., 29(10), 675-685 (1997).

3. R. Brond, D. Jeulin, P. Gateau, J. Jarrin and G. Serpe, Estimation of the transport properties of polymer composites by geodesic propagation, J. Microsc., 176, 167-177 (1994).

4. S. Bryson, Virtual spacetime: an environment for the visualization of curved spacetimes via geodesic ows, Technical Report, NASA NAS, Number RNR-92-009, March 1992.

5. V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Int. J. Comput. Vision, 22(1), 61-71 (1997).

6. L. Cohen and R. Kimmel, Global minimum for active contours models: a minimal path approach,Int. J. Comput. Vision, 24(1), 57-78 (1997).

7. G. Dahlquist and A.Bjorck, Numerical Methods. Prentice-Hall, Inc., Englewood Cli_s, NJ (1974).

8. M.P. Do Carmo, Di_erential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Clifis, NJ (1976).

9. I.D. Faux and M.J. Pratt, Computational Geometry for Design and Manufacturing, Ellis Horwood, England, (1979).

10. L. Grundig, L. Ekert and E. Moncrieff, Geodesic and Semi-geodesic Line Algorithms for Cutting Pattern Generation of Architectural Textile Structures, in: T.T. Lan, (Ed.), Proceedings of the Asia-Pacific Conference on Shell and Spatial Structures, Beijing (1996).

11. R.J. Haw, An application of geodesic curves to sail design, Comput. Graphics Forum, 4(2), 137-139 (1985).

12. R.J. Haw and R.J. Munchmeyer, Geodesic curves on patched polynomial surfaces, Comput. Graphics Forum, 2(4), 225-232 (1983).

13. R. Heikes and D.A. Randall, Numerical integration of the shallow-water equations of a twisted icosahedral grid. Part I: Basic design and results of tests, Mon. Weath. Rev., 123, 1862-1880 (1995).

14. R. Heikes and D.A. Randall, Numerical integration of the shallow-water equations of a twisted icosahedral grid. Part II: A detailed description of the grid and an analysis of numerical accuracy, Mon. Weath. Rev., 123, 1881-1887 (1995).

15. I. Hotz and H. Hagen, Visualizing geodesics, in: Proceedings IEEE visualization, Salt Lake City, UT, 311-318 (2000).

16. R. Kimmel, R. Malladi and N. Sochen, Images as embedded maps and minimal surfaces:movies, color, texture, and volumetric medical images, Int. J. Comput. Vision, 39(2), 111-129 (2000).

17. R. Kimmel, Intrinsic scale space for images on surfaces: the geodesic curvature ow, Graph.Models Image Process, 59(5), 365-372 (1997).

18. T. Lindeberg, Scale-space Theory in Computer Vision, Kluwer Academic, Dordrecht (1994).

19. N.M. Patrikalakis and L. Badris, O_sets of curves on rational B-spline surfaces, Eng. Comput., 5, 39-46 (1989).

20. D.J. Struik, Lectures on Classical Di_erential Geometry, Addison-Wesley, Cambridge, MA (1950).

21. D.L. Williamson, Integration of the barotropicvorticity equation on a spherical geodesic grid, Tellus, 20, 642-653 (1968).