The Extension of Geodesics

Abstract:

We introduced an extension of traditional geodesic. Local neighbourhood of smooth curve on manifold shows good propositions which make it possible to construct a geodesic mapping from the neighbourhood to a rectangular district of UV-plane. Length of subsurface extended from smooth curve is defined according to the parameterization induced by geodesic mapping. And extended geodesic is the curve with minimal length of subsurface it extends. We also proposed a constrained mass-spring based approach to solve extended geodesic on discrete mesh. Experimental result shows that it is a high precision approximation for problems of measurement where the path can not be reduced as a single curve.

• Periodical: Applied Mechanics and Materials (Volumes 333-335)
• Pages: 95-104
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.333-335.95
• Online since: July 2013
• Authors: Meng Jian Li, Wei Dong Geng
• Keywords: Extended Subsurface, Extension of Geodesic, Geodesic Mapping, Geodesics

References

[1] Maheshwari, A. and S. Wuhrer, Geodesic Paths on 3D Surfaces: Survey and Open Problems. (2009).

[2] Dijkstra, E.W., A note on two problems in connexion with graphs. Numerische Mathematik, (1959).

[3] Mitchell, J.S.B., D.M. Mount, and C.H. Papadimitriou, The discrete geodesic problem. SIAM Journal on Computing, 1987. 16: p.647.

DOI: 10.1137/0216045

[4] Surazhsky, V., et al., Fast exact and approximate geodesics on meshes. ACM Transactions on Graphics (TOG), 2005. 24(3): pp.553-560.

DOI: 10.1145/1073204.1073228

[5] Hotz, I. and H. Hagen. Visualizing geodesics. in Visualization 2000. Proceedings. 2000. IEEE.

[6] Kasap, E., M. Yapici, and F.T. Akyildiz, A numerical study for computation of geodesic curves. Applied Mathematics and Computation, 2005. 171(2): pp.1206-1213.

DOI: 10.1016/j.amc.2005.01.109

[7] Chen, S.G., Geodesic-like curves on parametric surfaces. Computer Aided Geometric Design, 2010. 27(1): pp.106-117.

DOI: 10.1016/j.cagd.2009.10.001

[8] Seong, J.K., W.K. Jeong, and E. Cohen, Curvature-based anisotropic geodesic distance computation for parametric and implicit surfaces. The Visual Computer, 2009. 25(8): pp.743-755.

DOI: 10.1007/s00371-009-0362-0

[9] Pennec, X., Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. Journal of Mathematical Imaging and Vision, 2006. 25(1): pp.127-154.

DOI: 10.1007/s10851-006-6228-4

[10] Sethian, J.A., A fast marching level set method for monotonically advancing fronts. Proceedings of the National Academy of Sciences, 1996. 93(4): pp.1591-1595.

DOI: 10.1073/pnas.93.4.1591

[11] Klingenberg, W. and D. Hoffman, A course in differential geometry1978: Springer-Verlag. 178.

[12] Ravi Kumar, G., et al., Geodesic curve computations on surfaces. Computer Aided Geometric Design, 2003. 20(2): pp.119-133.

DOI: 10.1016/s0167-8396(03)00023-2

[13] Baraff, D. and A. Witkin, Large steps in cloth simulation, in Proceedings of the 25th annual conference on Computer graphics and interactive techniques1998, ACM. pp.43-54.

DOI: 10.1145/280814.280821

[14] Breen, D.E., D.H. House, and P.H. Getto, A physically-based particle model of woven cloth. The Visual Computer, 1992. 8(5): pp.264-277.

DOI: 10.1007/bf01897114

[15] Bischoff, S., T. Weyand, and L. Kobbelt, Snakes on triangle meshes. Bildverarbeitung für die Medizin 2005, 2005: pp.208-212.

DOI: 10.1007/3-540-26431-0_43