Evolution of Generalized Space Curve as a Function of Its Local Geometry

Affiliation(s)

Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt.

Mathematics Department, Girls College of Science, University of Dammam, Dammam, KSA.

Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt.

Mathematics Department, Girls College of Science, University of Dammam, Dammam, KSA.

ABSTRACT

Kinematics of
moving generalized curves in a *n*-dimensional Euclidean space is formulated in
terms of intrinsic geometries. The evolution equations of the orthonormal frame
and higher curvatures are obtained. The integrability conditions for the
evolutions are given. Finally, applications in *R*^{2} are
given and plotted.

Cite this paper

Abdel-All, N. , Mohamed, S. and Al-Dossary, M. (2014) Evolution of Generalized Space Curve as a Function of Its Local Geometry.*Applied Mathematics*, **5**, 2381-2392. doi: 10.4236/am.2014.515230.

Abdel-All, N. , Mohamed, S. and Al-Dossary, M. (2014) Evolution of Generalized Space Curve as a Function of Its Local Geometry.

References

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http://dx.doi.org/10.1006/ciun.1993.1042

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[8] Kwon, D.Y. and Park, F.C. (1999) Evolution of Inelastic Plane Curves. Applied Mathematics Letters, 12, 115-119.

http://dx.doi.org/10.1016/S0893-9659(99)00088-9

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http://dx.doi.org/10.1016/j.aml.2005.02.004

[10] Latifi, D. and Razavi, A. (2008) Inextensible Flows of Curves in Minkowskian Space. Advanced Studies in Theoretical Physics, 2, 761-768.

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[13] Goldstein, R.E. and Petrich, D.M. (1991) The Korteweg-de Vries Hierarchy as Dynamics of Closed Curves in the Plane. Physical Review Letters, 67, 3203-3206.

http://dx.doi.org/10.1103/PhysRevLett.67.3203

[14] Nakayama, K., Segur, H. and Wadati, M. (1992) Integrability and the Motion of Curves. Physical Review Letters, 69, 2603-2606. http://dx.doi.org/10.1103/PhysRevLett.69.2603

[15] Li, Y.Y., Qu, C.Z. and Shu, S. (2010) Integrable Motions of Curves in Projective Geometries. Journal of Geometry and Physics, 60, 972-985.

http://dx.doi.org/10.1016/j.geomphys.2010.03.001

[16] Ögrenmis, A.O. and Yeneroglu, M. (2010) Inextensible Curves in the Galilean Space. IJPS, 5, 1424-1427.

[17] Do Carmo, M.P. (1976) Differential Geometry of Curves and Surfaces. Englewood Cliffs, New Jersey.

[18] Nakayama, K. and Wadati, M. (1993) Motion of Curves in the Plane. Journal of the Physical Society of Japan, 62, 473-479. http://dx.doi.org/10.1143/JPSJ.62.473

[1] Lu, H.Q., Todhunter, J.S. and Sze, T.W. (1993) Congruence Conditions for Nonplanar Developable Surfaces and Their Application to Surface Recognition. CVGIP, Image Understanding, 58, 265-285.

http://dx.doi.org/10.1006/ciun.1993.1042

[2] Kass, M., Witkin, A. and Terzopoulos, D. (1987) Snakes: Active Contour Models. Proceedings of the 1st International Conference on Computer Vision (ICCV’87), London, June 1987, 259-268.

[3] Desbrun, M. and Cani, M.P. (1998) Active Implicit Surface for Animation. Proceedings of the Graphics Interface 1998 Conference, Vancouver, 18-20 June 1998.

[4] Unger, D.J. (1991) Developable Surfaces in Elastoplastic Fracture Mechanics. International Journal of Fracture, 50, 33-38. http://dx.doi.org/10.1007/BF00032160

[5] Chirikjian, G.S. and Burdick, J.W. (1990) Kinematics of Hyper-Redundant Manipulators. Proceeding of the ASME Mechanisms Conference, Vol. 25, Chicago, 16-19 September 1990, 391-396.

[6] Gage, M. and Hamilton, R.S. (1986) The Heat Equation Shrinking Convex Plane Curves. Journal of Differential Geometry, 23, 69-96.

[7] Grayson, M.A. (1987) The Heat Equation Shrinks Embedded Plane Curves to Round Points. Journal of Differential Geometry, 26, 223-370.

[8] Kwon, D.Y. and Park, F.C. (1999) Evolution of Inelastic Plane Curves. Applied Mathematics Letters, 12, 115-119.

http://dx.doi.org/10.1016/S0893-9659(99)00088-9

[9] Kwon, D.Y., Park, F.C. and Chi, D.P. (2005) Inextensible Flows of Curves and Developable Surfaces. Applied Mathematics Letters, 18, 1156-1162.

http://dx.doi.org/10.1016/j.aml.2005.02.004

[10] Latifi, D. and Razavi, A. (2008) Inextensible Flows of Curves in Minkowskian Space. Advanced Studies in Theoretical Physics, 2, 761-768.

[11] Bobenko, A., Sullivan, J., Schröder, P. and Ziegler, G. (2008) Discrete Differential Geometry. Oberwolfach Seminars, Vol. 38, X+341 p.

[12] Rogers, C. and Schief, W.K. (2002) Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge.

[13] Goldstein, R.E. and Petrich, D.M. (1991) The Korteweg-de Vries Hierarchy as Dynamics of Closed Curves in the Plane. Physical Review Letters, 67, 3203-3206.

http://dx.doi.org/10.1103/PhysRevLett.67.3203

[14] Nakayama, K., Segur, H. and Wadati, M. (1992) Integrability and the Motion of Curves. Physical Review Letters, 69, 2603-2606. http://dx.doi.org/10.1103/PhysRevLett.69.2603

[15] Li, Y.Y., Qu, C.Z. and Shu, S. (2010) Integrable Motions of Curves in Projective Geometries. Journal of Geometry and Physics, 60, 972-985.

http://dx.doi.org/10.1016/j.geomphys.2010.03.001

[16] Ögrenmis, A.O. and Yeneroglu, M. (2010) Inextensible Curves in the Galilean Space. IJPS, 5, 1424-1427.

[17] Do Carmo, M.P. (1976) Differential Geometry of Curves and Surfaces. Englewood Cliffs, New Jersey.

[18] Nakayama, K. and Wadati, M. (1993) Motion of Curves in the Plane. Journal of the Physical Society of Japan, 62, 473-479. http://dx.doi.org/10.1143/JPSJ.62.473