JAMP  Vol.6 No.8 , August 2018
Evolutions of the Ruled Surfaces via the Evolution of Their Directrix Using Quasi Frame along a Space Curve
ABSTRACT
In this paper, evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented. These evolutions of the ruled surfaces depend on the evolutions of their directrix using quasi frame along a space curve.
Cite this paper
Soliman, M. , Abdel-All, N. , Hussien, R. and Youssef, T. (2018) Evolutions of the Ruled Surfaces via the Evolution of Their Directrix Using Quasi Frame along a Space Curve. Journal of Applied Mathematics and Physics, 6, 1748-1756. doi: 10.4236/jamp.2018.68149.
References
[1]   Chirikjian, G. and Burdick, J. (1994) A Modal Approach to Hyper-Redundant Manipulator Kinematics. IEEE Transactions on Robotics and Automation, 10, 343-354.
https://doi.org/10.1109/70.294209

[2]   Brockett, R. (1991) Notes on Motion Control. Harvard University ES258 Class Notes. Harvard University, Cambridge.

[3]   Jayender, J. and Vosburgh, K.G. (2010) New Kinematic Metric for Quantifying Surgical Skill for Flexible Instrument Manipulation. Information Processing in Computer-Assisted Interventions, Springer, Berlin.

[4]   Shahab, K. and Uwe, Z. (2005) Control of Contour Formations of Autonomous Vehicles by General Curve Evolution Theory. Proceedings of the 2005 Australasian Conference on Robotics and Automation, Sydney, 5-7 December 2005, 1-8.
http://hdl.handle.net/1885/76694

[5]   Da Rios, L.S. (1906) Sul moto d’un liquido indefinito con un filetto vorticoso. Gordon and Breach, 22, 117-135.

[6]   Galdi, G.P. (1971) Motion of Distorted Vortex Rings. Journal of the Physical Society of Japan, 31, 591-599.
https://doi.org/10.1143/JPSJ.31.591

[7]   Lamb Jr., G.L. (1977) Solitons on Moving Space Curves. Journal of Mathematical Physics, 18, 1654-1661.
https://doi.org/10.1063/1.523453

[8]   Lakshmanan, M., Ruijgrok, Th.W. and Thompson, C.J. (1976) On the Dynamics of a Continuum Spin System. Physica A, 577, 577-590.
https://doi.org/10.1016/0378-4371(76)90106-0

[9]   Doliwa, A. and Santini, P. (1994) An Elementary Geometric Characterisation of the Integrable Motions of a Curve. Physics Letters A, 185, 373-384.
https://doi.org/10.1016/0375-9601(94)90170-8

[10]   Abd-Ellah, H.N. (2015) Evolution of Translation Surfaces in Euclidean 3-Space. Applied Mathematics & Information Sciences, 9, 661-668.

[11]   Hussien, R.A. and Mohamed, S.G. (2016) Generated Surfaces via Inextensible Flows of Curves in R3. Journal of Applied Mathematics, 2016, Article ID: 6178961.
https://doi.org/10.1155/2016/6178961

[12]   Kwon, D.Y. and Park, F.C. (1999) Evolution of Inelastic Plane Curves. Applied Mathematics Letters, 12, 115-119.
https://doi.org/10.1016/S0893-9659(99)00088-9

[13]   Kwon, D.Y. and Park, F.C. (2005) Inextensible Flows of Curves and Developable Surfaces. Applied Mathematics Letters, 18, 1156-1162.
https://doi.org/10.1016/j.aml.2005.02.004

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

[15]   Korpinar, T., Altay, G. and Turhan, E. (2011) New Inextensible Flows of Tangent Developable Surfaces in Euclidian 3-Space E3. Revista Notas de Matemática, 7, 172-176.

[16]   Mukherjee, R. and Balakrishnan, R. (2008) Moving Curves of the Sine-Gordon Equation: New Links. Physics Letters A, 372, 6347-6362.
https://doi.org/10.1016/j.physleta.2008.08.070

[17]   Eisenhart, L.P. (1909) A Treatise on the Differential Geometry of Curves and Surfaces. Dover Pub. Inc., New York.

[18]   Bishop, R.L. (1975) There Is More than One Way to Frame a Curve. The American Mathematical Monthly, 82, 246-251.
https://doi.org/10.1080/00029890.1975.11993807

[19]   Ylmaz, S. and Turgut, M. (2010) A New Version of Bishop Frame and an Application to Spherical Images. Journal of Mathematical Analysis and Applications, 371, 764-776.

[20]   Bloomenthal, J. (1990) Calculation of Reference Frames along a Space Curve. In: Glassner, A., Graphics Gems, Academic Press, Cambridge, 567-571.
https://doi.org/10.1016/B978-0-08-050753-8.50124-8

[21]   Baur, O. and Grafarend, E.W. (2005) Orbital Rotations of a Satellite. Case Study: GOCE. Artificial Satellites, 40, 87-107.

[22]   Dede, M., Ekici, C. and Tozak, H. (2015) Directional Tubular Surfaces. International Journal of Algebra, 9, 527-535.
https://doi.org/10.12988/ija.2015.51274

[23]   Lipschutz, M.M. (1969) Theory and Problems of Differential Geometry. Schaum’s Outline Series.

[24]   VassiliouIan, P.J. and Lisle, G. (2000) Geometric Approaches to Differential Equations. Cambridge University Press, Cambridge.

[25]   Rogers, C. and Schief, W.K. (2002) Backlund and Darboux Transformations Geometry and Modern Application in Soliton Theory. Cambridge University Press, Cambridge.
https://doi.org/10.1017/CBO9780511606359

[26]   Hussien, R.A. and Youssef, T. (2016) Evolution of Special Ruled Surfaces via the Evolution of Their Directrices in Euclidean 3-Space E3. Applied Mathematics & Information Sciences, 10, 1949-1956.
https://doi.org/10.18576/amis/100536

 
 
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