AIMS Mathematics, 2020, 5(3): 2027-2039. doi: 10.3934/math.2020134.

Research article Special Issues

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Evolution of space curves and the special ruled surfaces with modified orthogonal frame

Department of Mathematics, Faculty of Arts and Sciences, Sakarya University, Sakarya, 54187 Turkey

Special Issues: 8th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2019)

In this study, we investigate the evolution of space curves and the special ruled surfaces with modified orthogonal frame. Firstly, we recall the relations between the Serret-Frenet frame and the modified orthogonal frame. Then we investigate the evolution of space curves relative to these relations. Moreover, we determine the first and second fundamental forms, the Gaussian and mean curvatures of the tangent, normal, and binormal ruled surfaces according to the modified frame. In these regards, we obtain the characterizations of the minimal and developable ruled surfaces based on modified orthogonal frame.
  Figure/Table
  Supplementary
  Article Metrics

Keywords modified orthogonal frame; evolution of curve; smoke ring equations; ruled surfaces; Gaussian curvature; mean curvature

Citation: Kemal Eren, Hidayet Huda Kosal. Evolution of space curves and the special ruled surfaces with modified orthogonal frame. AIMS Mathematics, 2020, 5(3): 2027-2039. doi: 10.3934/math.2020134

References

  • 1. M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, 1976.
  • 2. A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC, New York, 1998.
  • 3. K. Sprott, B. Ravani, Kinematic generation of ruled surfaces, Adv. Comput. Math., 17 (2002), 115-133.    
  • 4. B. van Sosin, M. Barton, G. Elber, Accessibility for line-cutting in freeform surfaces, Comput. Aided Design, 114 (2019), 202-214.    
  • 5. H. Y. Chen, H. Pottmann, Approximation by ruled surfaces, J. Comput. Appl. Math., 102 (1999), 143-156.    
  • 6. S. Flöry, H. Pottmann, Ruled Surfaces for Rationalization and Design in Architecture, In: LIFE in:formation. On Responsive Information and Variations in Architecture, 2010, 103-109.
  • 7. K. Nakayama, M. Wadati, Motion of curves in the plane, J. Phys. Soc. JPN, 62 (1993), 473-479.    
  • 8. K. Nakayama, H. Segur, M. Wadati, Integrability and the motion of curves, Phys. Rev. Lett., 69 (1992), 2603-2606.    
  • 9. N. H. Abdel-All, R. A. Hussien, T. Youssef, Evolution of curves via the velocities of the moving frame, J. Math. Comput. Sci., 2 (2012), 1170-1185.
  • 10. A. Doliwa, P. Santini, An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A, 185 (1994), 373-384.    
  • 11. D. Y. Kwon, F. C. Park, Evolution of inelastic plane curves, Appl. Math. Lett., 12 (1999), 115-119.
  • 12. H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 52 (1972), 477-485.
  • 13. G. L. Lamb, Solitons on moving space curves, J. Math. Phys., 18 (1977), 1654-1661.    
  • 14. C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations, In: Geometry and Modern Applications in Soliton Theory, Cambridge University Press, 2002.
  • 15. H. N. Abd-Ellah, Evolution of translation surfaces in Euclidean 3-space, Appl. Math. Inform. Sci., 9 (2015), 661-668.
  • 16. R. A. Hussien, S. G. Mohamed, Generated surfaces via inextensible flows of curves in R3, J. Appl. Math., 1 (2016), 6178961.
  • 17. D. Y. Kwon, F. C. Park, Inextensible flows of curves and developable surfaces, Appl. Math. Lett., 18 (2005), 1156-1162.    
  • 18. O. G. Yıldız, S. Ersoy, M. A. Masal, Note on inextensible flows of curves on oriented surface, Cubo, 16 (2014), 11-19.    
  • 19. K. Nakayama, M. Wadati, The motion of surfaces, J. Phys. Soc. JPN, 62 (1993), 1895-1901.    
  • 20. R. A. Hussien, T. Youssef, Evolution of special ruled surfaces via the evolution of their directrices in Euclidean 3-Space E3, Appl. Math. Inform. Sci., 10 (2016), 1949-1956.
  • 21. T. Sasai, The fundamental theorem of analytic space curves And apparent singularities of Fuchsian differential equations, Tohoku Math. J., 36 (1984), 17-24.    
  • 22. B. Bükcü, M. K. Karacan, On the modified orthogonal frame with curvature and torsion in 3- Space, Math. Sci. Appl. E-Notes, 4 (2016), 184-188.
  • 23. M. S. Lone, E. S Hasan, M. K. Karacan, et al., Mannheim curves with modified orthogonal frame in Euclidean 3-Space, Turk. J. Math., 43 (2019), 648-663.    
  • 24. M. S. Lone, E. S Hasan, M. K. Karacan, et al., On some curves with modified orthogonal frame in Euclidean 3-Space, Iran. J. Sci. Technol. Trans. A. Sci., 43 (2019), 1905-1916.    
  • 25. B. Bükcü, M. K. Karacan, Spherical curves with modified orthogonal frame, J. New Res. Sci., 10 (2016), 60-68.
  • 26. R. A. Hord, Torsion at an inflection point of a space curve, Am. Math. Mon., 79 (1972), 371-374.    
  • 27. T. Sasai, Geometry of analytic space curves with singularities and regular singularities of differential equations, Funkcial. Ekvac., 30 (1987), 283-303.

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved