Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

SURFACES GENERATED VIA THE EVOLUTION OF SPHERICAL IMAGE OF A SPACE CURVE

  • Soliman, M.A. (Department of Mathematics Assiut University) ;
  • H.Abdel-All, Nassar (Department of Mathematics Assiut University) ;
  • Hussien, R.A. (Department of Mathematics Assiut University) ;
  • Shaker, Taha Youssef (Department of Mathematics Assiut University)
  • Received : 2018.03.19
  • Accepted : 2018.08.21
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we linked the motion of spherical images with the motion of their curves. Surfaces generated by the evolution of spherical image of a space curve are constructed. Also geometric proprieties of these surfaces are obtained.

Keywords

References

  1. P. Pelce, Dynamics of Curved Fronts, Academic Press, New York, (1988).
  2. R.C. Brower et al., Geometrical models of interface evolution, Phys. Rev. A. 30 (1984), 3161-3174. https://doi.org/10.1103/PhysRevA.30.3161
  3. R. E. Goldstein and D. M. Petrich, The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), 3203-3206. https://doi.org/10.1103/PhysRevLett.67.3203
  4. K. Nakayama, H. Segur and M. Wadati, Integrability and the motion of curve, Phys. Rev. Lett. 69 (1992), 2603-2606. https://doi.org/10.1103/PhysRevLett.69.2603
  5. K. Nakayama, J. Hoppe and M. Wadati, On the level-set formulation of geometrical models, J. Phys. So. Japan. 64 (1995), 403-406. https://doi.org/10.1143/JPSJ.64.403
  6. Takeya Tsurumi et al., Motion of curves specified by accelerations, Physics Letters A. 224 (1997), 253-263. https://doi.org/10.1016/S0375-9601(96)00834-1
  7. D.Y. Kwon and F.C. Park, Evolution of inelastic plane curves, Appl. Math. Lett. 12 (1999), 115-119.
  8. D.Y. Kwon, F.C. Park and D.P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 18 (2005), 1156-1162. https://doi.org/10.1016/j.aml.2005.02.004
  9. Ahmad T. Ali and M.Turgut, Position vector of a timelike slant helix in Minkowski 3-Space, J. Math. Anal. Appl. 365 (2010), 559-569. https://doi.org/10.1016/j.jmaa.2009.11.026
  10. Ahmad T. Ali, Position vectors of slant helices in Euclidean space, J. Egypt. Math. Soc. 20 (2012) 1-6. https://doi.org/10.1016/j.joems.2011.12.005
  11. Ahmad T. Ali, New special curves and their spherical indicatrices, Global J. Adv. Res. Class. Mod. Geom. 1 (2012), 28-38.
  12. Murat Babaarslan et al., A note on Bertrand curves and constant slope surfaces according to Darboux frame, J. Adv. Math. Stud. 5 (2012) 87-96.
  13. S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. 28 (2004), 153-163.
  14. S. Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, J Geom. 74 (2002), 97-109. https://doi.org/10.1007/PL00012543
  15. L. Kula and Y. Yayli, On slant helix and its spherical indicatrix, Appl. Math. Comput. 169 (2005), 600-607.
  16. L. Kula, N. Ekmekci, Y. Yayh. and K. Ilarslan, Characterizations of slant helices in Euclidean 3-space, Turk J Math. 34 (2010) ,261-273.
  17. S. Yilmaz and M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl. 371 (2010) ,764-776. https://doi.org/10.1016/j.jmaa.2010.06.012
  18. E. Turhan and T. Korpynar, New Approach for binormal spherical image in terms of inextensible flow in $E^3$, Prespacetime Journal. 4 (2013), 342-355.
  19. E. Turhan and T. Korpynar, Time evolution equation for surfaces generated via binormail spherical image in terms of inextensible flow, J. Dyn. Syst. Geom. Theor. 12 (2014), 145-157.
  20. L. P. Eisenhart, A treatise on the differential geometry of curves and surfaces, New York, Dover, (1960).
  21. S. K. Chung, A study on the spherical indicatrix of a space curve in $E^3$, J. Korea Soc. Math. Educ. 20 (1982), 23-26.
  22. C. Rogers and W. K. Schief, Backlund and Darboux transformations geometry and modern application in soliton theory, Cambridge University press, Cambridge, (2002).