Computing Special Smarandache Curves According to Darboux Frame in Euclidean 4-Space

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M. Khalifa Saad
M. A. Abd-Rabo


In this paper, we study some special Smarandache curves and their differential geometric properties according to Darboux frame in Euclidean 4-space E4. Also, we compute some of these curves which lie fully on a hypersurface in E4. Moreover, we defray some computational examples in support our main results.

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  1. C. Ashbacher, Smarandache geometries, Smarandache Notions J. 8(1-3) (1997), 212-215.
  2. M. Khalifa Saad, Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space, J. Egypt. Math. Soc. 24 (2016), 416-423.
  3. M. Cetin, H. Kocayigit, On the quaternionic Smarandache curves in Euclidean 3- space, Int. J. Contemp. Math. Sci. 8(3) (2013), 139-150.
  4. M. Do Carmo, Differential Geometry of curves and surface, Englewood Cliffs, NJ, USA, Prentice Hall, 1976.
  5. H.S. Abdel-Aziz, M. Khalifa Saad, Smarandache curves of some special curves in the Galilean 3-space, Honam Math. J. 37(2) (2015), 253-264.
  6. H.S. Abdelaziz, M. Khalifa Saad, Computation of Smarandache curves according to Darboux frame in Minkowski 3-space, J. Egypt. Math. Soc. 25 (2017), 382-390.
  7. H.S. Abdelaziz, M. Khalifa Saad, Some geometric invariants of pseudo-spherical evolutes in the hyperbolic 3-space, Comput. Mater. Continua 57(3) (2018), 389-415.
  8. A.T. Ali, Special Smarandache curves in the Euclidean space, Int. J. Math. Comb. 2 (2010), 30-36.
  9. O. Bektas, S. Yuce, Smarandache curves according to Darboux frame in Euclidean space, Rom. J. Math. Comput. Sci. 3(1) (2013), 48-59.
  10. M. Cetin, Y. Tuncer, M. K. Karacan, Smarandache curves according to Bishop frame in Euclidean space, Gen. Math. Notes 20(2) (2014), 50-66.
  11. M. Turgut, S. Yilmaz, Smarandache curves in Minkowski space-time, Int. J. Math. Comb. 3 (2008), 51-55.
  12. K. Taskpru, M. Tosun, Smarandache curves according to Sabban frame on S 2 , Bol. Soc. Paran. Mat. 32(1) (2014), 51-59.
  13. O. Alessio, Differential geometry of intersection curves in R4 of three implicit surfaces, Comput. Aided Geom. Des. 26 (2009), 455-471.
  14. H.S. Abdel-Aziz, M. Khalifa Saad, A. A. Abdel-Salam, On Implicit Surfaces and Their Intersection Curve in Euclidean 4-Space, Houston J. Math. 40(2) (2014), 339-352.
  15. B. O'Neill, Elementary Differential Geometry, Burlington, MA, USA, Academic Press, 1966.
  16. M. Duldu, B. D ¨uld ¨ul, N. Kuruoˇglu and E. Ozdamar, Extension of the Darboux frame into Euclidean 4-space and its ¨ invariants, Turk. J. Math. 41 (2017), 1628-1639.