Smarandache-Based Ruled Surfaces with the Darboux Vector According to Frenet Frame in E3

Smarandache-Based Ruled Surfaces with the Darboux Vector According to Frenet Frame in E3

The paper introduces a new kind of special ruled surface. The base of each ruled surface is taken to be one of the Smarandache curves of a given curve according to Frenet frame, and the generator (ruling) is chosen to be the correspond- ing unit Darboux vector. The characteristics of these newly defined ruled surfaces are investigated by means of first and second fundamental forms and their corre- sponding curvatures. An example is provided by considering both the helix curve and the Viviani’s curve.

___

  • [1] P. do-Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliff, 1976.
  • [2] A. Gray E. Abbena, S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapman and Hall/CRC, New York, 2017.
  • [3] H. H. Hacısaliho ̆glu, Differential Geometry II, Ankara University Press, Ankara, 2000.
  • [4] D. J. Struik, Lectures on classical differential geometry, Addison-Wesley Publishing Company, 1961.
  • [5] M. Juza, Ligne De Striction Sur Unegeneralisation a Plusierurs Dimensions D’une Surface Regle, Czechoslovak Mathematical Journal 12 (1962) 243–250.
  • [6] G. Y. S ̧ent ̈urk, S. Yuce, Characteristic Properties of Ruled Surface with Darboux Frame in E3, Kuwait Journal of Science 42 (2) (2015) 14–33.
  • [7] Y. Tun ̧cer, Ruled Surfaces with the Bishop Frame in Euclidean 3 Space, General Mathematics Notes 26 (2015) 74–83.
  • [8] M. Masal, A. Z. Azak, Ruled Surfaces according to Bishop Frame in the Euclidean 3-Space, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 89 (2019) 415–424.
  • [9] R. L. Bishop There is More Than One Way to Frame a Curve, The American Mathematical Monthly 82 (1975) 246–251.
  • [10] S. Ouarab, A. O. Chahdi, M. Izid, Ruled Surfaces with Alternative Moving Frame in Euclidean 3-Space, International Journal of Mathematical Sciences and Engineering Applications 12 (2018) 43–58.
  • [11] S. Ouarab, A. O. Chahdi, Some Characteristic Properties of Ruled Surface with Frenet Frame of an Arbitrary Non-Cylindrical Ruled Surface in Euclidean 3-Space, International Journal of Applied Physics and Mathematics 10 (1) (2020) 16–24.
  • [12] M. Turgut, S. Yılmaz, Smarandache Curves in Minkowski Spacetime, International Journal of Mathematical Combinatorics 3 (2008) 51–55.
  • [13] A. T. Ali, Special Smarandache Curves in the Euclidean Space, International Journal of Mathe- matical Combinatorics 2 (2010) 30–36.
  • [14] S. Ouarab, Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E3, Abstract and Applied Analysis Hindawi 2021 Article ID: 5526536.
  • [15] S. Ouarab, Smarandache Ruled Surfaces according to Darboux Frame in E3, Journal of Mathe- matics 2021 Article ID: 9912624.
  • [16] S. Ouarab, NC-Smarandache Ruled Surface and NW-Smarandache Ruled Surface according to Alternative Moving Frame in E3, Journal of Mathematics 2021 Article ID: 9951434.
  • [17] ̈O. Bekta ̧s, S. Y ̈uce, Special Smarandache Curves According to Darboux Frame in E3, Romanian Journal of Mathematics and Computer Science 3 (2013) 48–59.
  • [18] A. Berk, A Structural Basis for Surface Discretization of Free Form Structures: Integration of Geometry, Materials and Fabrication, PhD Dissertation, Michigan University (2012) Ann Arbor, USA.
  • [19] M. C ̧ etin, H. Kocayi ̆git, On the Quaternionic Smarandache Curves in Euclidean 3-Space, Inter- national Journal of Contemporary Mathematical Sciences 8 (3) (2013) 139–150.
  • [20] H. Pottmann, A. Asperl, M. Hofer, A. Killian, Architectural Geometry, Bentley Institute Press, Exton, 2007.
  • [21] S. S ̧enyurt, S. Sivas, An Application of Smarandache Curve, Ordu University Journal of Science and Tecnology 3 (1) (2013) 46–60.
  • [22] J. Stillwell, Mathematics and Its History, Undergraduate Texts in Mathematics, Springer, New York, 2010.
  • [23] K. Ta ̧sk ̈opr ̈u, M. Tosun, Smarandache Curves on S2, Boletim da Sociedade Paranaense de Matem- atica 32 (1) (2014) 51–59.