Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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HARMONIC CURVATURE FUNCTIONS OF SOME SPECIAL CURVES IN GALILEAN 3-SPACE

  • Yilmaz, Beyhan (Department of Mathematics, Kahramanmaras Sutcu Imam University) ;
  • Metin, Seyma (Department of Mathematics, Ankara University) ;
  • Gok, Ismail (Department of Mathematics, Ankara University) ;
  • Yayli, Yusuf (Department of Mathematics, Ankara University)
  • Received : 2018.10.02
  • Accepted : 2019.02.23
  • Published : 2019.06.25
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Abstract

The aim of the paper is to characterize some curves with the help of their harmonic curvature functions. First of all, we have defined harmonic curvature function of an arbitrary curve and have re-determined the position vectors of helices in terms of their harmonic curvature functions in Galilean 3-space. Then, we have investigated the relation between rectifying curves and Salkowski (or anti-Salkowski) curves in Galilean 3-space. Furthermore, the position vectors of them are obtained via the serial approach of the curves. Finally, we have given some illustrated examples of helices and rectifying curves with some assumptions.

Keywords

HNSHCY_2019_v41n2_301_f0001.png 이미지

FIGURE 1. The curve α (s) in terms of its harmonic curvature function. To see that different examples about this subject, you can look Ali's study in [3].

HNSHCY_2019_v41n2_301_f0002.png 이미지

FIGURE 2. Rectifying Salkowski Curve γ (φ)

HNSHCY_2019_v41n2_301_f0003.png 이미지

FIGURE 3. Rectifying Anti Salkowski Curve α (ψ)

HNSHCY_2019_v41n2_301_f0004.png 이미지

FIGURE 4. κ(s) = 1 and τ(s) = s

HNSHCY_2019_v41n2_301_f0005.png 이미지

FIGURE 5. ${\kappa}(s)=\frac{1}{s}$ and τ(s) = 1

References

  1. A. O. Ogrenmis, H. Oztekin, and M. Ergut, Bertrand curves in Galilean space and their characterizations, Kragujevac Journal of Mathematics, 32 (2009), 139-147.
  2. A. O. Ogrenmis, M. Ergut, and M. Bektas, On the helices in the Galilean space $G_3$ , Iranian Journal of Science and Technology Transaction A: Science, 31(2) (2007), 177-181.
  3. A. T. Ali, Position vectors of curves in the Galilean space $G_3$ , Matematiqki vesnik, 64(3) (2012), 200-210.
  4. B. A. Rozenfel'd, Non-Euclidean spaces, Moscow, 1969.
  5. B. J. Pavkovic, The general solution of the Frenet system of differential equations for curves in the Galilean space $G_3$ , Rad HAZU Math, 450 (1990), 123-128.
  6. B. J. Pavkovic, I. Kamenarovic, The equiform differential geometry of curves in the Galilean space $G_3$ , Glasnik mat., 22(42) (1987), 449-457.
  7. B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110 (2003), 147-152. https://doi.org/10.1080/00029890.2003.11919949
  8. E. Salkowski, Zur transformation von raumkurven, Mathematische Annalen., 66(4) (1909), 517-557. https://doi.org/10.1007/BF01450047
  9. H. Oztekin, Normal and rectifying curves in Galilean Space $G_3$, Proceedings of IAM, 5(1) (2016), 98-109.
  10. I. M. Yaglom, A Simple non-Euclidean geometry and its physical basis, New York, 1979.
  11. J. Monterde, Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Computer Aided Geometric Design, 26 (2009), 271-278. https://doi.org/10.1016/j.cagd.2008.10.002
  12. M. O. Yun, L. S. Ye, A Curve Satisfying ${\tau}/{\kappa}=s$ with constant ${\kappa}>0$, American Journal of Undergraduate Research, 12(2) (2015), 57-62.
  13. O. Roschel, Die Geometrie Des Galileischen Raumes, Berichte der Math.-Stat. Sektionim Forschumgszentrum Graz, Ber., 256 (1986), 1-20.
  14. R. Penrose, Structure of space-time C. M. DeWitt (ed.), J. A. Wheeler (ed.), Batelle Rencontres 1967 Lectures in Math. Physics, Benjamin, (1968), 121-235.
  15. Z. Erjavec, On Generalization of Helices in the Galilean and the Pseudo-Galilean Space, Journal of Mathematics Research, 6 (3) (2014), 39-50. https://doi.org/10.5539/jmr.v6n3p39