Research Article
BibTex RIS Cite

Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$

Year 2022, , 8 - 15, 30.04.2022
https://doi.org/10.33187/jmsm.1007857

Abstract

In this present paper, rectifying curves are re-characterized in a shorter and simpler way using harmonic curvatures and some relations between rectifying curves and focal curves are found in terms of their harmonic curvature functions in $n-$dimensional Euclidean space. Then, a rectifying Salkowski curve, which is the focal curve of a given space curve is investigated. Finally, some figures related to the theory are given in the case $n=3$.

References

  • [1] D. S. Kim, H. S. Chung, K. H. Cho, Space curves satisfying t=k = as+b, Honam Mathematical J., 15(1) (1993), 5-9.
  • [2] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110(2) (2003), 147-152.
  • [3] B. Y. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sinica, 33(2) (2005), 77-90.
  • [4] B. Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3􀀀space, Tamkang J. Math., 48(2) (2017), 209-214.
  • [5] S. Cambie, W. Goemans, I. Van Den Bussche, Rectifying curves in the n􀀀dimensional Euclidean space, Turk. J. Math., 40 (2016), 210-223.
  • [6] K. ˙Ilarslan, E. Nesovic, M. Petrovic-Torgasev, Some characterizations of rectifying curves in the Minkowski 3􀀀space, Novi. Sad. J. Math., 33(2) (2003), 23-32.
  • [7] K. ˙Ilarslan, E. Nesovic, On rectifying curves as centrodes and extremal curves in the Minkowski in the Minkowski 3􀀀space, Novi. Sad. J. Math., 37(1) (2007), 53-64.
  • [8] E. O¨ zdamar, H. H. Hacisalihog˘lu, A characterization of inclined curves in Euclidean n􀀀space, Communication de la facult´e des sciences de L’Universit´e d’Ankara, 24 (1975), 15-22.
  • [9] C¸ . Camci, L. Kula, K. ˙Ilarslan, H. H. Hacisaliho˘glu, On the eplicit characterization of curves on a (n􀀀1)􀀀sphere in Sn, Int. Electron. J. Geom., 6(2) (2013), 63-69.
  • [10] F. Ertem Kaya, Y. Yaylı, H. H. Hacisaliho˘glu, Harmonic curvature of a strip in E3, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 59(2) (2010), 1-14.
  • [11] C¸ . Camcı, K. ˙Ilarslan,L. Kula, H. H. Hacisaliho˘glu, Harmonic curvatures and generalized helices in En, Chaos Solit. Fractals., 40 (2007), 1-7.
  • [12] ˙I. G¨ok, C¸ . Camcı, H. H. Hacisaliho˘glu, Vn􀀀slant helices in Euclidean n􀀀space En, Math. Commun., 14(2) (2009), 317-329.
  • [13] ˙I. G¨ok, C¸ . Camcı, H. H. Hacisaliho˘glu, Vn􀀀slant helices in Minkowski n􀀀space E1n , Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 58(1) (2009), 29-38.
  • [14] N. Ekmekc¸i, H. H. Hacisaliho˘glu and K. ˙Ilarslan, Harmonic curvatures in Lorentzian Space Bull. Malaysian Math. Sc. Soc. 23 (2000), 173-179.
  • [15] M. K¨ulahcı, M. Bektas¸ and M. Erg¨ut, On Harmonic curvatures of null curves of the AW(k)-type in Lorentzian Space Naturforsch, 63(a) (2008), 248-252.)
  • [16] E. ˙Iyig¨un, K. Arslan, On Harmonic curvatures of curves in Lorentzian N􀀀Space Commun. Fac. Sci. Univ. Ank. Series A1, 54(1) (2005), 29-34.
  • [17] R. Uribe-Vargas, On vertices, focal curvatures and differential geometry of space curves, Bull. Brazilian Math. Soc., 36 (2005), 285-307.
  • [18] G. O¨ ztu¨rk, K. Arslan, On focal curves in Euclidean n􀀀space Rn, Novi Sad J Math., 46(1) (2016), 35-44.
  • [19] G. O¨ zturk, B. Bulca, B. Bayram, K. Arslan, Focal representation of k􀀀slant Helices in Em+1, Acta Univ. Sapientiae, Mathematica, 7(2) (2015), 200-209.
  • [20] H. Gluk, Higher curvatures of curves in Euclidean space, Amer. Math. Month., 73 (1966), 699-704.
  • [21] H. H. Hacisaliho˘glu, Diferensiyel Geometri, Ankara University, Faculty of Science Press, 1993.
  • [22] E. Salkowski, Zur Transformation von Raumkurven, Mathematische Annalen, 4(66)(1909), 517–557.
  • [23] J. Monterde, Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geom. Des., 26 (2009), 271–278.
  • [24] A. S¸ enol, E. Zıplar, Y. Yaylı, ˙I. G¨ok, A new approach on helices in Euclidean n􀀀space, Math. Commun., 18 (2013), 241-256.
Year 2022, , 8 - 15, 30.04.2022
https://doi.org/10.33187/jmsm.1007857

Abstract

References

  • [1] D. S. Kim, H. S. Chung, K. H. Cho, Space curves satisfying t=k = as+b, Honam Mathematical J., 15(1) (1993), 5-9.
  • [2] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110(2) (2003), 147-152.
  • [3] B. Y. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sinica, 33(2) (2005), 77-90.
  • [4] B. Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3􀀀space, Tamkang J. Math., 48(2) (2017), 209-214.
  • [5] S. Cambie, W. Goemans, I. Van Den Bussche, Rectifying curves in the n􀀀dimensional Euclidean space, Turk. J. Math., 40 (2016), 210-223.
  • [6] K. ˙Ilarslan, E. Nesovic, M. Petrovic-Torgasev, Some characterizations of rectifying curves in the Minkowski 3􀀀space, Novi. Sad. J. Math., 33(2) (2003), 23-32.
  • [7] K. ˙Ilarslan, E. Nesovic, On rectifying curves as centrodes and extremal curves in the Minkowski in the Minkowski 3􀀀space, Novi. Sad. J. Math., 37(1) (2007), 53-64.
  • [8] E. O¨ zdamar, H. H. Hacisalihog˘lu, A characterization of inclined curves in Euclidean n􀀀space, Communication de la facult´e des sciences de L’Universit´e d’Ankara, 24 (1975), 15-22.
  • [9] C¸ . Camci, L. Kula, K. ˙Ilarslan, H. H. Hacisaliho˘glu, On the eplicit characterization of curves on a (n􀀀1)􀀀sphere in Sn, Int. Electron. J. Geom., 6(2) (2013), 63-69.
  • [10] F. Ertem Kaya, Y. Yaylı, H. H. Hacisaliho˘glu, Harmonic curvature of a strip in E3, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 59(2) (2010), 1-14.
  • [11] C¸ . Camcı, K. ˙Ilarslan,L. Kula, H. H. Hacisaliho˘glu, Harmonic curvatures and generalized helices in En, Chaos Solit. Fractals., 40 (2007), 1-7.
  • [12] ˙I. G¨ok, C¸ . Camcı, H. H. Hacisaliho˘glu, Vn􀀀slant helices in Euclidean n􀀀space En, Math. Commun., 14(2) (2009), 317-329.
  • [13] ˙I. G¨ok, C¸ . Camcı, H. H. Hacisaliho˘glu, Vn􀀀slant helices in Minkowski n􀀀space E1n , Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 58(1) (2009), 29-38.
  • [14] N. Ekmekc¸i, H. H. Hacisaliho˘glu and K. ˙Ilarslan, Harmonic curvatures in Lorentzian Space Bull. Malaysian Math. Sc. Soc. 23 (2000), 173-179.
  • [15] M. K¨ulahcı, M. Bektas¸ and M. Erg¨ut, On Harmonic curvatures of null curves of the AW(k)-type in Lorentzian Space Naturforsch, 63(a) (2008), 248-252.)
  • [16] E. ˙Iyig¨un, K. Arslan, On Harmonic curvatures of curves in Lorentzian N􀀀Space Commun. Fac. Sci. Univ. Ank. Series A1, 54(1) (2005), 29-34.
  • [17] R. Uribe-Vargas, On vertices, focal curvatures and differential geometry of space curves, Bull. Brazilian Math. Soc., 36 (2005), 285-307.
  • [18] G. O¨ ztu¨rk, K. Arslan, On focal curves in Euclidean n􀀀space Rn, Novi Sad J Math., 46(1) (2016), 35-44.
  • [19] G. O¨ zturk, B. Bulca, B. Bayram, K. Arslan, Focal representation of k􀀀slant Helices in Em+1, Acta Univ. Sapientiae, Mathematica, 7(2) (2015), 200-209.
  • [20] H. Gluk, Higher curvatures of curves in Euclidean space, Amer. Math. Month., 73 (1966), 699-704.
  • [21] H. H. Hacisaliho˘glu, Diferensiyel Geometri, Ankara University, Faculty of Science Press, 1993.
  • [22] E. Salkowski, Zur Transformation von Raumkurven, Mathematische Annalen, 4(66)(1909), 517–557.
  • [23] J. Monterde, Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geom. Des., 26 (2009), 271–278.
  • [24] A. S¸ enol, E. Zıplar, Y. Yaylı, ˙I. G¨ok, A new approach on helices in Euclidean n􀀀space, Math. Commun., 18 (2013), 241-256.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Beyhan Yılmaz 0000-0002-5091-3487

İsmail Gök 0000-0001-8407-133X

Yusuf Yaylı 0000-0003-4398-3855

Publication Date April 30, 2022
Submission Date October 10, 2021
Acceptance Date January 24, 2022
Published in Issue Year 2022

Cite

APA Yılmaz, B., Gök, İ., & Yaylı, Y. (2022). Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$. Journal of Mathematical Sciences and Modelling, 5(1), 8-15. https://doi.org/10.33187/jmsm.1007857
AMA Yılmaz B, Gök İ, Yaylı Y. Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$. Journal of Mathematical Sciences and Modelling. April 2022;5(1):8-15. doi:10.33187/jmsm.1007857
Chicago Yılmaz, Beyhan, İsmail Gök, and Yusuf Yaylı. “Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$”. Journal of Mathematical Sciences and Modelling 5, no. 1 (April 2022): 8-15. https://doi.org/10.33187/jmsm.1007857.
EndNote Yılmaz B, Gök İ, Yaylı Y (April 1, 2022) Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$. Journal of Mathematical Sciences and Modelling 5 1 8–15.
IEEE B. Yılmaz, İ. Gök, and Y. Yaylı, “Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$”, Journal of Mathematical Sciences and Modelling, vol. 5, no. 1, pp. 8–15, 2022, doi: 10.33187/jmsm.1007857.
ISNAD Yılmaz, Beyhan et al. “Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$”. Journal of Mathematical Sciences and Modelling 5/1 (April 2022), 8-15. https://doi.org/10.33187/jmsm.1007857.
JAMA Yılmaz B, Gök İ, Yaylı Y. Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$. Journal of Mathematical Sciences and Modelling. 2022;5:8–15.
MLA Yılmaz, Beyhan et al. “Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$”. Journal of Mathematical Sciences and Modelling, vol. 5, no. 1, 2022, pp. 8-15, doi:10.33187/jmsm.1007857.
Vancouver Yılmaz B, Gök İ, Yaylı Y. Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$. Journal of Mathematical Sciences and Modelling. 2022;5(1):8-15.

29237    Journal of Mathematical Sciences and Modelling 29238

                   29233

Creative Commons License The published articles in JMSM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.