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

Year 2022, Volume: 5 Issue: 1, 8 - 15, 30.04.2022

Beyhan Yılmaz İsmail Gök Yusuf Yaylı

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 $n=3$.

Keywords

Focal curve, Harmonic curvature, Rectifying curve.

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.