Some characterizations of non-null rectifying curves in dual Lorentzian 3-space $\mathbb{D}_{1}^{3}$

Roa Makki; Roa Makki

doi:10.3934/math.2021129

AIMS Mathematics

2021, Volume 6, Issue 3: 2114-2131. doi: 10.3934/math.2021129

Previous Article Next Article

Research article

Some characterizations of non-null rectifying curves in dual Lorentzian 3-space $\mathbb{D}_{1}^{3}$

Roa Makki ^,

Department of Mathematical Sciences, College of Applied Sciences, Umm Alqura University, Makkah, Saudi Arabia

Received: 15 August 2020 Accepted: 10 November 2020 Published: 09 December 2020
MSC : 53C50, 53C40

This paper gives several properties and characterization of non-null rectifying curves in dual Lorentzian 3-space $\mathbb{D% }_{1}^{3}$. In considering a causal character of a dual curve we give some parameterization of rectifying dual curves, and a dual differential equation of third order is constructed for every non-null dual curve. Then several well-known characterizations of spherical, normal and rectifying dual curves are consequences of this differential equation.
- Serret-Frenet formulae,
- rectifying dual curve,
- dual helices
Citation: Roa Makki. Some characterizations of non-null rectifying curves in dual Lorentzian 3-space $\mathbb{D}_{1}^{3}$[J]. AIMS Mathematics, 2021, 6(3): 2114-2131. doi: 10.3934/math.2021129

Related Papers:

Abstract

This paper gives several properties and characterization of non-null rectifying curves in dual Lorentzian 3-space $\mathbb{D% }_{1}^{3}$. In considering a causal character of a dual curve we give some parameterization of rectifying dual curves, and a dual differential equation of third order is constructed for every non-null dual curve. Then several well-known characterizations of spherical, normal and rectifying dual curves are consequences of this differential equation.

References

[1]	O. Bottema, B. Roth, Theoretical Kinematics, North-Holland Press, New York, 1979.
[2]	A. Karger, J. Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York, 1985.
[3]	H. Pottman, J. Wallner, Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg, 2001.
[4]	Y. Li, D. Pe, Evolutes of dual spherical curves for ruled surfaces, Math. Meth. Appl. Sci., 39 (2016), 3005–3015. doi: 10.1002/mma.3748
[5]	R. A. Abdel-Baky, Evolutes of hyperbolic dual spherical curve in dual Lorentzian 3-space, Int. J. Anal. Appl., 15 (2017), 114–124.
[6]	F. Tas, R. A. Abdel-Baky, On a spacelike line congruence which has the parameter ruled surfaces as principal ruled surfaces, Inter. Elect. J. Geom., 12 (2019), 135–143.
[7]	R. A. Abdel-Baky, Time-like line congruence in the dual Lorentzian 3-space D3 1, Inter. J. Geom. Methods Mod. Phys., 16 (2019), 201–211.
[8]	B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Mounthly, 110 (2003), 147–152. doi: 10.1080/00029890.2003.11919949
[9]	D. S. Kim, Chung, K. H. Cho, Space curves satisfying τ/κ = as + b, Honam Math. J., 15 (1993) 5–9.
[10]	S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk. J. Math., 28 (2004), 153–163.
[11]	M. Turgut, S. Yilmaz, Contributions to classical differential geometry of the curves in E3, Magna, 4 (2008), 5–9.
[12]	S. Deshmukh, B. Y. Chen, N. B. Turki, A differential equations for Frenet curves in Euclidean 3-space and its applications, Rom. J. Math. Comput. Sci., 8 (2018), 1–6.
[13]	K. Ilarslan, Ő. Boyacoğlu, Position vectors of a timelike and a null helix in Minkowski 3-space, Chaos, Solitons Fractals, 38 (2008), 1383–1389.
[14]	K. Ilarslan, Spacelike normal curves in Minkowski space E3 1, Turkish J. Math., 29 (2005), 53–63.
[15]	K. Ilarslan, E. Nesovic, M. Petrovic'-Torgasev, Some characterizations of rectifying curves in the Minkowski 3-space, Novi. Sad. J. Math., 33 (2003), 23–32.
[16]	A. Ucum, C. Camc, K. Ilarslan, General helices with spacelike slope axis in Minkowski 3-space, Asian-Eur. J. Math., 12 (2019), 1950076. doi: 10.1142/S1793557119500761
[17]	B. Ylmaz, I. Gok, Y. Yayl, Extended rectifying curves in Minkowski 3-space, Adv. Appl. Clifford Algebras, 26 (2016), 861–872. doi: 10.1007/s00006-015-0637-7
[18]	A. Uçum, C. Camci, K. Ilarslan, General helices with timelike slope axis in Minkowski 3-Space, Adv. Appl. Clifford Algebras, 26 (2016), 793–807. doi: 10.1007/s00006-015-0610-5
[19]	C. Camci, K. Ilarslan, A. Uçum, General helices with lightlike slope axis, Filomat, 32 (2018), 355–367. doi: 10.2298/FIL1802355C
[20]	N. Ayyıldız, A. C. Coken, A. Yucesan, On the dual Darboux rotation axis of the spacelike dual space curve, Demonstratio Math., 37 (2004), 197–202.
[21]	N. Ayyıldız, A. C. Coken, A. Yucesan, A characterization of dual Lorentzian spherical curves in the dual Lorentzian space, Taiwanese J. Math., 11 (2007), 999–1018.
[22]	E. Ozbey, M. Oral, A study on rectifyng curves in the dual Lorentzian space, Bull. Korean Math. Soc., 46 (2009), 967–978.
[23]	M. Onder, Z. Ekinci, On closed timelike and spacelike ruled surfaces, Math. Methods Appl. Sci., 40 (2017), 786–795. doi: 10.1002/mma.4011
[24]	Y. Li, Z. Wang, T. Zhao, Slant helix of order n and sequence of Darboux developable of principaldirectional curves, Math. Methods Appl. Sci., 43 (2020), 1–16. doi: 10.1002/mma.5729

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)