Research Article
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Year 2021, , 107 - 113, 30.09.2021
https://doi.org/10.32323/ujma.937479

Abstract

References

  • M. P. Do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications, 2016.
  • A. N. Pressley, Elementary Differential Geometry, Springer Science & Business Media, 2010. B.Y. Chen, When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Month., 110(2) (2003), 147–152.
  • B.Y. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sinica, 33(2) (2005), 77-90.
  • B.Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math., 48(2) (2017), 209-214.
  • S. Deshmukh, B.Y. Chen, S. Alshamari, On rectifying curves in Euclidean 3-space, Turk. J. Math., 42(2) (2018), 609-620.
  • K. {\.I}larslan, E. Nesovic, Some characterizations of rectifying curves in the Euclidean space $\mathbf{E}^4$, Turk. J. Math., 32(1) (2008), 21-30.
  • S. Cambie, W. Goemans, I. Van den Bussche, Rectifying curves in the $n$-dimensional Euclidean space,Turk. J. Math., 40(1) (2016), 210-223.
  • P. Lucas, J.A. Ortega-Yag{\"u}es, Rectifying curves in the three-dimensional sphere, J. Math. Anal. Appl., 421(2) (2015), 1855-1868.
  • P. Lucas, J.A. Ortega-Yag{\"u}es, Rectifying curves in the three-dimensional hyperbolic space, Medit. J. Math., 13(4) (2016), 2199-2214.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, T. M. Petrovi\'{c}, Some characterization of rectifying curves in the Minkowski 3-Space, Novi Sad J. Math., 33(2) (2003), 23-32.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, On rectifying curves as centrodes and extremal curves in the Minkowski 3-Space, Novi Sad J. Math., 37(1) (2007), 53-64.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time, Taiwanese J. Math., 12(5) (2008), 1035-1044.
  • T.A. Ali, M. Onder, Some characterizations of space-like rectifying curves in the Minkowski space-time, Glob. J. Sci. Fron. Res. Math. Des. Sci., 12(1) (2012), 57-63.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, Some relations between normal and rectifying curves in Minkowski space-time, Inter. Elec. J. Geom., 7(1) (2014), 26-35.
  • F. Hathout, A new class of curves generalizing helix and rectifying curves, arXiv: Diff. Geom., (2018).
  • Z. Iqbal, J. Sengupta, Non-null (spacelike or timelike) f-rectifying curves in the Minkowski 3-space $\mathbb{E}_1^3$, Eurasian Bul. Math., 3(1) (2020), 38-55.
  • Z. Iqbal, J. Sengupta, Null (lightlike) f-rectifying curves in the Minkowski 3-space $\mathbb{E}_1^3$, Fundam. J. Math. Appl., 3(1) (2020), 8-16.
  • Z. Iqbal, J. Sengupta, Differential geometric aspects of lightlike $f$-rectifying curves in Minkowski space-time, Diff. Geom. - Dyn. Syst., 22 (2020), 113-129. Z. Iqbal, J. Sengupta, On $f$-Rectifying Curves in the Euclidean 4-Space, Acta Univ. Sapientiae Matem. (Accepted Manuscript).
  • H. Gluck, Higher curvatures of curves in Euclidean space, Amer. Math. Mon., 73(7) (1966), 699--704.
  • H. Gluck, Higher curvatures of curves in Euclidean space II, Amer. Math. Mon., 74(9) (1967), 1049--1056.

A Study on $f$-Rectifying Curves in Euclidean $n$-Space

Year 2021, , 107 - 113, 30.09.2021
https://doi.org/10.32323/ujma.937479

Abstract

A rectifying curve in the Euclidean $n$-space $\mathbb{E}^n$ is defined as an arc-length parametrized curve $\gamma$ in $\mathbb{E}^n$ such that its position vector always lies in its rectifying space (i.e., the orthogonal complement of its principal normal vector field) in $\mathbb{E}^n$. In this paper, in analogy to this, we introduce the notion of an $f$-rectifying curve in $\mathbb{E}^n$ as a curve $\gamma$ in $\mathbb{E}^n$ parametrized by its arc-length $s$ such that its $f$-position vector field $\gamma_f$, defined by $\gamma_f(s) = \int f(s) d\gamma$, always lies in its rectifying space in $\mathbb{E}^n$, where $f$ is a nowhere vanishing real-valued integrable function in parameter $s$. The main purpose is to characterize and classify such curves in $\mathbb{E}^n$.

References

  • M. P. Do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications, 2016.
  • A. N. Pressley, Elementary Differential Geometry, Springer Science & Business Media, 2010. B.Y. Chen, When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Month., 110(2) (2003), 147–152.
  • B.Y. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sinica, 33(2) (2005), 77-90.
  • B.Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math., 48(2) (2017), 209-214.
  • S. Deshmukh, B.Y. Chen, S. Alshamari, On rectifying curves in Euclidean 3-space, Turk. J. Math., 42(2) (2018), 609-620.
  • K. {\.I}larslan, E. Nesovic, Some characterizations of rectifying curves in the Euclidean space $\mathbf{E}^4$, Turk. J. Math., 32(1) (2008), 21-30.
  • S. Cambie, W. Goemans, I. Van den Bussche, Rectifying curves in the $n$-dimensional Euclidean space,Turk. J. Math., 40(1) (2016), 210-223.
  • P. Lucas, J.A. Ortega-Yag{\"u}es, Rectifying curves in the three-dimensional sphere, J. Math. Anal. Appl., 421(2) (2015), 1855-1868.
  • P. Lucas, J.A. Ortega-Yag{\"u}es, Rectifying curves in the three-dimensional hyperbolic space, Medit. J. Math., 13(4) (2016), 2199-2214.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, T. M. Petrovi\'{c}, Some characterization of rectifying curves in the Minkowski 3-Space, Novi Sad J. Math., 33(2) (2003), 23-32.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, On rectifying curves as centrodes and extremal curves in the Minkowski 3-Space, Novi Sad J. Math., 37(1) (2007), 53-64.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time, Taiwanese J. Math., 12(5) (2008), 1035-1044.
  • T.A. Ali, M. Onder, Some characterizations of space-like rectifying curves in the Minkowski space-time, Glob. J. Sci. Fron. Res. Math. Des. Sci., 12(1) (2012), 57-63.
  • K. Ilarslan, E. Ne\v{s}ovi\'{c}, Some relations between normal and rectifying curves in Minkowski space-time, Inter. Elec. J. Geom., 7(1) (2014), 26-35.
  • F. Hathout, A new class of curves generalizing helix and rectifying curves, arXiv: Diff. Geom., (2018).
  • Z. Iqbal, J. Sengupta, Non-null (spacelike or timelike) f-rectifying curves in the Minkowski 3-space $\mathbb{E}_1^3$, Eurasian Bul. Math., 3(1) (2020), 38-55.
  • Z. Iqbal, J. Sengupta, Null (lightlike) f-rectifying curves in the Minkowski 3-space $\mathbb{E}_1^3$, Fundam. J. Math. Appl., 3(1) (2020), 8-16.
  • Z. Iqbal, J. Sengupta, Differential geometric aspects of lightlike $f$-rectifying curves in Minkowski space-time, Diff. Geom. - Dyn. Syst., 22 (2020), 113-129. Z. Iqbal, J. Sengupta, On $f$-Rectifying Curves in the Euclidean 4-Space, Acta Univ. Sapientiae Matem. (Accepted Manuscript).
  • H. Gluck, Higher curvatures of curves in Euclidean space, Amer. Math. Mon., 73(7) (1966), 699--704.
  • H. Gluck, Higher curvatures of curves in Euclidean space II, Amer. Math. Mon., 74(9) (1967), 1049--1056.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Zafar Iqbal

Joydeep Sengupta

Publication Date September 30, 2021
Submission Date May 15, 2021
Acceptance Date October 1, 2021
Published in Issue Year 2021

Cite

APA Iqbal, Z., & Sengupta, J. (2021). A Study on $f$-Rectifying Curves in Euclidean $n$-Space. Universal Journal of Mathematics and Applications, 4(3), 107-113. https://doi.org/10.32323/ujma.937479
AMA Iqbal Z, Sengupta J. A Study on $f$-Rectifying Curves in Euclidean $n$-Space. Univ. J. Math. Appl. September 2021;4(3):107-113. doi:10.32323/ujma.937479
Chicago Iqbal, Zafar, and Joydeep Sengupta. “A Study on $f$-Rectifying Curves in Euclidean $n$-Space”. Universal Journal of Mathematics and Applications 4, no. 3 (September 2021): 107-13. https://doi.org/10.32323/ujma.937479.
EndNote Iqbal Z, Sengupta J (September 1, 2021) A Study on $f$-Rectifying Curves in Euclidean $n$-Space. Universal Journal of Mathematics and Applications 4 3 107–113.
IEEE Z. Iqbal and J. Sengupta, “A Study on $f$-Rectifying Curves in Euclidean $n$-Space”, Univ. J. Math. Appl., vol. 4, no. 3, pp. 107–113, 2021, doi: 10.32323/ujma.937479.
ISNAD Iqbal, Zafar - Sengupta, Joydeep. “A Study on $f$-Rectifying Curves in Euclidean $n$-Space”. Universal Journal of Mathematics and Applications 4/3 (September 2021), 107-113. https://doi.org/10.32323/ujma.937479.
JAMA Iqbal Z, Sengupta J. A Study on $f$-Rectifying Curves in Euclidean $n$-Space. Univ. J. Math. Appl. 2021;4:107–113.
MLA Iqbal, Zafar and Joydeep Sengupta. “A Study on $f$-Rectifying Curves in Euclidean $n$-Space”. Universal Journal of Mathematics and Applications, vol. 4, no. 3, 2021, pp. 107-13, doi:10.32323/ujma.937479.
Vancouver Iqbal Z, Sengupta J. A Study on $f$-Rectifying Curves in Euclidean $n$-Space. Univ. J. Math. Appl. 2021;4(3):107-13.

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