Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$

Sümeyye Gür Mazlum; Süleyman Şenyurt; Mehmet Bektaş

doi:10.53570/jnt.1140546

Research Article

Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$

Year 2022, Issue: 40, 12 - 26, 30.09.2022

Sümeyye Gür Mazlum Süleyman Şenyurt Mehmet Bektaş

https://doi.org/10.53570/jnt.1140546

Cited By: 3

Abstract

In this study, we examine some properties of Salkowski curves in $\mathbb{E}^{3}$. We then make sense of the angle $(nt)$ in the parametric equation of the Salkowski curves. We provide the relationship between this angle and the angle between the binormal vector and the Darboux vector of the Salkowski curves. Through this angle, we obtain the unit vector in the direction of the Darboux vector of the curve. Finally, we calculate the modified orthogonal frames with both the curvature and the torsion and give the relationships between the Frenet frame and the modified orthogonal frames of the curve.

Keywords

Salkowski curve, modified orthogonal frame, Frenet frame

References

M. P. Do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications, 2016.
H. H. Hacısalihoğlu, Differantial Geometry, Ankara University Faculty of Science Press, Ankara, Türkiye, 2000.
L. Kula, N. Ekmekci, Y. Yayli, K. İlarslan, Characterizations of Slant Helices in Euclidean 3-Space, Turkish Journal of Mathematics 34 (2) (2010) 261-274.
A. T. Ali, Position Vectors of Slant Helices in Euclidean 3-Space, Journal of the Egyptian Mathematical Society 20 (1) (2012) 1-6.
E. Salkowski, Zur Transformation Von Raumkurven Mathematische Annalen 66 (4) (1909) 517-557.
J. Monterde, Salkowski Curves Revisited: A Family of Curves with Constant Curvature and Non-Consant Torsion, Computer Aided Geometric Design 26 (2009) 271-278.
S. Gur, S. Senyurt, Frenet Vectors and Geodesic Curvatures of Spheric Indicators of Salkowski Curve in E3, Hadronic Journal 33 (5) (2010) 485-512.
S. Şenyurt, B. Öztürk, Smarandache Curves of Salkowski Curve According to Frenet Frame, Turkish Journal of Mathematics and Computer Science 10 (2018) 190-201.
T. Sasai, The Fundamental Theorem of Analytic Space Curves and Apparent Singularities of Fuchsian Differential Equations, Tohoku Math Journal 36 (1984) 17-24.
T. Sasai, Geometry of Analytic Space Curves with Singularities and Regular Singularities of Differential Equations, Funkcialaj Ekvacioj 30 (1987) 283-303.
B. Bükcü, M. K. Karacan, On the Modified Orthogonal Frame with Curvature and Torsion in 3-Space, Mathematical Sciences and Applications E-Notes 1 (2016) 184-188.
B. Bükcü, M. K. Karacan, Spherical Curves with Modified Orthogonal Frame, Journal of New Results in Science 10 (2016) 60-68.
S. Uddin, M. S. Stankovic, M. Iqbal, S. K. Yadav, M. Aslam, Slant Helices in Minkowski 3-Space $E^3_1$ with Sasai's Modified Frame Fields, Filomat 36 (2022) 151-164.
H. K. Elsayied, A. A. Altaha, A. Elsharkawy, On Some Special Curves According to the Modified Orthogonal Frame in Minkowski 3-Space $E^3_1$, Kasmera 49 (2021) 2-15.
M. Arıkan, S. K. Nurkan, Adjoint Curve According to Modified Orthogonal Frame with Torsion in 3-Space, Uşak University Journal of Science and Natural Sciences 2 (2020) 54-64.
S. Şenyurt, S. Gür Mazlum, L. Grilli, Gaussian Curvatures of Parallel Ruled Surfaces, Applied Mathematical Sciences 14 (2020) 171-183.
A. Z. Azak, Involute-Evolute Curves According to Modified Orthogonal Frame, Journal of Science and Arts 21(2) (2021) 385-394.
K. Eren, H. H. Kosal, Evolution of Space Curves and the Special Ruled Surfaces With Modified Orthogonal Frame, AIMS Mathematics 5 (3) (2020) 2027-2039.
M. S. Lone, H. Es, M. K. Karacan, B. Bükcü, On Some Curves With Modified Orthogonal Frame in Euclidean 3-Space, Iranian Journal of Science and Technology, Transactions A: Science 43 (4) (2019) 1905-1916.
M. S. Lone, H. Es, M. K. Karacan, B. Bükcü, Mannheim Curves with Modified Orthogonal Frame in Euclidean 3-Space, Turkish Journal of Mathematics 43(2) (2019) 648-663.
Y. Li, S. Y. Liu, Z. G. Wang, Tangent Developables and Darboux Developables of Framed Curves, Topology and Its Applications 301 (2021) 107526.
Y. Li, D. Ganguly, S. Dey, A. Bhattacharyya, Conformal _-Ricci Solitons Within the Framework of Indefinite Kenmotsu Manifolds, AIMS Mathematics 7 (4) (2022), 5408-5430.
Y. Yayli, I. Gok, H. H. Hacisalihoglu, Extended Rectifying Curves as New Kind of Modified Darboux Vectors, TWMS Journal of Pure and Applied Mathematics 9 (2018) 18-31.
A. Kelleci, M. Bektai, M. Ergüt, The Hasimoto Surface According to Bishop Frame, Adıyaman University Journal of Science 9 (1) (2019) 13-22.
S. Gür Mazlum, S. Şenyurt, L. Grilli, The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space, Symmetry 14 (5) (2022) 1062.
S. Gür Mazlum, M. Bektaş, On the Modified Orthogonal Frames of the Non-Unit Speed Curves in Euclidean Space $\mathbb{E}^{3}$, Turkish Journal of Science (2022) In Press.
W. Fenchel, On the Difierential Geometry of Closed Space Curves, Bulletin of the American Mathematical Society 57 (1951) 44-54.

Year 2022, Issue: 40, 12 - 26, 30.09.2022

Sümeyye Gür Mazlum Süleyman Şenyurt Mehmet Bektaş

https://doi.org/10.53570/jnt.1140546

Cited By: 3

Abstract

References

M. P. Do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications, 2016.
H. H. Hacısalihoğlu, Differantial Geometry, Ankara University Faculty of Science Press, Ankara, Türkiye, 2000.
L. Kula, N. Ekmekci, Y. Yayli, K. İlarslan, Characterizations of Slant Helices in Euclidean 3-Space, Turkish Journal of Mathematics 34 (2) (2010) 261-274.
A. T. Ali, Position Vectors of Slant Helices in Euclidean 3-Space, Journal of the Egyptian Mathematical Society 20 (1) (2012) 1-6.
E. Salkowski, Zur Transformation Von Raumkurven Mathematische Annalen 66 (4) (1909) 517-557.
J. Monterde, Salkowski Curves Revisited: A Family of Curves with Constant Curvature and Non-Consant Torsion, Computer Aided Geometric Design 26 (2009) 271-278.
S. Gur, S. Senyurt, Frenet Vectors and Geodesic Curvatures of Spheric Indicators of Salkowski Curve in E3, Hadronic Journal 33 (5) (2010) 485-512.
S. Şenyurt, B. Öztürk, Smarandache Curves of Salkowski Curve According to Frenet Frame, Turkish Journal of Mathematics and Computer Science 10 (2018) 190-201.
T. Sasai, The Fundamental Theorem of Analytic Space Curves and Apparent Singularities of Fuchsian Differential Equations, Tohoku Math Journal 36 (1984) 17-24.
T. Sasai, Geometry of Analytic Space Curves with Singularities and Regular Singularities of Differential Equations, Funkcialaj Ekvacioj 30 (1987) 283-303.
B. Bükcü, M. K. Karacan, On the Modified Orthogonal Frame with Curvature and Torsion in 3-Space, Mathematical Sciences and Applications E-Notes 1 (2016) 184-188.
B. Bükcü, M. K. Karacan, Spherical Curves with Modified Orthogonal Frame, Journal of New Results in Science 10 (2016) 60-68.
S. Uddin, M. S. Stankovic, M. Iqbal, S. K. Yadav, M. Aslam, Slant Helices in Minkowski 3-Space $E^3_1$ with Sasai's Modified Frame Fields, Filomat 36 (2022) 151-164.
H. K. Elsayied, A. A. Altaha, A. Elsharkawy, On Some Special Curves According to the Modified Orthogonal Frame in Minkowski 3-Space $E^3_1$, Kasmera 49 (2021) 2-15.
M. Arıkan, S. K. Nurkan, Adjoint Curve According to Modified Orthogonal Frame with Torsion in 3-Space, Uşak University Journal of Science and Natural Sciences 2 (2020) 54-64.
S. Şenyurt, S. Gür Mazlum, L. Grilli, Gaussian Curvatures of Parallel Ruled Surfaces, Applied Mathematical Sciences 14 (2020) 171-183.
A. Z. Azak, Involute-Evolute Curves According to Modified Orthogonal Frame, Journal of Science and Arts 21(2) (2021) 385-394.
K. Eren, H. H. Kosal, Evolution of Space Curves and the Special Ruled Surfaces With Modified Orthogonal Frame, AIMS Mathematics 5 (3) (2020) 2027-2039.
M. S. Lone, H. Es, M. K. Karacan, B. Bükcü, On Some Curves With Modified Orthogonal Frame in Euclidean 3-Space, Iranian Journal of Science and Technology, Transactions A: Science 43 (4) (2019) 1905-1916.
M. S. Lone, H. Es, M. K. Karacan, B. Bükcü, Mannheim Curves with Modified Orthogonal Frame in Euclidean 3-Space, Turkish Journal of Mathematics 43(2) (2019) 648-663.
Y. Li, S. Y. Liu, Z. G. Wang, Tangent Developables and Darboux Developables of Framed Curves, Topology and Its Applications 301 (2021) 107526.
Y. Li, D. Ganguly, S. Dey, A. Bhattacharyya, Conformal _-Ricci Solitons Within the Framework of Indefinite Kenmotsu Manifolds, AIMS Mathematics 7 (4) (2022), 5408-5430.
Y. Yayli, I. Gok, H. H. Hacisalihoglu, Extended Rectifying Curves as New Kind of Modified Darboux Vectors, TWMS Journal of Pure and Applied Mathematics 9 (2018) 18-31.
A. Kelleci, M. Bektai, M. Ergüt, The Hasimoto Surface According to Bishop Frame, Adıyaman University Journal of Science 9 (1) (2019) 13-22.
S. Gür Mazlum, S. Şenyurt, L. Grilli, The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space, Symmetry 14 (5) (2022) 1062.
S. Gür Mazlum, M. Bektaş, On the Modified Orthogonal Frames of the Non-Unit Speed Curves in Euclidean Space $\mathbb{E}^{3}$, Turkish Journal of Science (2022) In Press.
W. Fenchel, On the Difierential Geometry of Closed Space Curves, Bulletin of the American Mathematical Society 57 (1951) 44-54.

There are 27 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Article
Authors	Sümeyye Gür Mazlum 0000-0003-2471-1627 Süleyman Şenyurt 0000-0003-1097-5541 Mehmet Bektaş 0000-0002-5797-4944
Publication Date	September 30, 2022
Submission Date	July 5, 2022
Published in Issue	Year 2022 Issue: 40

Cite

APA	Gür Mazlum, S., Şenyurt, S., & Bektaş, M. (2022). Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$. Journal of New Theory(40), 12-26. https://doi.org/10.53570/jnt.1140546
AMA	Gür Mazlum S, Şenyurt S, Bektaş M. Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$. JNT. September 2022;(40):12-26. doi:10.53570/jnt.1140546
Chicago	Gür Mazlum, Sümeyye, Süleyman Şenyurt, and Mehmet Bektaş. “Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$”. Journal of New Theory, no. 40 (September 2022): 12-26. https://doi.org/10.53570/jnt.1140546.
EndNote	Gür Mazlum S, Şenyurt S, Bektaş M (September 1, 2022) Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$. Journal of New Theory 40 12–26.
IEEE	S. Gür Mazlum, S. Şenyurt, and M. Bektaş, “Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$”, JNT, no. 40, pp. 12–26, September 2022, doi: 10.53570/jnt.1140546.
ISNAD	Gür Mazlum, Sümeyye et al. “Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$”. Journal of New Theory 40 (September 2022), 12-26. https://doi.org/10.53570/jnt.1140546.
JAMA	Gür Mazlum S, Şenyurt S, Bektaş M. Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$. JNT. 2022;:12–26.
MLA	Gür Mazlum, Sümeyye et al. “Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$”. Journal of New Theory, no. 40, 2022, pp. 12-26, doi:10.53570/jnt.1140546.
Vancouver	Gür Mazlum S, Şenyurt S, Bektaş M. Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$. JNT. 2022(40):12-26.