Browse > Article
http://dx.doi.org/10.5831/HMJ.2021.43.3.453

TUBULAR SURFACES WITH MODIFIED ORTHOGONAL FRAME IN EUCLIDEAN 3-SPACE  

Akyigit, Mahmut (Department of Mathematics, Faculty of Arts and Sciences, Sakarya University)
Eren, Kemal (Department of Mathematics, Faculty of Arts and Sciences, Sakarya University)
Kosal, Hidayet Huda (Department of Mathematics, Faculty of Arts and Sciences, Sakarya University)
Publication Information
Honam Mathematical Journal / v.43, no.3, 2021 , pp. 453-463 More about this Journal
Abstract
In this study, tubular surfaces that play an important role in technological designs in various branches are examined for the case of the base curve is not satisfying the fundamental theorem of the differential geometry. In order to give an alternative perspective to the researches on tubular surfaces, the modified orthogonal frame is used in this study. Firstly, the relationships between the Serret-Frenet frame and the modified orthogonal frame are summarized. Then the definitions of the tubular surfaces, some theorems, and results are given. Moreover, the fundamental forms, the mean curvature, and the Gaussian curvature of the tubular surface are calculated according to the modified orthogonal frame. Finally, the properties of parameter curves of the tubular surface with modified orthogonal frame are expressed and the tubular surface is drawn according to the Frenet frame and the modified orthogonal frame.
Keywords
Tubular surfaces; modified orthogonal frame; geodesic curve; asymptotic curve; line of curvature;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 M. K. Karacan, D. W. Yoon and Y. Tuncer, Tubular surfaces of Weingarten types in Minkowski 3-space, Gen. Math. Notes 22(1) (2014), 44-56.
2 T. Korpinar and E. Turhan, Tubular surfaces around timelike biharmonic curves in Lorentzian Heisenberg group Heis3, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 20(1) (2012), 431-446.
3 P. A. Blaga, On tubular surfaces in computer graphics, Stud. Univ. Babes-Bolyai Inform. 50 (2005), 81-90.
4 B. Bukcu and M. K. Karacan, On the modified orthogonal frame with curvature and torsion in 3-space, Math Sci. Appl. E-Notes 4(1) (2016), 184-188.
5 M. Dede, Tubular surfaces in Galilean space, Commun. Math. 18(1) (2013), 209-217.
6 M. Dede, C. Ekici, and H. Tozak, Directional tubular surfaces, Int. J.Algebra 9(12) (2015), 527-535.   DOI
7 P. M. Do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
8 F. Dogan and Y. Yayli, Tubes with Darboux frame, Int. J. Contemp. Math. Sci. 7(16) (2012), 751-758.
9 F. Dogan and Y. Yayli, On the curvatures of tubular surface with Bishop frame, Communications Fac. Sci. Univ. Ank. Ser. A1 60(1) (2011), 59-69.   DOI
10 R. L. Faber, Differential Geometry and Relativity Theory, An Introduction, Marcel Dekker, Inc., 1983.
11 M. S. Lone, E. S. Hasan, M. K. Karacan and B. Bukcu, On some curves with modified orthogonal frame in Euclidean 3-space, Iran. J. Sci. Technol. Trans. A Sci. 43(4) (2019), 1905-1916.   DOI
12 M. K. Karacan and Y. Yayli, On the geodesics of tubular surfaces in Minkowski 3-space, Bull. Malays. Math. Sci. Soc. 1 (2008), 1-10.
13 M. K. Karacan, D. W. Yoon and Y. Tuncer, Weingarten and linear Weingarten type tubular surfaces in E3, Math. Probl. Eng. 2011(3) (2011), 1-11.   DOI
14 S. Kiziltug, S. Kaya and O. Tarakci, Tube surfaces with type-2 Bishop frame of Weingarten types in E3, Int. J. Math. Anal. 7(1) (2013), 9-18.   DOI
15 M. S. Lone, E. S. Hasan, M. K. Karacan and B. Bukcu, Mannheim curves with modified orthogonal frame in Euclidean 3-space, Turkish J. Math. 43(2) (2019), 648-663.   DOI
16 T. Sasai, The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations, Tohoku Math. J. 36 (1984), 17-24.   DOI
17 B. O'Neill, Elementary Differential Geometry, Academic Press, Inc, New York, 1966.
18 K. E. Ozen, M. Guner and M. Tosun, A Note on the Acceleration and Jerk in Motion Along a Space Curve, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 28(1) (2020), 151-164.
19 H. Reichardt, Einfuhrung in die Differentialgeometrie, Springer, MR0116267 Zbl 0091, 34001, 1960.
20 D. W. Yoon, Some properties of parallel surfaces in Euclidean 3-spaces, Honam Math. J. 30(4) (2008), 637-644.   DOI
21 M. K. Karacan and Y. Tuncer, Tubular surfaces of Weingarten types in Galilean and Pseudo-Galilean, Bull. Math. Anal. Appl. 5(2) (2013), 87-100.
22 Z. Xu, R. Feng and J. Sun, Analytic and algebraic properties of canal surfaces, J. of Comp. and Appl. Mathematics 195(1) (2006), 220-228.   DOI
23 A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC, New York, 1998.