Abstract
In this paper, we study helices and the Bertrand curves. We obtain some of the classification results of these curves with respect to the modified orthogonal frame in Euclidean 3-spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ali AT, Lopez R (2011) Slant helices in Minkowski space \(E_{1}^{3}\). J Korean Math Soc 48:159–167
Balgetir H, Bektas M, Inoguchi J (2004) Null Bertrand curves in Minkowski 3-space and their characterizations. Note di Matematica 23(1):7–13
Barros M (1997) General helices and a theorem of Lancret. Proc Am Math Soc 125(5):1503–1509
Bertrand JM (1850) Mémoire sur la théorie des courbes á double courbure. J Math Pures Appl 15:332–350
Burke JF (1960) Bertrand curves associated with a pair of curves. Math Mag 34(1):60–62
Do Carmo M (1976) Differential geometry of curves and surfaces. Prentice-Hall, New Jersey
Ekmekci N (2000) On general helices and submanifolds of an indefinite-Riemann manifold. An Ştiint Univ Al I Cuza Iaşi Mat (NS) 46(2):263–270
Ilarslan K (2003) Characterizations of spacelike general helices in Lorentzian manifolds. Kragujev J Math 25:209–218
Izumiya S, Takeuchi N (2004) New special curves and developable surfaces. Turk J Math 28:153–163
Kula L, Yaylı Y (2005) On slant helix and its spherical indicatrix. Appl Math Comput 169(1):600–607
Kula L, Ekmekçi N, Yaylı Y, Ilarslan K (2010) Characterizations of slant helices in Euclidean 3-space. Turk J Math 34:261–273
Matsuda H, Yorozu SH (2003) Notes on Bertrand curves. Yokohama Math J 50:41–58
Monterde J (2009) Salkowski curves revisted: a family of curves with constant curvature and non-constant torsion. Comput Aided Geom Des 26:271–278
Nutbourne AW, Martin RR (1988) Differential geometry applied to the design of curves and surfaces. Ellis Horwood, Chichester
Ogrenmis AO, Ergut M, Bektas M (2007) On the helices in the Galilean space \(G_{3}\). Iran J Sci Technol Trans A 31(A2):177–181
Ogrenmis AO, Oztekin H, Ergut M (2009) Bertrand curves in Galilean space and their characterizations. Kragujev J Math 32:139–147
Oztekin HB, Bektas M (2010) Representation formulae for Bertrand curves in the Minkowski 3-space. Sci Magna 6(1):89–96
Salkowski E (1909) Zur transformation von raumkurven. Math Ann 66(4):517–557
Sasai T (1984) The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Math J 36:17–24
Schief WK (2003) On the integrability of Bertrand curves and Razzaboni surfaces. J Geom Phys 45:130–150
Struik DJ (1988) Lectures on classical differential geometry, 2nd edn. Addison Wesley, Boston
van-Brunt B, Grant K (1996) Potential application of Weingarten surfaces in CADG, part I: Weingarten surfaces and surface shape investigation. Comput Aided Geom Des 13:569–582
Yoon DW (2012) General helices of AW(k)-type in the lie group. J Appl Math 10. Article ID 535123. https://doi.org/10.1155/2012/535123
Acknowledgements
The authors would like to thank all the anonymous referees for their valuable comments and suggestions which helped to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lone, M.S., ES, H., Karacan, M.K. et al. On Some Curves with Modified Orthogonal Frame in Euclidean 3-Space. Iran J Sci Technol Trans Sci 43, 1905–1916 (2019). https://doi.org/10.1007/s40995-018-0661-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-018-0661-2