Content uploaded by Abel Cavasi

Author content

All content in this area was uploaded by Abel Cavasi on Feb 17, 2022

Content may be subject to copyright.

CREATIVE MATH. & INF.

23 (2014), No. 2, 175 - 182

Online version available at http://creative-mathematics.ubm.ro/

Print Edition: ISSN 1584 - 286X Online Edition: ISSN 1843 - 441X

The recurrence theorem of Frenet formulae

ABEL CAVAS¸I

ABSTRACT.

In this paper we generalize Frenet trihedron and we provide some other classes of trihedrons, called Frenet trihedrons of order n,n≥1.Moreover

we prove that those trihedrons can be deﬁned recurrently and we emphasize their role in the study of the generalized helices of order n. As

practical applications we present the inﬂuence of the recurrence theorem in some interdisciplinary domains like physics, chemistry and biology.

Acknowledgements. Many thanks to Mrs. Teacher of Mathematics Alina-Ramona Baias, who made considerable

effort to translate and adapt this work from Romanian.

REFERENCES

[1] Barros, M. General Helices and a Theorem of Lancret, Proc. Amer. Math. Soc., 125 (1997), No. 5, 1503–1509

[2] Frenet, F. (1847), Sur les courbes `a double courbure, Th`

ese, Toulouse. Abstract in J. de Math., 17 (1852)

[3] Hacisalihoglu, H. H., Differential Geometry, Ankara University, 12 (13) (2012), 73 Faculty of Science Press, 2000

[4] Izumiya, S. and Takeuchi, N., New special curves and developable surfaces, Turk. J. Math, 28, (2004), 531–537

[5] Liu, H. and Wang, F., Mannheim partner curves in 3-space, Journal of Geometry, col. 88, nr. 1-2, pp 120-126, (2008)

[6] Monterde, J., Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geomet. Design, 26

(2009), 271–278

[7] Ramis, C., Uzunoglu, B. and Yayli, Y., New Associated Curves k-Principle Direction Curves and NkSlant Helix, arXiv:1404.7369 [math.DG], (2014)

[8] Scoﬁeld, P. D., Curves of constant precession, Amer. Math., Monthly, Vol. 102, (1995), pp 531-537, MR 96d:53002

[9] Serret, J. A. Sur quelques formules relatives `a la th´eorie des courbes `a double courbure, J. de Math., 16 (1851)

[10] Uzunoglu, B., Gok, I. and Yayli, Y., A New Approach on Curves of Constant Precession, arXiv:1311.4730 [math.DG], (2013)

[11] Yayli, Y. and Saracoglu, S., Characterizations of Special Curves, arXiv:1202.0133 [math.DG], (2012)

[12] Yayli, Y. and Ziplar, E., On slant helices an general helices in euclidean n-space, Mathematica Aeterna, Vol. 1(2011), Nr. 08, 599–610

[13] Ziplar, E., Senol, A. and Yayli, Y., On Darboux Helices in Euclidean 3-Space, Global J. of Science Frontier Research, 12(13) (2012), 73–80

WEST UNIV ERSIT Y OF TI MIS¸OA RA

FACULTY OF MATHEMATICS AND COMPUTER SCIENCE

ROMANI A

E-mail address:abel.cavasi@gmail.com

Received: 20.05.2014; In revised form: 20.10.2010; Accepted: 25.10.2014

2010 Mathematics Subject Classiﬁcation. 53A04.

Key words and phrases. Frenet’s formulas, Darboux vector, recurrence theorem, helix.

175