ON TZITZEICA CURVES IN EUCLIDEAN 3-SPACE E^3

Bengü Bayram, Emrah Tunç, Kadri Arslan, Günay Öztürk

DOI Number
https://doi.org/10.22190/FUMI1803409B
First page
409
Last page
416

Abstract


In this study, we consider Tzitzeica curves (Tz-curves) in Euclidean 3-space E^3. We characterize such curves according to their curvatures. We show that there is no Tz-curve with constant curvatures (W-curves). We consider Salkowski (TC-curve) and Anti-Salkowski curves.


Keywords

Tz-curves, W-curves, TC-curves

Keywords


Tzitzeica curve, Frenet Frame, Salkowski curve

Full Text:

PDF

References


bibitem{AE} {sc M.E. Aydi n, M. Erg"{u}t}: textit{Non-null curves of Tzitzeica

type in Minkowski 3-space}. Romanian J. of Math. and Comp. Science

{bf 4(1)} (2014), 81-90.

bibitem{B} {sc N. Bila}: textit{Symmetry raductions for the Tzitzeica curve

equation}. Math. and Comp. Sci. Workin Papers {bf 16} (2012).

bibitem{BBC} {sc A. Bobe, W. G. Boskoff and M. G. Ciuca}: textit{Tzitzeica type

centro-affine invariants in Minkowski space}. An. St. Univ. Ovidius

Constanta {bf 20(2)} (2012), 27-34.

bibitem{Ch1} {sc B. Y. Chen}: textit{Geometry of warped products as Riemannian

submanifolds and related problems}. Soochow J. Math. {bf 28}

(2002), 125-156.

bibitem{Ch2} {sc B. Y. Chen}: textit{Convolution of Riemannian manifolds and its

applications}. Bull. Aust. Math. Soc. {bf 66} (2002), 177-191.

bibitem{Ch3} {sc B.Y. Chen}: textit{When does the position vector of a space

curve always lies in its rectifying plane?}. Amer. Math. Monthly

{bf 110} (2003), 147-152.

bibitem{C} {sc M. Crac{s}mareann}: textit{Cylindrical Tzitzeica curves implies

forced harmonic oscillators}. Balkan J. of Geom. and Its App. {bf

(1)} (2002), 37-42.

bibitem{CC} {sc O. Constantinescu, M. Crac{s}mareann}.: textit{A new Tzitzeica

hypersurface and cubic Finslerian metrics of Berwall type} Balkan J.

of Geom. and Its App. {bf 16(2)} (2011), 27-34.

bibitem{G} {sc A. Gray}: textit{Modern differential geometry of curves and surface}, CRS Press, Inc. 1993.

bibitem{Gl} {sc H. Gluck}: textit{Higher curvatures of curves in Euclidean space} Amer. Math. Monthly {bf 73} (1966), 699-704.

bibitem{GAO} {sc S. G"{u}rpi nar, K. Arslan, G. "{O}zt"{u}rk}: textit{A

Characterization of Constant-ratio Curves in Euclidean 3-space

$E^{3}$}. Acta Universitatis Apulensis textbf{44} (2015), 39--51.

bibitem{KB} {sc M. K. Karacan, B. B"{u}kc{c}"{u}}: textit{On the elliptic

cylindrical Tzitzeica curves in Minkowski 3-space}. Sci. Manga {bf

} (2009), 44-48.

bibitem{KAO} {sc B. Ki li c{c}, K. Arslan and G. "{O}zt"{u}rk}: textit{Tangentially cubic curves in Euclidean spaces}. Differential

Geometry-Dynamical Systems {bf 10} (2008), 186-196.

bibitem{KL} {sc F. Klein, S. Lie}: textit{Uber diejenigen ebenenen kurven

welche durch ein geschlossenes system von einfach unendlich vielen

vartauschbaren linearen Transformationen in sich "{u}bergehen

}Math. Ann. {bf 4} (1871), 50-84.

bibitem{M} {sc J. Monterde}: textit{Salkowski curves revisited: A family of

curves with constant curvature and non-constant torsion}. Computer

Aided Geometric Design. {bf 26} (2009) 271--278.

bibitem{OAH} {sc G. "{O}zt"{u}rk, K. Arslan and H. Hacisalihou{g}lu}: textit{A

characterization of ccr-curves in $mathbb{R}^{n}$}. Proc. Estonian

Acad. Sciences {bf 57} (2008), 217-224.

bibitem{S} {sc E. Salkowski}: textit{Zur transformation von raumkurven}.

Mathematische Annalen. {bf 66(4)} (1909) 517--557.

bibitem{V} {sc G. E. Vilcu}: textit{A geometric perspective on the generalized

Cobb-Douglas production function}. Appl. Math. Lett. {bf 24}

(2011), 777-783.




DOI: https://doi.org/10.22190/FUMI1803409B

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)