Filomat 2022 Volume 36, Issue 6, Pages: 2051-2062
https://doi.org/10.2298/FIL2206051N
Full text (
433 KB)
Cited by
On non-null relatively normal-slant helices in Minkowski 3-space
Nešović Emilija (Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Kragujevac, Serbia), nesovickg@sbb.rs
Öztürk Ufuk (Department Mathematics, Faculty of Science, University of Çankırı Karatekin, Çnkırı, Turkey), ozturkufuk06@gmail.com, uuzturk@asu.edu
Koç Öztürk Esra Betül (Department of Mathematics, Faculty of Arts and Sciences, Bolu Abant İzzet Baysal University, Bolu, Turkey), e.betul.e@gmail.com, ekocoztu@asu.edu
By using the Darboux frame |ξ, ζ, η| of a non-null curve lying on a
timelike surface in Minkowski 3-space, where ξ is the unit tangent vector of
the curve, η is the unit spacelike normal vector field restricted to the
curve and ζ = ±η × ξ, we define relatively normal-slant helices as the
curves satisfying the condition that the scalar product of the fixed vector
spanning their axis and the non-constant vector field ζ is constant. We give
the necessary and sufficient conditions for non-null curves lying on a
timelike surface to be relatively normal-slant helices. We consider the
special cases when non-null relatively-normal slant helices are geodesic
curves, asymptotic curves, or lines of the principal curvature. We show that
an asymptotic spacelike hyperbolic helix lying on the principal normal
surface over the helix and a geodesic spacelike general helix lying on the
timelike cylindrical ruled surface, are some examples of non-null relatively
normal-slant helices in E31.
Keywords: Slant helix, general helix, Darboux frame, timelike surface, Minkowski space
Show references
A. T. Ali, R. López, Slant helices in Minkowski space E31, J. Korean Math. Soc. 48(1) (2011), 159-167.
A. T. Ali, R. López, M. Turgut, k-type partially null and pseudo null slant helices in Minkowski 4-space, Math. Commun. 17(1) (2012), 93-103.
M. Barros, A. Ferrandez, P. Lucas, M. A. Merono, General helices in the 3-dimensional Lorentzian space forms, Rocky Mountain J. Math. 31 (2001), 373-388.
W. B. Bonnor, Null curves in a Minkowski space-time, Tensor (N.S.) 20(2) (1969), 229-242.
N. Chouaieb, A. Goriely, J. H. Maddocks, Helices, Proc. Natl. Acad. Sci. USA 103(25) (2006), 9398-9403.
F. Doğan, Isophote curves on timelike surfaces in Minkowski 3-space, An. Ştiinţ. Univ. Al. I. Cuza Ias，i. Mat. (N.S.) 63(1) (2012), 133-143.
F. Doğan, Y. Yayli, On isophote curves and their characterizations, Turk. J. Math. 39(5) (2015), 650-664.
M. Ergüt, H. B. Öztekin, S. Aykurt, Non-null k-slant helices and their spherical indicatrices in Minkowski 3-space, J. Adv. Res. Dyn. Control Syst. 2(2) (2010), 1-12.
A. Ferrández, A. Gimnez, P. Lucas, Null generalized helices in Lorentz-Minkowski spaces, Journal of Physics A: Mathematical and General 35(39) (2002), 8243-8251.
I. Gök, Ç. Camcı, H. H. Hacisalihoglu, Vn-slant helices in Euclidean n-space, Math. Commun. 14(2) (2009), 317-329.
T. J. Healey, Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids 7(4) (2002), 405-420.
S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. 28(2) (2004), 153-164.
K. İlarslan, E. Nešović, M. Petrovic-Torgasev, Some characterizations of rectifying curves in the Minkowski 3-space, Novi Sad J. Math. 33(2) (2003), 23-32.
M. J. Keil, J. Rodriguez, Methods for generating compound spring element curves, J. Geom. Graphics 3 (1999), 67-76.
L. Kula, Y. Yaylı, On slant helix and its spherical indicatrix, Appl. Math. Comput. 169(1) (2005), 600-607.
L. Kula, N. Ekmekci, Y. Yaylı, K. İlarslan, Characterizations of slant helices in Euclidean 3-space, Turk. J. Math. 34(2) (2010), 261-273.
W. Kühnel, Differential geometry: curves-surfaces-manifolds, American Mathematical Soc., 2005.
P. Lucas, J. A. Ortega-Yagües, Slant helices in the Euclidean 3-space revisited, Bull. Belg. Math. Soc. Simon Stevin 23(1) (2016), 133-150.
N. Macit, M. Düldül, Relatively normal-slant helices lying on a surface and their characterizations, Hacet. J. Math. Stat. 46(3) (2017), 397-408.
A. Menninger, Characterization of the slant helix as successor curve of the general helix, Int. Electron. J. Geom. 7(2) (2014), 84-91.
E. Nešović, E. Betül Koc， Öztürk, U. Öztürk, k-type null slant helices in Minkowski space-time, Math. Commun. 20(1) (2015), 83-95.
E. Nešović, E. B. Koc， Öztürk, U. Öztürk, k-type null cartan slant helices in Minkowski 3-space, Math. Methods Appl. Sci. 41(17) (2018), 7583-7598.
A. I. Norman, Y. Fei, D. L. Ho, S. C. Greer, Folding and unfolding of polymer helices in solution, Macromolecules 40(7) (2007), 2559-2567.
O. Z. Okuyucu, İ. Gök, Y. Yaylı, N. Ekmekci, Slant helices in three dimensional lie groups, Appl. Math. Comput. 221 (2013), 672-683.
B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
U. Ozturk, E. Nešović, On pseudo null and null cartan darboux helices in Minkowski 3-space, Kuwait J. Sci. 23(2) (2016), 64-82.
G. Öztürk, B. Bulca, B. Bayram, K. Arslan, Focal representation of k-slant Helices in Em+1, Acta Univ. Sapientiae Math. 7(2) (2015), 200-209.
V. L. Sawa, L. Y. Chew, Helicalised fractals, Chaos, Solitons & Fractals 75 (2015), 191-203.
M. Schiffer, A. B. Edmundson, Use of helical wheels to represent the structures of proteins and to identify segments with helical potentials, Biophys. J. 7(2)(1967), 125-135.
D. J. Struik, Differential geometry, Addison-Wesley, Reading, Massachusetts, 1961.
C. D. Toledo-Suarez, On the arithmetic of fractal dimension using hyper helices, Chaos, Solitons Fractals 39 (2009), 342-349.
A. Uçum, Ç. Camcı, K. İlarslan, General helices with timelike slope axis in Minkowski 3-space, Adv. Appl. Clifford Algebr. 26(2) (2016), 793-807.
A. Uçum, K. İlarslan, M. Sakaki, k-type bi-null slant helices in R52, J. Geom. 108(2) (2017), 1-12.
B. Uzunoğlu, İ. Gök, Y. Yaylı, A new approach on curves of constant precession, Appl. Math. Comput. 275 (2016), 317-323.
X. Yang, High accuracy approximation of helices by quintic curve, Comput. Aided Geomet. Design 20 (2003), 303-317.
J. Walrave, Curves and surfaces in Minkowski space, PhD Thesis, Leuven University, Leuven, 1995.