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Relatively normal-slant helices lying on a surface and their characterizations

Year 2017, Volume: 46 Issue: 3, 397 - 408, 01.06.2017

Abstract

In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame $\{T,U,V\}$ along the curve,
where $T$ is the unit tangent vector field of the curve, $U$ is the surface normal restricted to the curve and $V=T\times U$. We define a new curve on a surface by using the Darboux frame. This new curve whose vector field V makes a constant angle with a fixed direction is called as
relatively normal-slant helix. We give some characterizations for such curves and obtain their axis. Besides we give some relations between
some special curves (general helices, integral curves, etc.) and relatively normal-slant helices. Moreover, when a regular surface is given
by its implicit or parametric equation, we introduce the method for generating the relatively normal-slant helix with the chosen direction and constant angle on the given surface.

References

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  • Ali AT. New special curves and their spherical indicatrices, Global Journal of Advanced Research on Classical and Modern Geometries 2012; 1: 28-38.
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  • Ali AT, Turgut M. Position vector of a time-like slant helix in Minkowski 3-space, Journal of Mathematical Analysis and Applications 2010; 365 (2), 559-569.
  • Biton YY, Coleman BD, Swigon D. On bifurcations of equilibria of intrinsically curved, electrically charged, rod-like structures that model DNA molecules in solution, Journal of Elasticity 2007; 87 (2), 187-210.
  • Bukcu B, Karacan MK. The slant helices according to Bishop frame, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2009; 3(11), 1010-1013.
  • Doğan F, Yayl Y. On isophote curves and their characterizations, Turkish Journal of Mathematics 2015; 39: 650-664.
  • Gökçelik F, Gök . Null W-slant helices in $E^3_1$ , Journal of Mathematical Analysis and Applications 2014; 420 (1), 222-241.
  • Hananoi S, Ito N, Izumiya S. Spherical Darboux images of curves on surfaces, Beitr Algebra Geom. 2015; 56: 575-585.
  • Izumiya S, Takeuchi N. New special curves and developable surfaces, Turkish Journal of Mathematics 2004; 28: 153163.
  • Izumiya S, Takeuchi N. Generic properties of helices and Bertrand curves, Journal of Geometry 2002; 74: 97109.
  • Kahraman F, Gök ݝ, Hacisalihoğlu HH. On the quaternionic $B_2$ slant helices in the semi- Euclidean space $E^4_2$ , Applied Mathematics and Computation 2012; 218 (11), 6391-6400.
  • Kocayiğit H, Pekacar BB. Characterizations of slant helices according to quaternionic frame, Applied Mathematical Sciences 2013; 7(75), 3739 - 3748.
  • Kula L, Yaylı Y. On slant helix and its spherical indicatrix, Applied Mathematics and Computation 2005; 169: 600607.
  • Kula L, Ekmekci N, Yaylı Y,İlarslan K. Characterizations of slant helices in Euclidean 3-Space, Turkish Journal of Mathematics 2010; 34: 261273.
  • Macit N, Düldül M. Some new associated curve of a Frenet curve in $E^3$ and $E^4$,Turkish Journal of Mathematics 2014; 38: 10231037.
  • Okuyucu OZ, Gök I, Yayli Y, Ekmekci N. Slant helices in three dimensional Lie groups, Applied Mathematics and Computation 2013; 221, 672-683.
  • O'Neill B. Elementary differential geometry, Academic Press, 1966.
  • Özdamar E, Hacsalihoğlu HH. A characterization of inclined curves in Euclidean n-space, Comm. Fac. Sci. Univ. Ankara Sér. A1 Math. 1975; 24(3), 15-23.
  • Puig-Pey J, Galvez A, Iglesias A. Helical curves on surfaces for computer-aided geomet- ric design and manufacturing, Computational science and its applications(ICCSA) 2004. Lecture Notes in Computer Science,Part II, 771-778, Springer.
  • Struik DJ. Lectures on classical dierential geometry, Dover publications, 1961.
  • Şenol A, Zıplar E, Yaylı Y, Gök I. A new approach on helices in Euclidean n-space, Math. Commun. 2013; 18, 241-256.
  • Toledo-Suárez CD. On the arithmetic of fractal dimension using hyperhelices, Chaos, Solitons & Fractals 2009; 39(1), 342349.
  • Yang X. High accuracy approximation of helices by quintic curves, Computer Aided Geometric Design 2003; 20, 303317.
  • Ziplar E, Senol A, Yayli Y. On Darboux Helices in Euclidean 3-Space, Global Journal of Science Frontier Research Mathematics and Decision Sciences 2012; 12(13), 73-80.
Year 2017, Volume: 46 Issue: 3, 397 - 408, 01.06.2017

Abstract

References

  • Ali AT. Position vectors of slant helices in Euclidean Space, Journal of the Egyptian Mathematical Society 2012; 20: 1-6.
  • Ali AT. New special curves and their spherical indicatrices, Global Journal of Advanced Research on Classical and Modern Geometries 2012; 1: 28-38.
  • Ali AT, Turgut M. Some characterizations of slant helices in the Euclidean space $E^n$, Hacettepe Journal of Mathematics and Statistics 2010; 39(3), 327-336.
  • Ali AT, Turgut M. Position vector of a time-like slant helix in Minkowski 3-space, Journal of Mathematical Analysis and Applications 2010; 365 (2), 559-569.
  • Biton YY, Coleman BD, Swigon D. On bifurcations of equilibria of intrinsically curved, electrically charged, rod-like structures that model DNA molecules in solution, Journal of Elasticity 2007; 87 (2), 187-210.
  • Bukcu B, Karacan MK. The slant helices according to Bishop frame, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2009; 3(11), 1010-1013.
  • Doğan F, Yayl Y. On isophote curves and their characterizations, Turkish Journal of Mathematics 2015; 39: 650-664.
  • Gökçelik F, Gök . Null W-slant helices in $E^3_1$ , Journal of Mathematical Analysis and Applications 2014; 420 (1), 222-241.
  • Hananoi S, Ito N, Izumiya S. Spherical Darboux images of curves on surfaces, Beitr Algebra Geom. 2015; 56: 575-585.
  • Izumiya S, Takeuchi N. New special curves and developable surfaces, Turkish Journal of Mathematics 2004; 28: 153163.
  • Izumiya S, Takeuchi N. Generic properties of helices and Bertrand curves, Journal of Geometry 2002; 74: 97109.
  • Kahraman F, Gök ݝ, Hacisalihoğlu HH. On the quaternionic $B_2$ slant helices in the semi- Euclidean space $E^4_2$ , Applied Mathematics and Computation 2012; 218 (11), 6391-6400.
  • Kocayiğit H, Pekacar BB. Characterizations of slant helices according to quaternionic frame, Applied Mathematical Sciences 2013; 7(75), 3739 - 3748.
  • Kula L, Yaylı Y. On slant helix and its spherical indicatrix, Applied Mathematics and Computation 2005; 169: 600607.
  • Kula L, Ekmekci N, Yaylı Y,İlarslan K. Characterizations of slant helices in Euclidean 3-Space, Turkish Journal of Mathematics 2010; 34: 261273.
  • Macit N, Düldül M. Some new associated curve of a Frenet curve in $E^3$ and $E^4$,Turkish Journal of Mathematics 2014; 38: 10231037.
  • Okuyucu OZ, Gök I, Yayli Y, Ekmekci N. Slant helices in three dimensional Lie groups, Applied Mathematics and Computation 2013; 221, 672-683.
  • O'Neill B. Elementary differential geometry, Academic Press, 1966.
  • Özdamar E, Hacsalihoğlu HH. A characterization of inclined curves in Euclidean n-space, Comm. Fac. Sci. Univ. Ankara Sér. A1 Math. 1975; 24(3), 15-23.
  • Puig-Pey J, Galvez A, Iglesias A. Helical curves on surfaces for computer-aided geomet- ric design and manufacturing, Computational science and its applications(ICCSA) 2004. Lecture Notes in Computer Science,Part II, 771-778, Springer.
  • Struik DJ. Lectures on classical dierential geometry, Dover publications, 1961.
  • Şenol A, Zıplar E, Yaylı Y, Gök I. A new approach on helices in Euclidean n-space, Math. Commun. 2013; 18, 241-256.
  • Toledo-Suárez CD. On the arithmetic of fractal dimension using hyperhelices, Chaos, Solitons & Fractals 2009; 39(1), 342349.
  • Yang X. High accuracy approximation of helices by quintic curves, Computer Aided Geometric Design 2003; 20, 303317.
  • Ziplar E, Senol A, Yayli Y. On Darboux Helices in Euclidean 3-Space, Global Journal of Science Frontier Research Mathematics and Decision Sciences 2012; 12(13), 73-80.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Nesibe Macit This is me

Mustafa Düldül

Publication Date June 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 3

Cite

APA Macit, N., & Düldül, M. (2017). Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics, 46(3), 397-408.
AMA Macit N, Düldül M. Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics. June 2017;46(3):397-408.
Chicago Macit, Nesibe, and Mustafa Düldül. “Relatively Normal-Slant Helices Lying on a Surface and Their Characterizations”. Hacettepe Journal of Mathematics and Statistics 46, no. 3 (June 2017): 397-408.
EndNote Macit N, Düldül M (June 1, 2017) Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics 46 3 397–408.
IEEE N. Macit and M. Düldül, “Relatively normal-slant helices lying on a surface and their characterizations”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, pp. 397–408, 2017.
ISNAD Macit, Nesibe - Düldül, Mustafa. “Relatively Normal-Slant Helices Lying on a Surface and Their Characterizations”. Hacettepe Journal of Mathematics and Statistics 46/3 (June 2017), 397-408.
JAMA Macit N, Düldül M. Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics. 2017;46:397–408.
MLA Macit, Nesibe and Mustafa Düldül. “Relatively Normal-Slant Helices Lying on a Surface and Their Characterizations”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, 2017, pp. 397-08.
Vancouver Macit N, Düldül M. Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics. 2017;46(3):397-408.