Research Article
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AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3

Year 2018, Volume: 22 Issue: 6, 1863 - 1867, 01.12.2018

İlim Kişi

https://doi.org/10.16984/saufenbilder.434801

Abstract

We deal with AW(k)-type  Salkowski
(anti-Salkowski) curves with constant   in the Euclidean
3-space. We show that there is no AW(1)-type Salkowski curve and AW(1)-type
anti-Salkowski curve in . Also, we handle weak AW(2)-type and weak AW(3)-type
Salkowski (anti-Salkowski) curves. Also, we show that there is no weak AW(2)-type
Salkowski curve in .

Keywords

AW(k)-type Salkowski Curve

References

  • [1] K. Arslan and A. West, “Product submanifols with pointwise 3-planar normal sections,” Glasgow Math. J., vol.37, pp. 73-81, 1995.
  • [2] K. Arslan and C. Özgür, “Curves and surfaces of AW(k) type,” Geometry and topology of Submanifolds IX, World Scientific, 21-26, 1997.
  • [3] C. Özgür and F. Gezgin,.”On some curves of AW(k)-type,”. Differential Geometry-Dynamical Systems, vol. 7, pp. 74.80, 2005.
  • [4] B. Kılıç and K. Arslan, “On curves and surfaces of AW(k)-type,” BAÜ Fen Bil. Enst. Dergisi, vol. 6, no. 1, pp. 52-61, 2004.
  • [5] I. Kişi, S. Büyükkütük, Deepmala, and G. Öztürk, “AW(k)-type curves according to parallel transport frame in Euclidean space ,” Facta Universitaties, Series: Mathematics and Informatics, vol. 31, no. 4, pp. 885-905, 2016.
  • [6] I. Kişi and G. Öztürk, “AW(k)-type curves according to the Bishop frame,” arXiv:1305.3381v1 [math.DG], 2013.
  • [7] D. W. Yoon, “General helices of Aw(k)-type in the Lie group,” J. Appl. Math., pp.1-10, 2012.
  • [8] N. Chouaieb, A. Goriely, and J.H. Maddocks, “Helices,” PANS, 103, 9398.9403, 2006.
  • [9] A. A. Lucas and P. Lambin, “Diffraction by DNA, carbon nanotubes and other helical nanostructures,” Rep. Prog. Phys., vol. 68, 1181.1249, 2005.
  • [10] C. D. Toledo-Suarez, “On the arithmetic of fractal dimension using hyperhelices,” Chaos Solitons and Fractals, vol. 39, pp. 342.349, 2009.
  • [11] X. Yang, “High accuracy approximation of helices by quintic curve,” Comput. Aided Geometric Design, vol. 20, pp. 303.317, 2003.
  • [12] A. Gray, “Modern differential geometry of curves and surface,” CRS Press, Inc., 1993.
  • [13] H. Gluck, “Higher curvatures of curves in Euclidean space,” Amer. Math. Monthly, vol. 73, pp. 699-704, 1966.
  • [14] F. Klein and S. Lie, “Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren lin-earen transformationen in sich übergehen,” Math. Ann,. vol. 4, pp. 50-84, 1871.
  • [15] J. Monterde, “Curves with constant curvature ratios,” Bulletin of Mexican Mathematic Society,” vol. 13, pp. 177-186, 2007.
  • [16] G. Öztürk, K. Arslan, and H.H. Hacisalihoğlu, “A characterization of ccr-curves in ,” Proc. Estonian Acad. Sciences, vol. 57, pp. 217-224, 2008.
  • [17] E. Salkowski, “Zur transformation von raumkurven,” Math. Ann., vol. 66, pp. 517-557, 1909.
  • [18] A.T. Ali, “Spacelike Salkowski and anti-Salkowski curves with spacelike principal normal in Minkowski 3-space,” Int. J. Open Problems Comp. Math., vol. 2, pp. 451.460, 2009.
  • [19] A.T. Ali, “Timelike Salkowski and anti-Salkowski curves in Minkowski 3-space,” J. Adv. Res. Dyn. Cont. Syst., vol. 2, pp. 17.26, 2010.
  • [20] F. Kaymaz and F. K. Aksoyak, “Some special curves and Manheim curves in three dimensional Euclidean space,” Mathematical Sciences and applications E-Notes, vol. 5, no. 1, pp. 34-39, 2017.
  • [21] J. Monterde, “Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion,” Computer Aided Geometric Design, vol. 26, no. 3, pp. 271-278, 2009.
  • [22] M. P. Do Carmo, “Differential geometry of curves and surfaces”, PrenticeHall, Englewood Cliffs, N. J. 1976.

Year 2018, Volume: 22 Issue: 6, 1863 - 1867, 01.12.2018

İlim Kişi

https://doi.org/10.16984/saufenbilder.434801

Abstract

References

  • [1] K. Arslan and A. West, “Product submanifols with pointwise 3-planar normal sections,” Glasgow Math. J., vol.37, pp. 73-81, 1995.
  • [2] K. Arslan and C. Özgür, “Curves and surfaces of AW(k) type,” Geometry and topology of Submanifolds IX, World Scientific, 21-26, 1997.
  • [3] C. Özgür and F. Gezgin,.”On some curves of AW(k)-type,”. Differential Geometry-Dynamical Systems, vol. 7, pp. 74.80, 2005.
  • [4] B. Kılıç and K. Arslan, “On curves and surfaces of AW(k)-type,” BAÜ Fen Bil. Enst. Dergisi, vol. 6, no. 1, pp. 52-61, 2004.
  • [5] I. Kişi, S. Büyükkütük, Deepmala, and G. Öztürk, “AW(k)-type curves according to parallel transport frame in Euclidean space ,” Facta Universitaties, Series: Mathematics and Informatics, vol. 31, no. 4, pp. 885-905, 2016.
  • [6] I. Kişi and G. Öztürk, “AW(k)-type curves according to the Bishop frame,” arXiv:1305.3381v1 [math.DG], 2013.
  • [7] D. W. Yoon, “General helices of Aw(k)-type in the Lie group,” J. Appl. Math., pp.1-10, 2012.
  • [8] N. Chouaieb, A. Goriely, and J.H. Maddocks, “Helices,” PANS, 103, 9398.9403, 2006.
  • [9] A. A. Lucas and P. Lambin, “Diffraction by DNA, carbon nanotubes and other helical nanostructures,” Rep. Prog. Phys., vol. 68, 1181.1249, 2005.
  • [10] C. D. Toledo-Suarez, “On the arithmetic of fractal dimension using hyperhelices,” Chaos Solitons and Fractals, vol. 39, pp. 342.349, 2009.
  • [11] X. Yang, “High accuracy approximation of helices by quintic curve,” Comput. Aided Geometric Design, vol. 20, pp. 303.317, 2003.
  • [12] A. Gray, “Modern differential geometry of curves and surface,” CRS Press, Inc., 1993.
  • [13] H. Gluck, “Higher curvatures of curves in Euclidean space,” Amer. Math. Monthly, vol. 73, pp. 699-704, 1966.
  • [14] F. Klein and S. Lie, “Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren lin-earen transformationen in sich übergehen,” Math. Ann,. vol. 4, pp. 50-84, 1871.
  • [15] J. Monterde, “Curves with constant curvature ratios,” Bulletin of Mexican Mathematic Society,” vol. 13, pp. 177-186, 2007.
  • [16] G. Öztürk, K. Arslan, and H.H. Hacisalihoğlu, “A characterization of ccr-curves in ,” Proc. Estonian Acad. Sciences, vol. 57, pp. 217-224, 2008.
  • [17] E. Salkowski, “Zur transformation von raumkurven,” Math. Ann., vol. 66, pp. 517-557, 1909.
  • [18] A.T. Ali, “Spacelike Salkowski and anti-Salkowski curves with spacelike principal normal in Minkowski 3-space,” Int. J. Open Problems Comp. Math., vol. 2, pp. 451.460, 2009.
  • [19] A.T. Ali, “Timelike Salkowski and anti-Salkowski curves in Minkowski 3-space,” J. Adv. Res. Dyn. Cont. Syst., vol. 2, pp. 17.26, 2010.
  • [20] F. Kaymaz and F. K. Aksoyak, “Some special curves and Manheim curves in three dimensional Euclidean space,” Mathematical Sciences and applications E-Notes, vol. 5, no. 1, pp. 34-39, 2017.
  • [21] J. Monterde, “Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion,” Computer Aided Geometric Design, vol. 26, no. 3, pp. 271-278, 2009.
  • [22] M. P. Do Carmo, “Differential geometry of curves and surfaces”, PrenticeHall, Englewood Cliffs, N. J. 1976.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

İlim Kişi 0000-0003-2557-9023

Publication Date December 1, 2018
Submission Date June 19, 2018
Acceptance Date September 5, 2018
Published in Issue Year 2018 Volume: 22 Issue: 6

Cite

APA Kişi, İ. (2018). AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3. Sakarya University Journal of Science, 22(6), 1863-1867. https://doi.org/10.16984/saufenbilder.434801
AMA Kişi İ. AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3. SAUJS. December 2018;22(6):1863-1867. doi:10.16984/saufenbilder.434801
Chicago Kişi, İlim. “AW(k)-Type Salkowski Curves in Euclidean 3-Space IE^3”. Sakarya University Journal of Science 22, no. 6 (December 2018): 1863-67. https://doi.org/10.16984/saufenbilder.434801.
EndNote Kişi İ (December 1, 2018) AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3. Sakarya University Journal of Science 22 6 1863–1867.
IEEE İ. Kişi, “AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3”, SAUJS, vol. 22, no. 6, pp. 1863–1867, 2018, doi: 10.16984/saufenbilder.434801.
ISNAD Kişi, İlim. “AW(k)-Type Salkowski Curves in Euclidean 3-Space IE^3”. Sakarya University Journal of Science 22/6 (December 2018), 1863-1867. https://doi.org/10.16984/saufenbilder.434801.
JAMA Kişi İ. AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3. SAUJS. 2018;22:1863–1867.
MLA Kişi, İlim. “AW(k)-Type Salkowski Curves in Euclidean 3-Space IE^3”. Sakarya University Journal of Science, vol. 22, no. 6, 2018, pp. 1863-7, doi:10.16984/saufenbilder.434801.
Vancouver Kişi İ. AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3. SAUJS. 2018;22(6):1863-7.
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