Research Article
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Singularities of the Ruled Surfaces According to RM Frame and Natural Lift Curves

Year 2022, Volume: 43 Issue: 2, 308 - 315, 29.06.2022

Emel Karaca

https://doi.org/10.17776/csj.1057212
https://izlik.org/JA49KA38XC

Abstract

In this study, the ruled surface generated by the natural lift curve in IR^3 is obtained by using the isomorphism between unit dual sphere, DS^2 and the subset of the tangent bundle of unit 2-sphere, T\bar{M}. Then, exploitting E. Study mapping and the isomorphism mentioned below, each natural lift curve on T\bar{M} is corresponded to the ruled surface in IR^3. Moreover, the singularities of this ruled surface are examined according to RM vectors and these ruled surfaces have been classified. Some examples are given to support the main results.

Keywords

Ruled surface Dual space Natural lift curve RM vectors

References

  • [1] Wang W., Juttler B., Zheng D., Liu Y., Computation of Rotation Minimizing Frame, ACM Trans. Graph., 27(1) (2008) 1-18.
  • [2] Bishop R.L., There is More Than One Way to Frame A Curve, Amer. Math. Monthly, 8(246) (1975) 1, 2.
  • [3] Do Carmo M.P., Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs, NJ, (1976).
  • [4] Dede M., Why Flc- Frame is Better than Frenet Frame? DOI: 10.13140/RG.2.2.23418.24002 (2019).
  • [5] Ravani R., Meghdari A., Ravani B., Rational Frenet-Serret Curves and Rotation Minimizing Frames in Spatial Motion Design, IEEE International Conference on Intelligent Engineering Systems, INES (2004) 186-192.
  • [6] Wang W., Joe B., Robust Computation of The Rotation Minimizing Frame for Sweep Surface Modeling, Computer Aided Design, 29(5) (1997) 379-391.
  • [7] Farouki R.T., Rational Rotation-Minimizing Frames Recent Advances and Open Problems, Applied Mathematics and Computation, 272(1) (2016) 80-91.
  • [8] Thorpe J.A., Elementary Topics in Differential Geometry. New York, HeidelbergBerlin: Springer Verlag, (1979).
  • [9] Ergün E., Çalışkan M., On Natural Lift of A Curve, Pure Mathematical Sciences, 2 (2012) 81-85.
  • [10] Ergün E., Bilici M., Çalışkan M., The Frenet Vector Fields and The Curvatures of The Natural Lift Curve, The Bulletin Society for Mathematical Services and Standarts, 2 (2012) 38-43.
  • [11] Mert T., Timelike Ruled Surface in de-Sitter 3-Space, Turkish Journal of Mathematics and Computer Science, 2(12) (2020) 166-175.
  • [12] Mert T., Atçeken M., Spacelike Ruled Surface in de-Sitter 3-Space, Asian Journal of Mathematics and Computer Research, 27(4) (2020) 37-53.
  • [13] Mert T., Atçeken M., Special Ruled Surfaces in de-Sitter 3-Space, Fundamental Journal of Mathematics and Applications, 4(3) (2021) 195-209.
  • [14] Altın M., Kazan A., Karadağ H.B., Ruled Surfaces Constructed by Planar Curves in Euclidean 3-Space with Density, Celal Bayar University Journal of Science, 16(1) (2020) 81-88.
  • [15] Altın M., Kazan A., Yoon Woon D., 2-Ruled hypersurfaces in Euclidean 4-space, Journal of Geometry and Physics, 166 (2021) 1-13.
  • [16] Fischer I.S., Dual-Number Methods in Kinematics, Statics and Dynamics. Boca Raton, London, New York, Washington DC: CRC Press, (1999).
  • [17] Karakaş B., Gündoğan H., A Relation among and Non-Cylindirical Ruled Surfaces, Mathematical Communications, 8 (2003) 9-14.
  • [18] Hathout F., Bekar M., Yaylı Y., Ruled Surfaces and Tangent Bundle of Unit 2-Sphere, Int. J. of Geo. M. M. Phy., 14(10) (2017).
  • [19] Karaca E., Çalışkan M., Ruled Surfaces and Tangent Bundle of Unit 2-Sphere of Natural Lift Curves, Gazi University Journal of Science, 33(5) (2020) 751-759.
  • [20] Bruce J.W., Giblin P.J., Curves and Singularities. 2nd ed. Cambridge: Cambridge Univ. Press, (1992) 1, 2,5.
  • [21] Izumiya S., Takeuchi N., New Special Curves and Developable Surfaces, Turk J Math, 28 (2004) 153-163.
  • [22] Bekar M., Hathout F., Yaylı Y., Legendre Curves and The Singularities of Ruled Surfaces Obtained by Using Rotation Minimizing Frame, Ukranian Mathematical Journal, 5(73) (2021) 589-601.

Year 2022, Volume: 43 Issue: 2, 308 - 315, 29.06.2022

Emel Karaca

https://doi.org/10.17776/csj.1057212
https://izlik.org/JA49KA38XC

Abstract

References

  • [1] Wang W., Juttler B., Zheng D., Liu Y., Computation of Rotation Minimizing Frame, ACM Trans. Graph., 27(1) (2008) 1-18.
  • [2] Bishop R.L., There is More Than One Way to Frame A Curve, Amer. Math. Monthly, 8(246) (1975) 1, 2.
  • [3] Do Carmo M.P., Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs, NJ, (1976).
  • [4] Dede M., Why Flc- Frame is Better than Frenet Frame? DOI: 10.13140/RG.2.2.23418.24002 (2019).
  • [5] Ravani R., Meghdari A., Ravani B., Rational Frenet-Serret Curves and Rotation Minimizing Frames in Spatial Motion Design, IEEE International Conference on Intelligent Engineering Systems, INES (2004) 186-192.
  • [6] Wang W., Joe B., Robust Computation of The Rotation Minimizing Frame for Sweep Surface Modeling, Computer Aided Design, 29(5) (1997) 379-391.
  • [7] Farouki R.T., Rational Rotation-Minimizing Frames Recent Advances and Open Problems, Applied Mathematics and Computation, 272(1) (2016) 80-91.
  • [8] Thorpe J.A., Elementary Topics in Differential Geometry. New York, HeidelbergBerlin: Springer Verlag, (1979).
  • [9] Ergün E., Çalışkan M., On Natural Lift of A Curve, Pure Mathematical Sciences, 2 (2012) 81-85.
  • [10] Ergün E., Bilici M., Çalışkan M., The Frenet Vector Fields and The Curvatures of The Natural Lift Curve, The Bulletin Society for Mathematical Services and Standarts, 2 (2012) 38-43.
  • [11] Mert T., Timelike Ruled Surface in de-Sitter 3-Space, Turkish Journal of Mathematics and Computer Science, 2(12) (2020) 166-175.
  • [12] Mert T., Atçeken M., Spacelike Ruled Surface in de-Sitter 3-Space, Asian Journal of Mathematics and Computer Research, 27(4) (2020) 37-53.
  • [13] Mert T., Atçeken M., Special Ruled Surfaces in de-Sitter 3-Space, Fundamental Journal of Mathematics and Applications, 4(3) (2021) 195-209.
  • [14] Altın M., Kazan A., Karadağ H.B., Ruled Surfaces Constructed by Planar Curves in Euclidean 3-Space with Density, Celal Bayar University Journal of Science, 16(1) (2020) 81-88.
  • [15] Altın M., Kazan A., Yoon Woon D., 2-Ruled hypersurfaces in Euclidean 4-space, Journal of Geometry and Physics, 166 (2021) 1-13.
  • [16] Fischer I.S., Dual-Number Methods in Kinematics, Statics and Dynamics. Boca Raton, London, New York, Washington DC: CRC Press, (1999).
  • [17] Karakaş B., Gündoğan H., A Relation among and Non-Cylindirical Ruled Surfaces, Mathematical Communications, 8 (2003) 9-14.
  • [18] Hathout F., Bekar M., Yaylı Y., Ruled Surfaces and Tangent Bundle of Unit 2-Sphere, Int. J. of Geo. M. M. Phy., 14(10) (2017).
  • [19] Karaca E., Çalışkan M., Ruled Surfaces and Tangent Bundle of Unit 2-Sphere of Natural Lift Curves, Gazi University Journal of Science, 33(5) (2020) 751-759.
  • [20] Bruce J.W., Giblin P.J., Curves and Singularities. 2nd ed. Cambridge: Cambridge Univ. Press, (1992) 1, 2,5.
  • [21] Izumiya S., Takeuchi N., New Special Curves and Developable Surfaces, Turk J Math, 28 (2004) 153-163.
  • [22] Bekar M., Hathout F., Yaylı Y., Legendre Curves and The Singularities of Ruled Surfaces Obtained by Using Rotation Minimizing Frame, Ukranian Mathematical Journal, 5(73) (2021) 589-601.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Emel Karaca 0000-0003-0703-939X

Submission Date January 13, 2022
Acceptance Date May 5, 2022
Publication Date June 29, 2022
DOI https://doi.org/10.17776/csj.1057212
IZ https://izlik.org/JA49KA38XC
Published in Issue Year 2022 Volume: 43 Issue: 2

Cite

APA Karaca, E. (2022). Singularities of the Ruled Surfaces According to RM Frame and Natural Lift Curves. Cumhuriyet Science Journal, 43(2), 308-315. https://doi.org/10.17776/csj.1057212

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