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

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.

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

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 Natural Sciences
Authors

Emel Karaca 0000-0003-0703-939X

Publication Date June 29, 2022
Submission Date January 13, 2022
Acceptance Date May 5, 2022
Published in Issue Year 2022Volume: 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