Research article Special Issues

Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean $ 3 $-space

  • Received: 10 February 2023 Revised: 24 March 2023 Accepted: 30 March 2023 Published: 12 April 2023
  • MSC : 53A04, 53A05, 53A17

  • In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding.

    Citation: Yanlin Li, Ali. H. Alkhaldi, Akram Ali, R. A. Abdel-Baky, M. Khalifa Saad. Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean $ 3 $-space[J]. AIMS Mathematics, 2023, 8(6): 13875-13888. doi: 10.3934/math.2023709

    Related Papers:

  • In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding.



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    [1] M. Khalifa Saad, R. Abdel-Baky, On ruled surfaces according to quasi-frame in Euclidean 3-space, Aust. J. Math. Anal. Appl., 17 (2020), 11.
    [2] R. Abdel-Baky, M. Khalifa Saad, Osculating surfaces along a curve on a surface in Euclidean 3-space, Journal of Mathematical and Computational Science, 12 (2022), 84.
    [3] R. Abdel-Baky, M. Khalifa Saad, Singularities of non-developable ruled surface with space-like ruling, Symmetry, 14 (2022), 716. http://dx.doi.org/10.3390/sym14040716 doi: 10.3390/sym14040716
    [4] V. Arnol'd, S. Gusein-Zade, A. Varchenko, Singularities of differentiable maps, Boston: Birkhäuser, 1988. http://dx.doi.org/10.1007/978-1-4612-3940-6
    [5] J. Bruce, On singularities, envelopes and elementary differential geometry, Math. Proc. Cambridge, 89 (1981), 43–48. http://dx.doi.org/10.1017/S0305004100057935 doi: 10.1017/S0305004100057935
    [6] J. Bruce, P. Giblin, Generic geometry, Am. Math. Mon., 90 (1983), 529–545. http://dx.doi.org/10.1080/00029890.1983.11971276
    [7] J. Bruce, P. Giblin, Curves and singularities, 2 Eds., Cambridge: Cambridge University Press, 1992.
    [8] M. Do Carmo, Differential geometry of curves and surfaces, New Jersey: Prentice-Hall, 1976.
    [9] D. Mond, Singularities of the tangent developable surface of a space curve, Quart. J. Math., 40 (1989), 79–91. http://dx.doi.org/10.1093/qmath/40.1.79 doi: 10.1093/qmath/40.1.79
    [10] H. Pottmann, M. Hofer, Geometry of the squared-distance functions to curves and surfaces, In: Visualization and mathematics III, Berlin: Springer, 2003,221–242. http://dx.doi.org/10.1007/978-3-662-05105-4_12
    [11] S. Izumiya, N. Takeuchi, Special curves and ruled surfaces, Beitr. Algebr. Geom., 44 (2003), 203–212.
    [12] M. Aldossary, R. Abdel-Baky, On the Bertrand offsets for ruled and developable surfaces, Boll. Unione. Mat. Ital., 8 (2015), 53–64. http://dx.doi.org/10.1007/s40574-015-0025-1 doi: 10.1007/s40574-015-0025-1
    [13] S. Izumiya, N. Takeuchi, Geometry of ruled surfaces, Proceedings of Applicable Mathematics in the Golden Age, 2003,305–338.
    [14] Y. Li, S. Liu, Z. Wang, Tangent developables and Darboux developables of framed curves, Topol. Appl., 301 (2020), 107526. http://dx.doi.org/10.1016/j.topol.2020.107526 doi: 10.1016/j.topol.2020.107526
    [15] Y. Li, K. Eren, K. Ayvacı, S. Ersoy, The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, AIMS Mathematics, 8 (2023), 2226–2239. http://dx.doi.org/10.3934/math.2023115 doi: 10.3934/math.2023115
    [16] Y. Li, Z. Chen, S. Nazra, R. Abdel-Baky, Singularities for timelike developable surfaces in Minkowski 3-space, Symmetry, 15 (2023), 277. http://dx.doi.org/10.3390/sym15020277 doi: 10.3390/sym15020277
    [17] Y. Li, M. Aldossary, R. Abdel-Baky, Spacelike circular surfaces in Minkowski 3-space, Symmetry, 15 (2023), 173. http://dx.doi.org/10.3390/sym15010173 doi: 10.3390/sym15010173
    [18] Y. Li, A. Abdel-Salam, M. Khalifa Saad, Primitivoids of curves in Minkowski plane, AIMS Mathematics, 8 (2023), 2386–2406. http://dx.doi.org/10.3934/math.2023123 doi: 10.3934/math.2023123
    [19] Y. Li, O. Tuncer, On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space, Math. Method. Appl. Sci., in press. http://dx.doi.org/10.1002/mma.9173
    [20] Y. Li, M. Erdoğdu, A. Yavuz, Differential geometric approach of Betchow-Da Rios soliton equation, Hacet. J. Math. Stat., 52 (2023), 114–125. http://dx.doi.org/10.15672/hujms.1052831 doi: 10.15672/hujms.1052831
    [21] Y. Li, A. Abolarinwa, A. Alkhaldi, A. Ali, Some inequalities of Hardy type related to Witten-Laplace operator on smooth metric measure spaces, Mathematics, 10 (2022), 4580. http://dx.doi.org/10.3390/math10234580 doi: 10.3390/math10234580
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