Hypersurfaces With a Common Geodesic Curve in 4D Euclidean space E4

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Sahar H. Nazra


In this paper, we attain the problem of constructing hypersurfaces from a given geodesic curve in 4D Euclidean space E4. Using the Serret–Frenet frame of the given geodesic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be geodesic. We illustrate this method by presenting some examples.

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  1. R.A. Abdel-Baky, N. Alluhaibi, Surfaces Family With a Common Geodesic Curve in Euclidean 3-Space E3, Int. J. Math. Anal. 13 (2019), 433–447. https://doi.org/10.12988/ijma.2019.9846.
  2. R.A. Al-Ghefari, R.A. Abdel-Baky, An Approach for Designing a Developable Surface With a Common Geodesic Curve, Int. J. Contemp. Math. Sci. 8 (2013), 875–891. https://doi.org/10.12988/ijcms.2013.39101.
  3. M. Altin, A. Kazan, H.B. Karadag, Hypersurface Families With Smarandache Curves in Galilean 4-Space, Commun. Fac. Sci. Univ. Ankara Ser. A1. Math. Stat. 70 (2021), 744–761. https://doi.org/10.31801/cfsuasmas.794779.
  4. G.S. Atalay, F. Guler, E. Bayram, E. Kasap, An Approach for Designing a Surface Pencil Through a Given Geodesic Curve, (2015). http://arxiv.org/abs/1406.0618.
  5. E. Bayram, E. Kasap, Parametric Representation of a Hypersurface Family With a Common Spatial Geodesic, (2014). http://arxiv.org/abs/1305.0411.
  6. E. Bayram, E. Kasap, Hypersurface Family with a Common Isoasymptotic Curve, Geometry. 2014 (2014), 623408. https://doi.org/10.1155/2014/623408.
  7. M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, 1976.
  8. G. Farin, Curves and Surfaces for Computer Aided Geometric Design, 2nd ed., Academic Press, New York, 1990.
  9. J. Hoschek, D. Lasser, Fundamentals of Computer Aided Geometric Design, A.K. Peters, Wellesley, MA, 1993.
  10. J. Zhou, Visualization of Four-Dimensional Space and Its Applications, Ph.D. Thesis, Purdue University, 1991.
  11. E. Kasap, F.T. Akyildiz, Surfaces With Common Geodesic in Minkowski 3-Space, Appl. Math. Comput. 177 (2006), 260–270. https://doi.org/10.1016/j.amc.2005.11.005.
  12. E. Kasap, Family of Surface With a Common Null Geodesic, Int. J. Phys. Sci. 4 (2009), 428-433.
  13. R. Makki, Hypersurfaces With a Common Asymptotic Curve in the 4D Galilean Space G4, Asian-Eur. J. Math. 15 (2022), 2250199. https://doi.org/10.1142/s1793557122501996.
  14. G.J. Wang, K. Tang, C.L. Tai, Parametric Representation of a Surface Pencil With a Common Spatial Geodesic, Computer-Aided Design. 36 (2004), 447–459. https://doi.org/10.1016/s0010-4485(03)00117-9.
  15. D.W. Yoon, Z.K. Yuzbas, An Approach for Curve in the 4D Galilean Space G4, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 25 (2018), 229-241.
  16. Z.K. Yuzbas, M. Bektas, On the Construction of a Surface Family With Common Geodesic in Galilean Space G3. Open Phys. 14 (2016), 360-363. https://doi.org/10.1515/phys-2016-0041.
  17. Z.K. Yuzbas, D.W. Yoon, On Constructions of Surfaces Using a Geodesic in Lie Group, J. Geom. 110 (2019), 29. https://doi.org/10.1007/s00022-019-0487-x.