Ruled Invariants and Ruled Surfaces Created with Spherical Curves in Galilean Space

Year 2026, Volume: 15 Issue: 1 , 394 - 407 , 24.03.2026

Keziban Orbay , Duygu Aydoğan , Tevfik Şahin

https://izlik.org/JA82UR37GX

Abstract

This study aims to examine the ruled invariants of ruled surfaces in three-dimensional Galilean space G_3, where the direction vector is determined by a curve lying on the Galilean central unit sphere. The main goal is to derive ruled invariants of such surfaces by employing a geometric approach rooted in the properties of spherical curves. To achieve this, we first compute the orthonormal frame and corresponding derivative equations of a curve lying on the surface of the Galilean central unit sphere. Then, structure functions and ruled invariants are defined and obtained in Galilean geometry sense. The study covers all three types of ruled surfaces in Galilean space. Additionally, the relationships between the Frenet frames of the curves and those of the ruled surfaces are examined in a systematic manner. The findings provide insight into the intrinsic properties of ruled surfaces and contribute to the broader understanding of geometry in non-Euclidean settings.

Keywords

Galilean space , Galilean sphere , Ruled surface , Ruled invariants

Ethical Statement

The study is complied with research and publication ethics.

Supporting Institution

No supporting institution

References

• D. Capecchi, “The foundation of classical mechanics,” Encyclopedia, vol. 1, no. 2, pp. 482–495, 2021. https://doi.org/10.3390/encyclopedia1020032
• J. Hakim, The Story of Science: Aristotle Leads the Way. Washington and New York: Smithsonian Books, 2004.
• J. Hakim, The Story of Science: Einstein Adds a New Dimension. Washington and New York: Smithsonian Books, 2007.
• A. N. Bernal and M. Sanchez, “Leibnizian, Galilean and Newtonian structures of spacetime,” Journal of Mathematical Physics, vol. 44, no. 3, pp. 1129–1149, 2003. https://doi.org/10.1063/1.1540724
• I. M. Yaglom, A Simple Non-Euclidean Geometry and Physical Basis. New York: Springer Verlag, 1979.
• O. Röschel, Die Geometrie Des Galileischen Raumes. Habilitationsschrift, Leoben, 1984.
• T. Şahin, F. Karakuş and K. Orbay, “Parallel transport with respect to Frenet and Darboux frames in the Galilean space,” Journal of Science and Arts, vol. 1, no. 50, pp. 13–24, 2020.
• Z. Erjavec, “On generalization of helices in the Galilean and the pseudo-Galilean space,” Journal of Mathematics Research, vol. 6, no. 3, pp. 39–46, 2014. https://doi.org/10.5539/jmr.v6n3p39
• M. Dede, C. Ekici and A. C. Çöken, “On the parallel surfaces in Galilean space,” Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 6, pp. 605–615, 2013.
• M. P. do Carmo, Differential Geometry of Curves and Surfaces. New Jersey: Prentice-Hall, 1976.
• M. Dede, “Tubular surfaces in Galilean space,” Mathematical Communications, vol. 18, no. 1, pp. 209–217, 2013.
• Z. M. Sipus and B. Divjak, “Translation surfaces in the Galilean space,” Glasnik Matematicki, vol. 46, no. 2, pp. 457–471, 2011. https://doi.org/10.3336/gm.46.2.14
• H. Liu and Y. Yuan, “Pitch functions of ruled surfaces and b-scrolls in Minkowski 3-space,” Journal of Geometry and Physics, vol. 62, no. 1, pp. 47–52, 2012. https://doi.org/10.1016/j.geomphys.2011.09.004
• Y. Yu, H. Liu and S. D. Jung, “Structure and characterization of ruled surfaces in Euclidean 3-space,” Applied Mathematics and Computation, vol. 233, pp. 252–259, 2014. https://doi.org/10.1016/j.amc.2014.01.010
• H. Liu, Y. Liu and S. D. Jung, “Ruled invariants and associated ruled surfaces of a space curve,” Applied Mathematics and Computation, vol. 348, pp. 479–486, 2019. https://doi.org/10.1016/j.amc.2018.11.091
• H. Liu, Y. Yu and S. D. Jung, “Invariants of non-developable ruled surfaces in Euclidean 3-space,” Beiträge zur Algebra und Geometrie, vol. 55, pp. 189–199, 2014. https://doi.org/10.1007/s13366-013-0167-1
• O. Giering, “Kinematik und Differentialgeometrie der Regelflächen,” Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, vol. 12, pp. 195–224, 1960.
• H. Liu, Y. Liu and S. D. Jung, “Ruled invariants and total classification of non-developable ruled surfaces,” Journal of Geometry, vol. 113, pp. 1–13, 2022. https://doi.org/10.1007/s00022-022-00642-w
• B. J. Pavković and I. Kamenarović, “The equiform differential geometry of curves in the Galilean space G3,” Glasnik Mathematics, vol. 22 (42), pp. 449–457, 1987.
• B. Divjak and Z. M. Sipus, “Special curves on the ruled surfaces in Galilean and pseudo-Galilean spaces,” Acta Mathematica Hungarica, vol. 98, no. 3, pp. 203–215, 2003. https://doi.org/10.1023/A:1016334524962