NONE 20260310115740997 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Hassan Alzoubi Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDAN Waseem Almashaleh Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDAN In this paper, we study and define the notion of bundles of lines in the 3-dimensional Euclidean space, we also investigate their essential geometric aspects. Starting from a regular surface, we construct an orthonormal moving frame on the initial surface and employ it to describe the local behavior of associated line bundles also, and derive the fundamental equations of the bundles of lines. Then, we utilize the tools of differential forms to prove some basic theorems relevant to this study. Using the formalism of differential forms and the basic theorems, several foundational theorems related to these bundles are established, providing a unified framework for studying families of lines generated by surfaces which open new perspectives for applications in classical differential geometry and providing also, a geometric framework for further analysis and applications. 03 10 2026 03 10 2026 29 4 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2026.25.4 https://wseas.com/journals/mathematics/2026/a085106-002(2026).pdf H. Hopf, Differential geometry in the large, Lecture notes in mathematics, 2 Eds., Springer, Berlin 1982. M. Hohmeyer, S. Teller, Determining the Lines Through Four Lines, J. Graph. Tools. 4 (3), 11–22. https://doi.org/10.1080/10867651.1999.10487 506. S. Ken, Plücker coordinates tutorial, Ray Tracing News, 11 (1998). A. T. Ricolfi, An Invitation to Modern Enumerative Geometry, SISSA Springer Series V. 3, Springer (2022). C. C. Hsiung, A first course in differential geometry, J. Wiley, N. York, 1981. C. C. Hsiung, Some global theorems in hypersurfaces, Can. J. Math., Vol. 9 (1957), pp 5–15. M. Mhailan, M. Abu Hammad, M. Al Horani, R. Khalil, On fractional vector analysis, J. Math. Comput. Sci. 10, (2020), 2320-2326. D. Abu Jude, M. Abu Hammad, Applications of conformable fractional pareto probability distribution. Int. J. Adv. Soft Comp. App. 14 No. 2 115-124 (2022). S. AlZu'bi, ,M. Elbes, & A. Mughaid, Emotion unveiled: A deep learning odyssey in facial expression analysis for intelligent HCI. Int. J. Adv. Soft Comp. App., 16(2), (2024) 315-333. H. Al-Mimi, N. A. Hamad, M. M. Abualhaj, M. Sh. Daoud, A. Al-dahoud, M. Rasmi, An Enhanced Intrusion Detection System for Protecting HTTP Services from Attacks, Int. J. Adv. Soft Comp. App, 15, 2 (2023), 67-84. doi: 10.15849/IJASCA.230720.05. H. Alzaareer, H. Al-Zoubi, W. Almashaleh, and M. Alsabagh, Quadric Surfaces in Terms of Coordinate Finite II-type, WSEAS Transactions on Mathematics, Vol. 24 (2025), 69–74. https://doi.org/10.37394/23206.2025.24.9. H. Al-Zoubi, T. Hamadneh, and A. Alkhatib, Quadric surfaces of coordinate finite type Gauss map in the Euclidean 3-space, Indian J. Math., Vol. 64 No. 3, (2022), pp 385–399.