Lorentzian construction of embankment-type ruled surfaces using the orthogonal modified frame in Minkowski 3-space

Mona Bin-Asfour; Ghaliah Alhamzi; Emad Solouma; Sayed Saber; Mona Bin-Asfour; Ghaliah Alhamzi; Emad Solouma; Sayed Saber

doi:10.3934/math.2026497

AIMS Mathematics

2026, Volume 11, Issue 4: 12108-12131. doi: 10.3934/math.2026497

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Research article

Lorentzian construction of embankment-type ruled surfaces using the orthogonal modified frame in Minkowski 3-space

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, 11623, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65779, Saudi Arabia

Received: 22 January 2026 Revised: 14 April 2026 Accepted: 17 April 2026 Published: 29 April 2026
MSC : 53A35, 53B30, 65D18

This study presents a new geometric formulation for constructing embankment–type ruled surfaces in the Lorentzian framework of Minkowski 3-space $ \mathrm{E}^3_1 $, by means of the orthogonal modified frame (OMF). The OMF serves as an orthogonal moving frame that is fully compatible with the Minkowski metric, providing a consistent representation of the causal behavior of both spacelike and timelike curves. Within this framework, three distinct surface families are established namely, the OMF embankment surface, the OMF embankment–like surface, and the OMF tubembankment–like surface. Each class is developed together with its explicit parametric expression and associated geometric invariants. The corresponding first and second fundamental forms are derived, from which closed analytical expressions for the Gaussian curvature and mean curvature are obtained. These computations lead to precise differential conditions governing the developability and minimality of the generated surfaces. Representative examples illustrate the smooth curvature distribution and the Lorentzian metric consistency of the OMF–based models, demonstrating clear advantages over traditional Euclidean constructions. Overall, the proposed formulation offers an efficient and unified framework for analyzing and modeling ruled surfaces in Lorentzian geometry, with prospective applications in relativistic motion theory, geometric design, and computer-aided kinematic simulation.
- orthogonal modified frame,
- Minkowski 3-space,
- ruled surfaces,
- embankment surfaces,
- Lorentzian geometry,
- Gaussian curvature
Citation: Mona Bin-Asfour, Ghaliah Alhamzi, Emad Solouma, Sayed Saber. Lorentzian construction of embankment-type ruled surfaces using the orthogonal modified frame in Minkowski 3-space[J]. AIMS Mathematics, 2026, 11(4): 12108-12131. doi: 10.3934/math.2026497

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Abstract

This study presents a new geometric formulation for constructing embankment–type ruled surfaces in the Lorentzian framework of Minkowski 3-space $ \mathrm{E}^3_1 $, by means of the orthogonal modified frame (OMF). The OMF serves as an orthogonal moving frame that is fully compatible with the Minkowski metric, providing a consistent representation of the causal behavior of both spacelike and timelike curves. Within this framework, three distinct surface families are established namely, the OMF embankment surface, the OMF embankment–like surface, and the OMF tubembankment–like surface. Each class is developed together with its explicit parametric expression and associated geometric invariants. The corresponding first and second fundamental forms are derived, from which closed analytical expressions for the Gaussian curvature and mean curvature are obtained. These computations lead to precise differential conditions governing the developability and minimality of the generated surfaces. Representative examples illustrate the smooth curvature distribution and the Lorentzian metric consistency of the OMF–based models, demonstrating clear advantages over traditional Euclidean constructions. Overall, the proposed formulation offers an efficient and unified framework for analyzing and modeling ruled surfaces in Lorentzian geometry, with prospective applications in relativistic motion theory, geometric design, and computer-aided kinematic simulation.

References

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