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RESEARCH ARTICLE

Geometric analysis of ruled surfaces constructed from integral curves in three-dimensional Euclidean space

Ayman Elsharkawy1* ,  Hasnaa Baizeed2 ,  Clemente Cesarano3 ,  SeyedehFahimeh Hashemi3
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1 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Gharbia, Egypt
2 Department of Mathematics, Faculty of Science, Beni Suef University, Beni Suef, Egypt
3 Section of Mathematics, International Telematic University Uninettuno, Roma, Italy
IJOCTA, 025450196 https://doi.org/10.36922/IJOCTA025450196
Received: 5 November 2025 | Revised: 1 December 2025 | Accepted: 16 December 2025 | Published online: 9 January 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

Ruled surfaces, defined by the motion of a straight line along a space curve, represent a fundamental class of surfaces in differential geometry with significant applications in engineering design, architectural modeling, and computer graphics. Despite their classical nature, the construction of ruled surfaces from integral curves, solutions to differential systems derived from Frenet frames, remains relatively unexplored in the literature. This paper presents a detailed geometric study of a new class of ruled surfaces constructed from integral curves associated with the Frenet frame of regular space curves with positive curvature. We focus on surfaces whose base curves are given by the integral binormal and integral normal curves of a given spatial curve. Explicit expressions for the fundamental forms, curvature properties, and striction curves are derived for six distinct types of surfaces. Necessary and sufficient conditions under which these surfaces are minimal or developable are established. A numerical example illustrates the theoretical results, highlighting potential applications in geometric modeling. This work extends the theory of ruled surfaces in differential geometry by introducing families based on integral curves and providing a complete geometric characterization via fundamental forms and curvature analysis.

Graphical abstract
Keywords
Ruled surfaces
Integral curves
Frenet frame
Euclidean space
Gaussian curvature
Mean curvature
Funding
None.
Conflict of interest
The authors declare no competing interests.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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