ORIGINAL RESEARCH article
Front. Phys.
Sec. Statistical and Computational Physics
Geometry of Generated Quasi-Ruled Surfaces from their Quasi-Parallel Curves according to the q-frame in R3
Provisionally accepted
- 1King Faisal University, Al-Ahsa, Saudi Arabia
- 2King Faisal University, Al Ahsa, Saudi Arabia
Select one of your emails
You have multiple emails registered with Frontiers:
Notify me on publication
Please enter your email address:
If you already have an account, please login
You don't have a Frontiers account ? You can register here
Parallel curves have been used extensively in a variety of fields during the last decade, including architecture, computer graphics, aerospace, and medicine. Despite these applications, the theoretical development of parallel curves in differential geometry has been relatively limited. In this study, we investigate parallel curves using the q-frame in three-dimensional Euclidean space, rather than the traditional Frenet frame. We introduce the new expression "quasi-parallel curves" as a generalization of classical parallel curves. We also examine several geometric properties of these curves and provide rigorous proofs. In differential geometry, ruled surfaces are of great importance. They are defined by the movement of generators, which generate straight lines on the surface. Furthermore, a directrix (base curve) is any curve that intersects all generators (rulings). Recently, we introduced the new expression of ruled surface called "quasi-ruled surface", which describes the ruled surfaces whose base curve is expressed via the q-frame. In [1] we explored the construction and properties of a quasi-ruled surface using quasi-focal curves as directrices. The rulings were expressed in terms of the q-frame associated with the quasi-focal curve. We studied various types of quasi-ruled surfaces, including the generalized osculating, normal, and rectifying variants. We derived conditions for developability and minimality, and some illustrative examples with visualizations are provided. In this paper, we investigate a new class of quasi-ruled surfaces whose base curve is both the original curve and its quasi-parallel curve in R3. We describe the geometric characteristics of these surfaces and derive the first and second fundamental forms, as well as the Gaussian and mean curvatures. Specific types of quasi-ruled surfaces are analyzed, and conditions for their developability and minimality are established. Finally, we provide several quasi-ruled surfaces generated from the helix curve as a base curve and others generated from its quasi-parallel curve as a base curve, and determine the parametrization of the resulting surfaces. All generated surfaces, together with their quasi-parallel and original curves, are visualized using Mathematica 13.
Keywords: Parallel curves, Q-frame, quasi-parallel curves, quasi-ruled surface, Ruled surfaces
Received: 20 Sep 2025; Accepted: 26 Dec 2025.
Copyright: © 2025 Gaber, Eliwa, Almujhim, Alfadhel and Almusayyib. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
* Correspondence: Samah Gaber
Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.