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Keywords = Combescure related curves

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15 pages, 352 KB  
Article
Solutions of Da Rios Vortex Filament Equation of Cartan Null Curves with Combescure Transformation
by Yanlin Li, Osman Keçilioğlu, Kazım İlarslan and Qingyou Sun
Mathematics 2025, 13(21), 3411; https://doi.org/10.3390/math13213411 - 26 Oct 2025
Viewed by 683
Abstract
In this study, Cartan null curves connected via the Combescure transformation are investigated within the framework of Minkowski 3-space, and the necessary conditions for establishing such connections are derived. The relationships between the Frenet vectors and curvatures of these curve pairs are also [...] Read more.
In this study, Cartan null curves connected via the Combescure transformation are investigated within the framework of Minkowski 3-space, and the necessary conditions for establishing such connections are derived. The relationships between the Frenet vectors and curvatures of these curve pairs are also analyzed. Furthermore, when a ruled surface generated by a Cartan null curve provides a solution to the Da Rios equation, the conditions under which the ruled surface generated was by the corresponding Cartan null curve, related through the Combescure transformation, also satisfies the equation. All obtained results are supported with illustrative examples. Full article
(This article belongs to the Special Issue Recent Studies in Differential Geometry and Its Applications)
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16 pages, 497 KB  
Article
Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space
by Yanlin Li, Osman Keçilioğlu and Kazım İlarslan
Axioms 2025, 14(4), 253; https://doi.org/10.3390/axioms14040253 - 27 Mar 2025
Cited by 5 | Viewed by 1258
Abstract
In this study, the existence of Bertrand curves (in the classical sense, i.e., curves with a common principal normal vector field) in four-dimensional Euclidean space is demonstrated using a novel approach. The necessary conditions for a regular curve to be a Bertrand curve [...] Read more.
In this study, the existence of Bertrand curves (in the classical sense, i.e., curves with a common principal normal vector field) in four-dimensional Euclidean space is demonstrated using a novel approach. The necessary conditions for a regular curve to be a Bertrand curve pair are obtained. Furthermore, the relationship between Bertrand curves and Combescure-related curves (pairs of curves with parallel Frenet vectors) is established, and several geometric properties are derived. Additionally, examples are constructed for both Bertrand curve pairs and Combescure-related curve pairs, and their orthogonal projections onto three-dimensional subspaces of four-dimensional space are visualized. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)
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