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Search Results (6)

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Keywords = slant helix

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13 pages, 345 KB  
Article
Slant Helices and Darboux Helices in Myller Configuration
by Yanlin Li, Akın Alkan, Mehmet Önder and Yuquan Xie
Axioms 2025, 14(5), 353; https://doi.org/10.3390/axioms14050353 - 5 May 2025
Cited by 3 | Viewed by 607
Abstract
In this paper, we study slant helices (or ξ_2-helices) and Darboux helices in the Myller configuration M. We demonstrate that a curve in M is a slant helix if and only if it is a Darboux helix. We present [...] Read more.
In this paper, we study slant helices (or ξ_2-helices) and Darboux helices in the Myller configuration M. We demonstrate that a curve in M is a slant helix if and only if it is a Darboux helix. We present the alternative frame for a curve in M. Furthermore, we derive the differential equations that characterize the curves in M using both the Frenet-type frame and the alternative frame. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)
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36 pages, 362 KB  
Article
The Differential Geometry of a Space Curve via a Constant Vector in ℝ3
by Azeb Alghanemi, Ghadah Matar and Amani Saloom
Axioms 2025, 14(3), 190; https://doi.org/10.3390/axioms14030190 - 4 Mar 2025
Viewed by 1441
Abstract
The differential geometry of space curves is a fascinating area of research for mathematicians and physicists, and this refers to its crucial applications in many areas. In this paper, a new method is derived to study the differential geometry of space curves. More [...] Read more.
The differential geometry of space curves is a fascinating area of research for mathematicians and physicists, and this refers to its crucial applications in many areas. In this paper, a new method is derived to study the differential geometry of space curves. More specifically, the position vector of a constant vector in R3 is given in the Frenet apparatus of a space curve, and it is implemented to study the differential geometry of the given space curve. Easy and neat proofs of various well-known results are given using this new method. Also, new results and the properties of space curves are obtained in light of this new method. More specifically, the position vectors of helices are given in simple forms. Moreover, a new frame associated with a smooth curve is obtained, as well as new curvatures associated with the new frame. The new frame and its curvatures are investigated and used to give the position vector of slant helix in a simple and memorable form. Furthermore, some non-trivial examples are given to illustrate some of the results obtained in this article. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)
14 pages, 3569 KB  
Article
Framed Natural Mates of Framed Curves in Euclidean 3-Space
by Yanlin Li and Mahmut Mak
Mathematics 2023, 11(16), 3571; https://doi.org/10.3390/math11163571 - 17 Aug 2023
Cited by 19 | Viewed by 2746
Abstract
In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we [...] Read more.
In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we present the relationships between a framed curve and its framed natural mates. In particular, we establish some necessary and sufficient conditions for the framed natural mates of specific framed curves, such as framed spherical curves, framed helices, framed slant helices, and framed rectifying curves. Finally, we support the concept with some examples. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Submanifolds)
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11 pages, 853 KB  
Article
On the Geometrical Properties of the Lightlike Rectifying Curves and the Centrodes
by Jianguo Sun, Yanping Zhao and Xiaoyan Jiang
Mathematics 2021, 9(23), 3103; https://doi.org/10.3390/math9233103 - 1 Dec 2021
Cited by 2 | Viewed by 1861
Abstract
This paper mainly focuses on some notions of the lightlike rectifying curves and the centrodes in Minkowski 3-space. Some geometrical characteristics of the three types of lightlike curves are obtained. In addition, we obtain the conditions of the centrodes of the lightlike curves [...] Read more.
This paper mainly focuses on some notions of the lightlike rectifying curves and the centrodes in Minkowski 3-space. Some geometrical characteristics of the three types of lightlike curves are obtained. In addition, we obtain the conditions of the centrodes of the lightlike curves are the lightlike rectifying curves. Meanwhile, a detailed analysis between the N-type lightlike slant helices and the centrodes of lightlike curves is provided in this paper. We give the projections of the lightlike rectifying curves to the timelike planes. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications Part II)
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9 pages, 826 KB  
Article
Geometrical Properties of the Pseudonull Hypersurfaces in Semi-Euclidean 4-Space
by Jianguo Sun, Xiaoyan Jiang and Fenghui Ji
Mathematics 2021, 9(11), 1274; https://doi.org/10.3390/math9111274 - 1 Jun 2021
Cited by 5 | Viewed by 2877
Abstract
In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example [...] Read more.
In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
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11 pages, 320 KB  
Article
On Helices and Bertrand Curves in Euclidean 3-Space
by Murat Babaarslan and Yusuf Yayli
Math. Comput. Appl. 2013, 18(1), 1-11; https://doi.org/10.3390/mca18010001 - 1 Apr 2013
Cited by 10 | Viewed by 1929
Abstract
In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show [...] Read more.
In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. Similarly, in case of a space curve is a slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve and a circular helix. Full article
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