Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space

Honda, Shun’ichi; Takahashi, Masatomo; Yu, Haiou

doi:10.1007/s00022-023-00673-7

Cited by 3 publications

(1 citation statement)

References 11 publications

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“…In this regard, recently, the concepts of framed helix [10], framed slant helix [11], framed clad helix [12], framed rectifying curve [3,13], framed normal curve [14], and framed Bertrand and Mannheim curves [15] have been introduced. References [16][17][18] are additional noteworthy studies that contribute to the theory of framed curves. Furthermore, a group of researchers, known as Li et al and referenced in [19][20][21][22][23][24], conducted theoretical research and development on submanifold theory, soliton theory, etc.…”

Section: Introductionmentioning

confidence: 99%

Framed Natural Mates of Framed Curves in Euclidean 3-Space

Li,

Mak

2023

Mathematics

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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.

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Section: Introductionmentioning

confidence: 99%

Framed Natural Mates of Framed Curves in Euclidean 3-Space

Li,

Mak

2023

Mathematics

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Framed clad helices in Euclidean 3-space

Mak

2023

Filomat

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In this study, we introduce framed clad helices, which are a generalization of clad (i.e. C-slant or 2-slant) helices in Euclidean 3-space. They also include framed helices and framed slant helices. After, we give a characterization of the framed clad helices using the alternative adapted frame, which is more useful than the adapted frame of a framed curve. Moreover, we prove the existence of framed spherical images of any framed curve using alternative adapted frames. Additionally, we obtain interesting results regarding the relationship between a framed clad helix and its framed spherical images. Finally, we support the concept with some nice graphics.

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Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space

Wu,

Zhou,

Yao

et al. 2024

Mathematics

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In this paper, we define the generalized Bertrand curves of non-light-like framed curves in Lorentz–Minkowski 3-space; their study is essential for understanding many classical and modern physics problems. Here, we consider two non-light-like framed curves as generalized Bertrand pairs. Our generalized Bertrand pairs can include Bertrand pairs with either singularities or not, and also include Mannheim pairs with singularities. In addition, we discuss their properties and prove the necessary and sufficient conditions for two non-light-like framed curves to be generalized Bertrand pairs.

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Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space

Cited by 3 publications

References 11 publications

Framed Natural Mates of Framed Curves in Euclidean 3-Space

Framed Natural Mates of Framed Curves in Euclidean 3-Space

Framed clad helices in Euclidean 3-space

Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space

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