Helix surfaces and slant helices in the three-dimensional anti-De Sitter space

Lucas, Pascual; Ortega-Yagües, José Antonio

doi:10.1007/s13398-016-0361-8

Cited by 6 publications

(10 citation statements)

References 23 publications

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“…in [20]. After this study, several kind of slant helices have been defined and studied (see [21]- [24] for details). In this section, we define and consider the slant helical trajectories according to PAF and investigate some special cases of them.…”

Section: Slant Helical Trajectories According To Pafmentioning

confidence: 99%

A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum

Özen

Tosun

2021

Journal of Mathematical Sciences and Modelling

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The theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close relationship between the differential geometry of the trajectory and the kinematics of the particle moving on it. One of the most important elements of the particle kinematics is the jerk vector of the moving particle. Recently, a new resolution of the jerk vector, along the tangential direction and two special radial directions, has been presented byÖzen et al. (JTAM 57(2)( 2019)). By means of these two special radial directions, we introduce a new moving frame for the trajectory of a moving particle with non vanishing angular momentum in this study. Then, according to this frame, some characterizations for the trajectory to be a rectifying curve, an osculating curve, a normal curve, a planar curve and a general helix are given. Also, slant helical trajectories are defined with respect to this frame. Afterwards, the necessary and sufficient conditions for the trajectory to be a slant helical trajectory (according to this frame) are obtained and some special cases of these trajectories are investigated. Moreover, we provide an illustrative numerical example to explain how this frame is constructed. This frame is a new contribution to the field and it may be useful in some specific applications of differential geometry, kinematics and robotics in the future.

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Section: Slant Helical Trajectories According To Pafmentioning

confidence: 99%

A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum

Özen

Tosun

2021

Journal of Mathematical Sciences and Modelling

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“…Until now, several kinds of slant helices have been defined and studied in the literature. There can be found some of them in [11][12][13][14]. In this section, we take into consideration the slant helical trajectories according to PAFORS and discuss some special cases of them.…”

Section: Slant Helical Trajectories According To Paforsmentioning

confidence: 99%

Some Characterizations on Geodesic, Asymptotic and Slant Helical Trajectories According to PAFORS

Özen

Tosun

2021

Maltepe Journal of Mathematics

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In this paper, we study the geodesic, asymptotic, and slant helical trajectories according to PAFORS in three-dimensional Euclidean space and give some characterizations on them. Also, we explain how we determine the helix axis for slant helical trajectories (according to PAFORS). Moreover, we develop a method that enables us to find the slant helical trajectory (if exists) lying on a given implicit surface which accepts a given fixed unit direction as an axis and a given angle as the constant angle. This method also gives information when the desired slant helical trajectory does not exist. The results obtained here involve some differential and partial differential equations or they are based on these equations. The aforementioned results are new contributions to the field and they may be useful in some specific applications of particle kinematics and differential geometry.

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“…Moreover, if P is one of the submanifolds of the above list we show that there exists a Killing vector field X ∈ K(R 3 ν ) such that P ⊆ M (X, ε) for some ε ∈ R . In Section 4 we recover the notion of slant helix introduced in [10,11] for a nonflat space, but without the condition that the extension of the Killing vector field to the ambient space be of constant length, and we apply it to the case of R 3 or R 3 1 . The new curves are called general slant helices because the Killing vector field is of constant length only along the curve.…”

Section: )mentioning

confidence: 99%

“…More recently, and following an idea used in [3] to define general helices in 3D-Lorentzian backgrounds (based on a previous work of Barros [2]), the authors defined in [10,11] slant helices in the three-dimensional sphere S 3 and anti-De Sitter space H 3 1 . In [2,3], the constant vector is replaced by a Killing vector field along the curve with constant length.…”

Section: Introductionmentioning

confidence: 99%