• Bulletin of the Korean Mathematical Society
• /
• v.50 no.3
• /
• pp.885-899
• /
• 2013

In this paper, we study null helices, null slant helices and Cartan slant helices in $\mathb{E}^3_1$. Using some associated curves, we characterize the null helices and the Cartan slant helices and construct them. Also, we study a space-like curve with the principal normal vector field which is a degenerate plane curve.

• Communications of the Korean Mathematical Society
• /
• v.35 no.4
• /
• pp.1269-1281
• /
• 2020

In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric 3-manifold equipped with the canonical affine connection ${\hat{\nabla}}^t$ is a slant helix with pseudo-Hermitian curvature ${\hat{\kappa}}={\mid}q{\mid}\;sin\;{\theta}$ and pseudo-Hermitian torsion ${\hat{\tau}}=q\;cos\;{\theta}$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric 3-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve γ in a quasi-Sasakian 3-manifold M is a slant curve with curvature κ = |(t - α) cos θ + q| sin θ and torsion τ = α + {(t - α) cos θ + q} cos θ. These curves are not helices, in general. Note that if the ambient space M is an α-Sasakian 3-manifold, then γ is a slant helix.

• Communications of the Korean Mathematical Society
• /
• v.33 no.1
• /
• pp.293-303
• /
• 2018

In this paper, we consider biharmonic slant curves in S-space forms. We obtain a main theorem, which gives us four different cases to find curvature conditions for these curves. We also give examples of slant curves in ${\mathbb{R}}^{2n+s}(-3s)$.

• Kyungpook Mathematical Journal
• /
• v.56 no.3
• /
• pp.1003-1016
• /
• 2016

In this paper, we introduce a new kind of slant helix for null curves called null $W_n$-slant helix and we give a definition of new harmonic curvature functions of a null curve in terms of $W_n$ in (n + 2)-dimensional Lorentzian space $M^{n+2}_1$ (for n > 3). Also, we obtain a characterization such as: "The curve ${\alpha}$ s a null $W_n$-slant helix ${\Leftrightarrow}H^{\prime}_n-k_1H_{n-1}-k_2H_{n-3}=0$" where $H_n,H_{n-1}$ and $H_{n-3}$ are harmonic curvature functions and $k_1,k_2$ are the Cartan curvature functions of the null curve ${\alpha}$.

• Honam Mathematical Journal
• /
• v.45 no.1
• /
• pp.130-145
• /
• 2023

In this paper, we define some integral curves connected with a framed curve in Euclidean 3-space. These curves include framed generalized principal-direction curve, framed generalized binormal-direction curve, framed principal-donor curve and framed Darboux-direction curve. We obtain some relations between the framed curvatures of new defined framed curves and framed curvatures of given framed curve. By using the obtained relationships we give some characterizations for such curves. We also give methods for constructing framed helix and framed slant helix from planar curves.

• Honam Mathematical Journal
• /
• v.45 no.4
• /
• pp.610-618
• /
• 2023

In this paper, we obtain (k, m)-type slant helices for a null Cartan curve with the Bishop frame in Minkowski space E⁴₁.

• Journal of the Korean Mathematical Society
• /
• v.54 no.4
• /
• pp.1331-1343
• /
• 2017

A curve ${\gamma}$ immersed in the three-dimensional sphere ${\mathbb{S}}^3$ is said to be a slant helix if there exists a Killing vector field V(s) with constant length along ${\gamma}$ and such that the angle between V and the principal normal is constant along ${\gamma}$. In this paper we characterize slant helices in ${\mathbb{S}}^3$ by means of a differential equation in the curvature ${\kappa}$ and the torsion ${\tau}$ of the curve. We define a helix surface in ${\mathbb{S}}^3$ and give a method to construct any helix surface. This method is based on the Kitagawa representation of flat surfaces in ${\mathbb{S}}^3$. Finally, we obtain a geometric approach to the problem of solving natural equations for slant helices in the three-dimensional sphere. We prove that the slant helices in ${\mathbb{S}}^3$ are exactly the geodesics of helix surfaces.

• Communications of the Korean Mathematical Society
• /
• v.35 no.3
• /
• pp.967-977
• /
• 2020

In this article, we find the necessary and sufficient condition for a proper biharmonic Frenet curve in the Lorentzian Sasakian space forms 𝓜³₁(H) except the case constant curvature -1. Next, we find that for a slant curve in a 3-dimensional Sasakian Lorentzian manifold, its ratio of "geodesic curvature" and "geodesic torsion -1" is a constant. We show that a proper biharmonic Frenet curve is a slant pseudo-helix with 𝜅² - 𝜏² = -1 + 𝜀₁(H + 1)𝜂(B)² in the Lorentzian Sasakian space forms x1D4DC³₁(H) except the case constant curvature -1. As example, we classify proper biharmonic Frenet curves in 3-dimensional Lorentzian Heisenberg space, that is a slant pseudo-helix.

• Korean Journal of Mathematics
• /
• v.26 no.4
• /
• pp.545-559
• /
• 2018

In this paper, we study the position vector of a developable q-slant ruled surface in the Euclidean 3-space $E^3$ in means of the Frenet frame of a q-slant ruled surface. First, we determinate the natural representations for the striction curve and ruling of a q-slant ruled surface. Then we obtain general parameterization of a developable q-slant ruled surface with respect to the conical curvature of the surface. Finally, we introduce some examples for the obtained result.

• Kyungpook Mathematical Journal
• /
• v.56 no.3
• /
• pp.939-949
• /
• 2016

In this paper, we give an algorithm to construct a space curve in Euclidean 3-space ${\mathbb{E}}^3$ from a plane curve which is called PDP-helix of order d. The notion of the PDP-helices is a generalization of a general helix and a slant helix in ${\mathbb{E}}^3$. It is naturally shown that the PDP-helix of order 1 and order 2 are the same as the general helix and the slant helix, respectively. We give a characterization of the PDP-helix of order d. Moreover, we study some geometric properties of that of order 3.