• Honam Mathematical Journal
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• v.37 no.4
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• pp.487-504
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• 2015

For a smooth plane curve, the curvature can be characterized by the rate of change of the angle between the tangent vector and a fixed vector. In this article we prove that the curvature of a space curve can also be given by the rate of change of the locally defined angle between the tangent vector at a point and the nearby point. By using height functions, we introduce turning angle of a space curve and characterize the curvature by the rate of change of the turning angle. The main advantage of the turning angle is that it can be used to characterize the curvature of discrete curves. For this purpose, we introduce a discrete turning angle and a discrete curvature called visual curvature for space curves. We can show that the visual curvature is an approximation of curvature for smooth curves.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.391-401
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• 2022

In this study, we investigate the explicit characterization of spherical curves using the Flc (Frenet like curve) frame in Euclidean 3-space. Firstly, the axis of curvature and the osculating sphere of a polynomial space curve are calculated using Flc frame invariants. It is then shown that the axis of curvature is on a straight line. The position vector of a spherical curve is expressed with curvatures connected to the Flc frame. Finally, a differential equation is obtained from the third order, which characterizes a spherical curve.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.1
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• pp.37-53
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• 2021

In this paper we define scaled visual curvature and visual Frenet frame that can be visually accepted for discrete space curves. Scaled visual curvature is relatively simple compared to multi-scale visual curvature and easy to control the influence of noise. We adopt scaled minimizing directions of height functions on each neighborhood. Minimizing direction at a point of a curve is a direction that makes the point a local minimum. Minimizing direction can be given by a small noise around the point. To reduce this kind of influence of noise we exmine the direction whether it makes the point minimum in a neighborhood of some size. If this happens we call the direction scaled minimizing direction of C at p ∈ C in a neighborhood Br(p). Normal vector of a space curve is a second derivative of the curve but we characterize the normal vector of a curve by an integration of minimizing directions. Since integration is more robust to noise, we can find more robust definition of discrete normal vector, visual normal vector. On the other hand, the set of minimizing directions span the normal plane in the case of smooth curve. So we can find the tangent vector from minimizing directions. This lead to the definition of visual tangent vector which is orthogonal to the visual normal vector. By the cross product of visual tangent vector and visual normal vector, we can define visual binormal vector and form a Frenet frame. We examine these concepts to some discrete curve with noise and can see that the scaled visual curvature and visual Frenet frame approximate the original geometric invariants.

• Honam Mathematical Journal
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• v.41 no.2
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• pp.301-319
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• 2019

The aim of the paper is to characterize some curves with the help of their harmonic curvature functions. First of all, we have defined harmonic curvature function of an arbitrary curve and have re-determined the position vectors of helices in terms of their harmonic curvature functions in Galilean 3-space. Then, we have investigated the relation between rectifying curves and Salkowski (or anti-Salkowski) curves in Galilean 3-space. Furthermore, the position vectors of them are obtained via the serial approach of the curves. Finally, we have given some illustrated examples of helices and rectifying curves with some assumptions.

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.609-622
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• 2021

In the Galilean space G₃, we study a special kind of tube surfaces, called tube-like surfaces. They are defined by sweeping a space curve along another central space curve. In this setting, we investigate some equations in terms of Gaussian and mean curvatures, showing some relevant theorems. Our theoretical results are illustrated with some plotted examples.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.211-218
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• 2014

In 3-dimensional Euclidean space, the geometric figures of a regular curve are completely determined by the curvature function and the torsion function of the curve, and surfaces are the fundamental curved spaces for pioneering study in modern geometry as well as in classical differential geometry. In this paper, we define parametrizations for surface by using parametric functions whose images are in the normal plane of each point on a given curve, and then obtain some results relating the Gaussian curvature of the surface with curvature and torsion of the given curve. In particular, we find some conditions for the surface to have either nonpositive Gaussian curvature or nonnegative Gaussian curvature.

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

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.1003-1016
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• 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 ${\alpha}$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.635-645
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• 2012

In this study, we investigate the curvature functions of ruled surface with lightlike ruling for a null curve with Cartan frame in Minkowski 3-space. Also, we give relations between the curvature functions of this ruled surface and curvature functions of central normal surface. Finally, we use the curvature theory of the ruled surface for determine differential properties of a robot end-effector motion.

• Bulletin of the Korean Mathematical Society
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• v.61 no.2
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• pp.401-419
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• 2024

Let K, H, K_II and H_II be the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature of a timelike tubular surface T_γ(α) with the radius γ along a timelike curve α(s) in Minkowski 3-space E³₁. We prove that T_γ(α) must be a (K, H)-Weingarten surface and a (K, H)-linear Weingarten surface. We also show that T_γ(α) is (X, Y)-Weingarten type if and only if its central curve is a circle or a helix, where (X, Y) ∈ {(K, K_II), (K, H_II), (H, K_II), (H, H_II), (K_II , H_II)}. Furthermore, we prove that there exist no timelike tubular surfaces of (X, Y)-linear Weingarten type, (X, Y, Z)-linear Weingarten type and (K, H, K_II, H_II)-linear Weingarten type along a timelike curve in E³₁, where (X, Y, Z) ∈ {(K, H, K_II), (K, H, H_II), (K, K_II, H_II), (H, K_II, H_II)}.