• Honam Mathematical Journal
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• v.34 no.3
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• pp.381-390
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• 2012

In this paper, we give some characterizations for a quaternionic general helix by means of curvatures of a curve in a 4-dimensional Euclidean space.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.419-431
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• 2022

Mannheim introduced the concept of a pair of curves, called as Mannheim partner curves, in 1878. Until now, Mannheim partner curves have been studied widely in the literature. In this study, we take into account of this concept according to Positional Adapted Frame (PAF) for the particles moving in the 3-dimensional Euclidean space. We introduce a new type special trajectory pairs which are called Mannheim partner P-trajectories in the Euclidean 3-space. The relationships between the PAF elements of this pair are investigated. Also, the relations between the Serret-Frenet basis vectors of Mannheim partner P-trajectories are given. Afterwards, we obtain the necessary conditions for one of these trajectories to be an osculating curve and for other to be a rectifying curve. Moreover, we provide an example including an illustrative figure.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.593-611
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• 2016

This paper is concerned with the classifications of quadric surfaces of first and second kinds in Euclidean 3-space satisfying the Jacobi condition with respect to their curvatures, the Gaussian curvature K, the mean curvature H, second mean curvature $H_{II}$ and second Gaussian curvature $K_{II}$. Also, we study the zero and non-zero constant curvatures of these surfaces. Furthermore, we investigated the (A, B)-Weingarten, (A, B)-linear Weingarten as well as some special ($C^2$, K) and $(C^2,\;K{\sqrt{K}})$-nonlinear Weingarten quadric surfaces in $E^3$, where $A{\neq}B$, A, $B{\in}{K,H,H_{II},K_{II}}$ and $C{\in}{H,H_{II},K_{II}}$. Finally, some important new lemmas are presented.

• Journal of the Korea Society of Computer and Information
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• v.17 no.1
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• pp.161-169
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• 2012

In general, Euclidean minimum spanning tree is a tree connecting input nodes with minimum connecting cost. But the tree may not be optimal when applied to real world problems of three dimension. In this paper, three dimension Euclidean minimum spanning tree is proposed, connecting all input nodes of 3-dimensional space with minimum cost. In experiments for 30,000 input nodes with 100% space ratio, the tree produced by the proposed method can reduce 90.0% connection cost tree, compared with the tree by two dimension Prim's minimum spanning tree. In two dimension plane, the proposed tree increases 251.2% connecting cost, which is pointless in 3-dimensional real world. Therefore, the proposed method can work well for many connecting problems in real world space of three dimensions.

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.523-535
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• 2015

We study surfaces in Euclidean space which are obtained as the sum of two curves or that are graphs of the product of two functions. We consider the problem of finding all these surfaces with constant Gauss curvature. We extend the results to non-degenerate surfaces in Lorentz-Minkowski space.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.581-590
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• 2018

In this paper, we consider surfaces in the 3-dimensional Euclidean space $\mathbb{E}^3$ which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We present an important family of surfaces, namely, tubes in $\mathbb{E}^3$. We show that tubes are of infinite III-type.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.310-324
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• 2022

In this paper, we define timelike curves in R⁴₂ and characterize such curves in terms of Frenet frame. Also, we examine the timelike helices of R⁴₂, taking into account their curvatures. In addition, we study timelike slant helices, timelike B₁-slant helices, timelike B₂-slant helices in four dimensional semi-Euclidean space, R⁴₂. And then we obtain an approximate solution for the timelike B₁ slant helix with Taylor matrix collocation method.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.67-83
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• 2014

The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and $n_2$ of the curve, [1]. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifying curves in $\mathbb{R}^3_1$. Moreover, by the aid of these characterizations, we have investigated semi real quaternionic rectifying curves in semi-quaternionic space $\mathbb{Q}_v$.

• Honam Mathematical Journal
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• v.45 no.4
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• pp.668-677
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• 2023

In this study, we bring forth a new general formula for inextensible flows of Euclidean curves as regards modified orthogonal frame (MOF) in E³. For an inextensible curve flow, we provide the necessary and sufficient conditions, which are denoted by a partial differential equality containing the curvatures and torsion.

• The Pure and Applied Mathematics
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• v.19 no.1
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• pp.1-5
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• 2012

We study surfaces of revolution in the three dimensional Euclidean space $\mathbb{R}^3$ with two distinct axes of revolution. As a result, we prove that if a connected surface in the three dimensional Euclidean space $\mathbb{R}^3$ admits two distinct axes of revolution, then it is either a sphere or a plane.