• Honam Mathematical Journal
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• v.37 no.1
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• pp.41-52
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• 2015

It is well known that the area U of the triangle formed by three tangents to a parabola X is half of the area T of the triangle formed by joining their points of contact. Recently, it was proved that this property is a characteristic one of parabolas. That is, among strictly locally convex $C^{(3)}$ curves in the plane $\mathbb{R}^2$ parabolas are the only ones satisfying the above area property. In this article, we study strictly locally convex curves in the plane $\mathbb{R}^2$. As a result, generalizing the above mentioned characterization theorem for parabolas we present some conditions which are necessary and sufficient for a strictly locally convex $C^{(3)}$ curve in the plane to be an open part of a parabola.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.83-90
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• 2000

We establish a sufficient condition for a simple closed curve in the Euclidean plain $\mathbb{R}^2$, the unit sphere $S^2$ or the hyperbolic plane $H^2$ to be the boundary of a metric ball.

• Honam Mathematical Journal
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• v.36 no.3
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• pp.475-492
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• 2014

In this study, we have investigated the possibility of whether any spacelike Frenet plane of a given space curve in Minkowski 3-space $\mathbb{E}_1^3$ also is any spacelike Frenet plane of another space curve in the same space. We have obtained some characterizations of a given space curve by considering nine possible case.

• Journal of the Chungcheong Mathematical Society
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• v.18 no.2
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• pp.195-203
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• 2005

In this paper, we introduced the tiling, for closed plane curves ${\alpha}(s)$, and we discussed the properties of tiling. Also if ${\alpha}(s)$ was arbitrary plane closed curve equipped by tiling ${\Im}$ then we studied the effect of retraction and tiling retraction on it.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1485-1500
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• 2021

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying curves in an (m + 2)-dimensional space and spherical curves in an (m + 1)-dimensional space.

• Proceedings of the KSME Conference
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• 2007.05a
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• pp.1807-1812
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• 2007

A bulge testing system was developed to measure mechanical properties of thin film materials. A bulge pressure test system for pressurizing the bulge window of the film and a micro out-of-plane ESPI(Electronic Speckle Pattern Interferometric) system for measuring deflection of the film were included in the testing system developed. For the out-of-plane ESPI system, whole field speckle fringe pattern, corresponding to the out-of-plane deflection of the bulged film, was 3-dimensionally visualized using 4-bucket phase shifting algorithm and least square phase unwrapping algorithm. The bulge pressure for loading and unloading was controlled at a constant rate. From the pressure-deflection curve measured by this testing system, ain-plane stress-strain curve could be determined. In this study, elastic modulus of an electrolytic copper film 18 ${\mu}m$ was determined. The modulus was calculated from determining the plain-strain biaxial elastic modulus at the respective unloading slopes of the stress-strain curve and for the Poisson's ratio of 0.34.

• Korean Journal of Computational Design and Engineering
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• v.6 no.1
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• pp.31-39
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• 2001

Presented in this paper are algorithms to compute the positions of vertices and equations of edges of the Voronoi diagram of a circle set. The circles are located in a Euclidean plane, the radii of the circles are not necessarily equal and the circles are not necessarily disjoint. The algorithms correctly and efficiently work when the correct topology of the Voronoi diagram was given. Given three circle generators, the position of the Voronoi vertex is computed by treating the plane as a complex plane, the Z-plane, and transforming it into another complex plane, the W-plane, via the Mobius transformation. Then, the problem is formulated as a simple point location problem in regions defined by two lines and two circles in the W-plane. And the center of the inverse-transformed circle in Z-plane from the line in the W-plane becomes the position of the Voronoi vertex. After the correct topology is constructed with the geometry of the vertices, the equations of edge are computed in a rational quadratic Bezier curve farm.

• Journal of the Korean Mathematical Society
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• v.57 no.1
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• pp.145-169
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• 2020

Let k be a field of characteristic 0. Let ${\mathfrak{C}}=Spec(k[x,y]/{\langle}f{\rangle})$ be a weighted homogeneous plane curve singularity with tangent space ${\pi}_{\mathfrak{C}}:T_{{\mathfrak{C}}/k}{\rightarrow}{\mathfrak{C}$. In this article, we study, from a computational point of view, the Zariski closure ${\mathfrak{G}}({\mathfrak{C}})$ of the set of the 1-jets on ${\mathfrak{C}}$ which define formal solutions (in F[[t]]² for field extensions F of k) of the equation f = 0. We produce Groebner bases of the ideal ${\mathcal{N}}_1({\mathfrak{C}})$ defining ${\mathfrak{G}}({\mathfrak{C}})$ as a reduced closed subscheme of $T_{{\mathfrak{C}}/k}$ and obtain applications in terms of logarithmic differential operators (in the plane) along ${\mathfrak{C}}$.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.583-595
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• 2016

Archimedes showed that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area S of the region bounded by the parabola X and chord AB is four thirds of the area T of triangle ${\Delta}ABP$. It is well known that the area U formed by three tangents to a parabola is half of the area T of the triangle formed by joining their points of contact. Recently, the first and third authors of the present paper and others proved that among strictly locally convex curves in the plane ${\mathbb{R}}^2$, these two properties are characteristic ones of parabolas. In this article, in order to generalize the above mentioned property $S={\frac{4}{3}}T$ for parabolas we study strictly locally convex curves in the plane ${\mathbb{R}}^2$ satisfying $S={\lambda}T+{\nu}U$, where ${\lambda}$ and ${\nu}$ are some functions on the curves. As a result, we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be an open arc of a parabola.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.601-609
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• 2011

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ to have pointwise 1-type Gauss map.