• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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• pp.197-206
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• 2010

We study the restriction estimate of Fourier transform to arbitrary convex curves in $R^2$ with no regularity assumption. Assuming that the convex curve has the lower bound of curvatures, we extend the restriction results from smooth convex curves to arbitrary convex curves. Our work has been motivated by the lecture notes of Terence Tao. The bilinear approach and geometric observations play an important role.

• Proceedings of the KSME Conference
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• 2007.05b
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• pp.2757-2762
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• 2007

A cyclone design concept named Convex cyclone was developed to reduce pressure losses. Contrary to conventional cylinder-on-con type cyclone, inner wall of Convex cyclone are defined with a continuous curve and it has convex shape body. The discontinuity of inner diameter variation rate of cylinder-on-con type cyclone cause additional pressure loss. Continuous wall of Convex cyclone prevent additional pressure loss. In order to verify Convex cyclone design concept, we make a comparative experiments between Stairmand HE and Convex cyclone. Experimental Convex cyclone designed based on Stairmand HE model, and inner wall are defined with circular arch. The experimental result clearly shows that Convex cyclone can achieve maximum 50% pressure loss reduction with a few percent of collection efficiency drop. In addition, the experimental results indicated the existence of optimum convexity, minimum pressure loss, of cyclone wall.

• 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.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.801-815
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• 2021

Ellipses have a lot of interesting geometric properties. It is quite natural to ask whether such properties of ellipses and some related ones characterize ellipses. In this paper, we study some chord properties and area properties of ellipses. As a result, using the curvature and the support function of a strictly convex curve, we establish four characterization theorems of ellipses and hyperbolas centered at the origin.

• 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.

• The Pure and Applied Mathematics
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• v.13 no.2 s.32
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• pp.151-155
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• 2006

Let $\gamma$ be a $C_2$ curve in the open unit disk $\mathbb{D}. Flinn and Osgood proved that $K_{\mathbb{D}}(z,\gamma){\geq}1$ for all $z{\in}{\gamma}$ if and only if the curve ${\Large f}o{\gamma}$ is convex for every convex conformal mapping $\Large f$ of $\mathbb{D}, where $K_{\mathbb{D}}(z,\;\gamma)$ denotes the hyperbolic curvature of $\gamma$ at the point z. In this paper we establish a generalization of the Flinn-Osgood characterization for a curve with the hyperbolic curvature at least 1.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1994.10a
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• pp.866-871
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• 1994

B-spline curves and surfaces are effective solutions for design and modelling of the freeform models. The control methods of these are using with control points, knot vectors and weight of NURBS. Using control point is easy and resonable for representation of general models. But the movement of control points bring out the reformation of original convex hull. The B-splines with nonuniform knot vector provide good result of the shape modification without convex hull reforming. B-splines are constructed with control points and basis functions. Nonuniform knot vectors effect on the basis functions. And the blending curves describe the prorities of knot vectors. Applying of these, users will forecast of the reformed curves after modifying controllabler primitives.

• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.1107-1119
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• 2024

Consider a two dimensional smooth convex body with a marked point on the boundary of it, sitting tangentially at the marked point over a base curve in 𝔼², ℍ² or 𝕊² and the image of this body by the reflection with respect to the tangent line of the base curve at the marked point. When we roll these two bodies simultaneously along the base curve, the trajectories of the marked point bound a closed region. We show that the area of the closed region is independent of the shape of the base curve if the base curve is not highly curved with respect to the boundary curve of the convex body.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.407-417
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• 2022

Archimedes proved 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 of the region bounded by the parabola X and the chord AB is four thirds of the area of the triangle ∆ABP. This property was proved to be a characteristic of parabolas, so called the Archimedean characterization of parabolas. In this article, we study strictly convex curves in the plane ℝ². As a result, first using a functional equation we establish a characterization theorem for quadrics. With the help of this characterization we give another proof of the Archimedean characterization of parabolas. Finally, we present two related conditions which are necessary and sufficient for a strictly convex curve in the plane to be an open arc of a parabola.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.901-909
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• 2014

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. In this article, we study some properties of U and T for strictly convex plane curves. As a result, we establish a characterization for parabolas.