• Title, Summary, Keyword: parabola

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CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE

  • Kim, Dong-Soo;Kim, Dong Seo
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.571-579
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    • 2015
  • Archimedes showed that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle ${\Delta}ABP$, where P is the point on the parabola at which the tangent is parallel to the chord AB. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With the aid of this characterization of parabolas, using centroid of triangles associated with a curve we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be a parabola.

ON THE ARCHIMEDEAN CHARACTERIZATION OF PARABOLAS

  • Kim, Dong-Soo;Kim, Young Ho
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.2103-2114
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    • 2013
  • Archimedes knew that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle ${\Delta}ABP$ where P is the point on the parabola at which the tangent is parallel to AB. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which are necessary and sufficient for a strictly convex curve in the plane to be a parabola.

Dynamic Representations of Parabolas in a Microworld (포물선의 동적 표현과 마이크로월드)

  • Kim, Hwa-Kyung
    • The Mathematical Education
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    • v.47 no.1
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    • pp.49-59
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    • 2008
  • In this paper, we discuss two representations of a curve. One is a static representation as set of points, the other is a dynamic representation using time parameter. And we suggest needs of designing a computer microword where we can represent a curve both statically and dynamically. We also emphasize the importance of translation activity from a static representation to a dynamic representation. For this purpose, we first consider constructionism and 'computers and mathematics education' as a theoretical backgrounds. We focus the curve of a parabola in this paper since this is common in mathematics curriculum and is related to realistic situation such as throwing ball. And we survey the mathematics curriculum about parabola representation. And we introduce JavaMAL microworld that is integrated microworld between LOGO and DGS. In this microworld, we represent a parabola using a dynamic action, and connect this dynamic parabola action to recursive patterns. Finally, we remake a parabola for a realistic situation using this dynamic representation. And we discuss the educational meaning of dynamic representation and its computer microworld.

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Areas associated with a Strictly Locally Convex Curve

  • Kim, Dong-Soo;Kim, Dong Seo;Kim, Young Ho;Bae, Hyun Seon
    • 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.

AREA OF TRIANGLES ASSOCIATED WITH A CURVE II

  • Kim, Dong-Soo;Kim, Wonyong;Kim, Young Ho;Park, Dae Heui
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.1
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    • pp.275-286
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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. In this article, we consider whether this property and similar ones characterizes parabolas. As a result, we present three conditions which are necessary and sufficient for a strictly convex curve in the plane to be an open part of a parabola.

Heart Beat Interval Estimation Algorithm for Low Sampling Frequency Electrocardiogram Signal (낮은 샘플링 주파수를 가지는 심전도 신호를 이용한 심박 간격 추정 알고리즘)

  • Choi, Byunghun
    • The Transactions of The Korean Institute of Electrical Engineers
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    • v.67 no.7
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    • pp.898-902
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    • 2018
  • A novel heart beat interval estimation algorithm is presented based on parabola approximation method. This paper presented a two-step processing scheme; a first stage is finding R-peak in the Electrocardiogram (ECG) by Shannon energy envelope estimator and a secondary stage is computing the interpolated peak location by parabola approximation. Experimental results show that the proposed algorithm performs better than with the previous method using low sampled ECG signals.

ON TRIANGLES ASSOCIATED WITH A CURVE

  • Kim, Dong-Soo;Kim, Dong Seo;Kim, Young Ho
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.925-933
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    • 2015
  • It is well-known that the area of parabolic region between a parabola and any chord $P_1P_2$ on the parabola is four thirds of the area of triangle ${\Delta}P_1P_2P$. Here we denote by P the point on the parabola where the tangent is parallel to the chord $P_1P_2$. In the previous works, the first and third authors of the present paper proved that this property is a characteristic one of parabolas. In this paper, with respect to triangles ${\Delta}P_1P_2PQ$ where Q is the intersection point of two tangents to X at $P_1$ and $P_2$ we establish some characterization theorems for parabolas.

Center of Gravity and a Characterization of Parabolas

  • KIM, DONG-SOO;PARK, SOOKHEE;KIM, YOUNG HO
    • Kyungpook Mathematical Journal
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    • v.55 no.2
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    • pp.473-484
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    • 2015
  • Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord AB on the parabola, let us denote by P the point on the parabola where the tangent is parallel to AB and by V the point where the line through P parallel to the axis of the parabola meets the chord AB. Then the center G of gravity of the section lies on PV called the axis of the parabolic section with $PG=\frac{3}{5}PV$. In this paper, we study strictly locally convex plane curves satisfying the above center of gravity properties. As a result, we prove that among strictly locally convex plane curves, those properties characterize parabolas.

A Simple Simulation of Parabola-Shaped Clouds in the Lee of a Low Bell-Shaped Mountain Using the ARPS

  • Lee, Seung-Jae;Lee, Hwa-Woon;Kang, Sung-Dae
    • Journal of Environmental Science International
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    • v.16 no.5
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    • pp.541-548
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    • 2007
  • A three-dimensional linear model and the Advanced Regional Prediction System (ARPS) were used to simulate parabola-shaped disturbances and clouds in the lee of a bell-shaped mountain. The ARPS model was compared in the x-y plane against the linear model's analytic solution. Under similar conditions with the linear theory, the ARPS produced well-developed parabola-shaped mountain disturbances and confirmed the features are accounted for in the linear regime. A parabola-shaped cloud in the lee of an isolated bell-shaped mountain was successfully simulated in the ARPS after 6 hours of integration time with the prescribed initial and boundary conditions, as well as a microphysical scheme.