평행사변형의 내접 타원에 대한 연구

박경수(Park, Gyeongsu),박정현(Park, Jeonghyeon),조영민(Cho, Youngmin) 한국과학영재교육학회 2021 과학영재교육 Vol.13 No.1

본 연구는 한국과학창의재단 과학영재 창의연구(R&E)에서 수행한 연구 결과를 바탕으로 이루어졌다. 박정현, 박경수, 조영민(2020)의 연구를 통해 삼각형 내부의 모든 점이 삼각형의 내접 타원의 초점이 될 수 있음을 알게 되었다. 그렇다면 평행사변형의 내접 타원의 초점은 어떤 점이 될 수 있을까?라는 의문점을 갖게 되었다. 본 연구에서는 박정현 외(2020)의 연구 방법을 확장하여 탐구를 진행하였다. 즉, 평행사변형의 각 변이 내접 타원의 접선이라는 아이디어를 적용하여 탐구를 진행하였다. 본 연구를 통해 다음과 같은 연구 결과를 얻을 수 있었다. 첫 번째, 평행사변형의 내접 타원의 초점이 될 수 있는 필요충분조건을 찾을 수 있었다. Geogebra 프로그램에서 타원에 외접하는 다양한 평행사변형을 그리고, 이들의 공통점을 찾음으로써 평행사변형의 내접 타원의 초점이 될 필요충분조건을 찾을 수 있었다. 두 번째, 네 변의 길이가 모두 같은 평행사변형인 마름모의 내접 타원의 초점이 될 수 있는 점은 마름모의 두 대각선을 이룸을 알 수 있었다. 이를 통해 정사각형 또한 내접 타원의 초점이 될 수 있는 점은 두 대각선 위에 있음을 알 수 있으며, 정사각형과 마름모의 내접 타원은 무수히 많이 존재함을 알 수 있었다. 세 번째, 마름모가 아닌 평행사변형의 내접 타원의 초점이 될 수 있는 점은 쌍곡선을 이룸을 알 수 있었다. 평행사변형의 각 중 90도 보다 크지 않은 각을 a라 하고, 평행사변형의 두 대각선의 교점을 원점이라 할 때, 내접 타원의 초점이 될 수 있는 점들은 표준형에서 원점을 중심으로 (45-a/2)도 만큼 회전한 쌍곡선을 이룸을 발견하였다. 이를 통해 마름모가 아닌 직사각형 및 평행사변형의 내접 타원 또한 무수히 많이 존재함을 알 수 있었다. 마지막으로 평행사변형의 내접 타원을 그리는 방법을 찾을 수 있었다. 본 연구 과정에서 발견한 평행사변형의 내접 타원의 초점이 이루는 곡선을 이용하여 평행사변형의 내접 타원을 그리는 방법을 찾을 수 있었다. This study was based on the research results conducted as a R&E project for the gifted students with a financial support from the Korea Foundation for the Advancement of Science and Creativity. Through the research of Park, Park, & Cho (2020), it was found that all points inside the triangle can be the focal point of the inscribed ellipse of the triangle. Then, what could be the focus of the inscribed ellipse of the parallelogram? In this study, the research method of Park, et al (2020) was expanded to investigate. In other words, the research was conducted by applying the idea that each side of a parallelogram is a tangent of an inscribed ellipse. Through this study, the following research results were obtained. First, we found the necessity and sufficiency to become the focal point of the inscribed ellipse of the parallelogram. By drawing various parallelograms circumscribed to an ellipse in the Geogebra program and finding their common points, we found find the necessity and sufficiency to be the focal point of the inscribed ellipse of the parallelogram. Second, we found that the point which can be the focal point of the inscribed ellipse of a rhombus forms two diagonal lines of a rhombus. Third, we found that the point that can be the focal point of the inscribed ellipse of a parallelogram, not a rhombus, forms a hyperbolic curve. When the angle of the parallelogram that is not bigger than 90 degrees is called a, and if we choose the origin as a intersection of the two diagonals of the parallelogram, the points that can be the focal points of the inscribed ellipse form a hyperbolic curve rotated clockwise by (45-a/2) degrees in standard form around the origin. Through this, it can be seen that there are countless inscribed ellipses of rectangular and parallelogram shapes, and rhombus. Finally, I was able to find a way to draw an inscribed ellipse of a parallelogram. The method of drawing the inscribed ellipse of the parallelogram was found by using the curve formed by the focal point of the inscribed ellipse of the parallelogram.

From Gnomon to Parallelogram: A Geometry of Interpretation in Dubliners

( Hee Whan Yun ) 한국제임스조이스학회 2013 제임스조이스저널 Vol.19 No.1

This paper tries to clarify the concept of gnomon and examine its possibility as an interpretative strategy in reading Dubliners. Gnomon is the part of a parallelogram which remains after a similar parallelogram has been taken away from one of its corners. Gnomon, therefore, is an incomplete parallelogram, a figure that would be whole were it not missing one of its corners. The gaps, ellipses, omissions, absences and silences in Joycean text frequently obstruct reader``s interpretation. Such a textual "uncertainty" can make readers feel frustrated in their deciphering process. A gnomonic reading is an effort to fill in those missing, unwritten parts of the text, trying to discover subtly programmed as well as deftly hidden keys and clues for interpretation. Such a reading, geometrically speaking, can be likened to making complete an incomplete parallelogram, that is, a gnomon. Gnomonic approach can be highly creative, bringing about new, radical, alternative interpretations. It requires on the reader``s part, however, to strike a balance between what is said and what is unsaid in the text. Otherwise a gnomonic reading can simply lead to idiosyncratic and irresponsible misreading. Understanding of gnomon as a narrative concept, as well as its applicability to reading, still remain controversial. But its practicality as a reading strategy is very challenging as well as promising because it could open up different textual interpretations hitherto unknown, as my gnomonic reading of Dubliners would hopefully show.

From Gnomon to Parallelogram: A Geometry of Interpretation in Dubliners

윤희환 한국제임스조이스학회 2013 제임스조이스저널 Vol.19 No.1

This paper tries to clarify the concept of gnomon and examine its possibility as an interpretative strategy in reading Dubliners. Gnomon is the part of a parallelogram which remains after a similar parallelogram has been taken away from one of its corners. Gnomon, therefore, is an incomplete parallelogram, a figure that would be whole were it not missing one of its corners. The gaps, ellipses, omissions, absences and silences in Joycean text frequently obstruct reader’s interpretation. Such a textual “uncertainty” can make readers feel frustrated in their deciphering process. A gnomonic reading is an effort to fill in those missing, unwritten parts of the text, trying to discover subtly programmed as well as deftly hidden keys and clues for interpretation. Such a reading, geometrically speaking, can be likened to making complete an incomplete parallelogram, that is, a gnomon. Gnomonic approach can be highly creative, bringing about new, radical, alternative interpretations. It requires on the reader’s part, however, to strike a balance between what is said and what is unsaid in the text. Otherwise a gnomonic reading can simply lead to idiosyncratic and irresponsible misreading. Understanding of gnomon as a narrative concept, as well as its applicability to reading, still remain controversial. But its practicality as a reading strategy is very challenging as well as promising because it could open up different textual interpretations hitherto unknown, as my gnomonic reading of Dubliners would hopefully show.

Perimeter centroids of quadrilaterals

김원용,김동수,김상욱,임수연 호남수학회 2017 호남수학학술지 Vol.39 No.3

For a quadrilateral $P$, we consider the centroid $G_0$ of the vertices of $P$, the perimeter centroid $G_1$ of the edges of $P$ and the centroid $G_2$ of the interior of $P$, respectively. We denote by $M$ the intersection point of two diagonals of $P$. If $P$ is a parallelogram, then we have $G_0=G_1=G_2=M$. Conversely, one of $G_0=M$ and $G_2=M$ implies that $P$ is a parallelogram. In this paper, we show that $G_1=M$ is also a characteristic property of parallelograms.

PERIMETER CENTROIDS OF QUADRILATERALS

( Wonyong Kim ),( Dong-soo Kim ),( Sangwook Kim ),( So Yeon Lim ) 호남수학회 2017 호남수학학술지 Vol.39 No.3

For a quadrilateral P, we consider the centroid G₀ of the vertices of P, the perimeter centroid G₁ of the edges of P and the centroid G₂ of the interior of P, respectively. We denote by M the intersection point of two diagonals of P. If P is a parallelogram, then we have G₀ = G₁ = G₂ = M. Conversely, one of G₀ = M and G₂ = M implies that P is a parallelogram. In this paper, we show that G₁ = M is also a characteristic property of parallelograms.

PERIMETER CENTROIDS OF QUADRILATERALS

Kim, Wonyong,Kim, Dong-Soo,Kim, Sangwook,Lim, So Yeon The Honam Mathematical Society 2017 호남수학학술지 Vol.39 No.3

For a quadrilateral P, we consider the centroid $G_0$ of the vertices of P, the perimeter centroid $G_1$ of the edges of P and the centroid $G_2$ of the interior of P, respectively. We denote by M the intersection point of two diagonals of P. If P is a parallelogram, then we have $G_0=G_1=G_2=M$. Conversely, one of $G_0=M$ and $G_2=M$ implies that P is a parallelogram. In this paper, we show that $G_1=M$ is also a characteristic property of parallelograms.

평면도형 높이에 대한 학생 이해도와 오류 유형

이광호,이현주,이주영,송윤오 한국교원대학교 뇌기반교육연구소 2014 Brain & Learning Vol.4 No.2

The purpose of the research is to firstly, understand 5th graders conceptions of heights in triangles and parallelograms. Secondly, analyze and categorize the frequently shown error types of the concept of heights in triangles and parallelograms. Thirdly, think about the cause and counterplan of the students’ misconceptions of heights from the perspective of concept definition and concept image of Vinner model. 433 (19 classes) 5th graders of 6 elementary schools in Chungcheong area of South Korea were tested to draw the heights on a questionnaire containing 14 triangles and 12 parallelograms with varied positions and shapes. From the study, we could conclude as followings. There is a need to include wordings such as ‘the side opposite the vertex’ and ‘a line containing the base(extended base)’ within the definition of heights. The meaning of the words used in the definition and used in everyday life should be clearly discriminated. When teaching the concept of heights in plane figures, teachers should utilize various example and counter-example images based on the errors students frequently make.

평행사변형 기구를 이용한 평면 병렬형 병진운동 기구 개발

김한성(Han Sung Kim) Korean Society for Precision Engineering 2007 한국정밀공학회지 Vol.24 No.8

In this paper, two types of novel planar Translational Parallel Manipulators (TPMs) by using parallelogram mechanism are conceived. One is made up of two Pa-P (Parallelogram-Prismatic) legs connecting the base to the moving platform. The other consists of two P-Pa legs, which is the kinematic inversion of the former. Since connecting links in a parallelogram mechanism are subject to only tensile/compressive load and all the heavy actuators are mounted at the base, the proposed manipulators can be applied for planar positioning/assembly tasks requiring high stiffness and high speed. The position, velocity, and statics are analyzed, and the design methodology using prescribed workspace and velocity transmission capability is presented. Finally, two types of prototype manipulators have been developed.

PERIMETER CENTROIDS AND CIRCUMSCRIBED QUADRANGLES

Ahn, Seung Ho,Jeong, Jeong Sook,Kim, Dong-Soo The Honam Mathematical Society 2017 호남수학학술지 Vol.39 No.1

For a quadrangle P, we consider the centroid $G_0$ of the vertices of P, the perimeter centroid $G_1$ of the edges of P and the centroid $G_2$ of the interior of P, respectively. If $G_0$ is equal to $G_1$ or $G_2$, then the quadrangle P is a parallelogram. We denote by M the intersection point of two diagonals of P. In this note, first of all, we show that if M is equal to $G_0$ or $G_2$, then the quadrangle P is a parallelogram. Next, we investigate various quadrangles whose perimeter centroid coincides with the intersection point M of diagonals. As a result, for an example, we show that among circumscribed quadrangles rhombi are the only ones whose perimeter centroid coincides with the intersection point M of diagonals.

Perimeter centroids and circumscribed quadrangles

안승호,정정숙,김동수 호남수학회 2017 호남수학학술지 Vol.39 No.1

For a quadrangle $P$, we consider the centroid $G_0$ of the vertices of $P$, the perimeter centroid $G_1$ of the edges of $P$ and the centroid $G_2$ of the interior of $P$, respectively. If $G_0$ is equal to $G_1$ or $G_2$, then the quadrangle $P$ is a parallelogram. We denote by $M$ the intersection point of two diagonals of $P$. In this note, first of all, we show that if $M$ is equal to $G_0$ or $G_2$, then the quadrangle $P$ is a parallelogram. Next, we investigate various quadrangles whose perimeter centroid coincides with the intersection point $M$ of diagonals. As a result, for an example, we show that among circumscribed quadrangles rhombi are the only ones whose perimeter centroid coincides with the intersection point $M$ of diagonals.