• International Journal of CAD/CAM
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• 제2권1호
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• pp.45-54
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• 2002

Given a set of three generator circles in a plane, we want to find a circumcircle of these generators. This problem is a part of well-known Apollonius' $10^{th}$ Problem and is frequently encountered in various geometric computations such as the Voronoi diagram for circles. It turns out that this seemingly trivial problem is not at all easy to solve in a general setting. In addition, there can be several degenerate configurations of the generators. For example, there may not exist any circumcircle, or there could be one or two circumcircle(s) depending on the generator configuration. Sometimes, a circumcircle itself may degenerate to a line. We show that the problem can be reduced to a point location problem among the regions bounded by two lines and two transformed circles via $M{\ddot{o}}bius$ transformations in a complex space. The presented algorithm is simple and the required computation is negligible. In addition, several degenerate cases are all incorporated into a unified framework.

• 통합자연과학논문집
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• 제5권1호
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• pp.38-41
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• 2012

In this paper we show that for any triangle and any point on the circumcircle the envelope of the Wallace-Simson lines with signed angle ${\alpha}$ is a parabola. The proof is obtained naturally using polar coordinates. We also present the reparametrization of the envelope which is a linear normal curve.

• 한국CDE학회논문집
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• 제6권1호
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• pp.24-30
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• 2001

An efficient and robust algorithm to compute the exact Voronoi diagram of a circle set is presented. The circles are located in a two dimensional Euclidean space, the radii of the circles are non-negative and not necessarily equal, and the circles are allowed to intersect each other. The idea of the algorithm is to use the topology of the point set Voronoi diagram as a seed so that the correct topology of the circle set Voronoi diagram can be obtained through a number of edge flipping operations. Then, the geometries of the Voronoi edges of the circle set Voronoi diagram are computed. In particular, this paper discusses the topological aspect of the algorithm, and the following paper discusses the geometrical aspect. The main advantages of the proposed algorithm are in its robustness, speed, and the simplicity in its concept as well as implementation. Since the algorithm is based on the result of the point set Voronoi diagram and the flipping operation is the only topological operation, the algorithm is always as stable as the Voronoi diagram construction algorithm of a point set.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제25권2호
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• pp.381-402
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• 2011

넓이 개념은 수학의 발생 초기에 형성된 중요한 개념의 하나이며, 역사적으로 넓이를 구하는 문제들이 연구의 중요한 시발점이 된 경우도 많았다. 본 연구에서는 중등학교 수학교과서에서 다루는 삼각형의 넓이 공식을 다양한 방법으로 변형시켜, 삼각형의 몇몇 요소들(변들, 각들, 중선들, 둘레, 외접원의 반지름)로 구성된 새로운 넓이 공식을 유도하여 제시하였다. 본 연구에서 제시된 몇몇 공식들은 과학고등학교 R&E 프로그햄의 진행 과정에서 얻어졌다. 본 연구를 통해 얻어진 결과들은 고등학교 수준의 수학 영재교육에서 수학적 발명을 지향하는 교수-학습 과정에 활용될 수 있을 것으로 기대된다.