• Communications of the Korean Mathematical Society
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• v.31 no.3
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• pp.637-645
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• 2016

For a polygon P, we consider the centroid $G_0$ of the vertices of P, the centroid $G_1$ of the edges of P and the centroid $G_2$ of the interior of P, respectively. When P is a triangle, the centroid $G_0$ always coincides with the centroid $G_2$. For the centroid $G_1$ of a triangle, it was proved that the centroid $G_1$ of a triangle coincides with the centroid $G_2$ of the triangle if and only if the triangle is equilateral. In this paper, we study the relationships between the centroids $G_0$, $G_1$ and $G_2$ of a quadrangle P. As a result, we show that parallelograms are the only quadrangles which satisfy either $G_0=G_1$ or $G_0=G_2$. Furthermore, we establish a characterization theorem for convex quadrangles satisfying $G_1=G_2$, and give some examples (convex or concave) which are not parallelograms but satisfy $G_1=G_2$.

• School Mathematics
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• v.7 no.4
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• pp.391-402
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• 2005

The purpose of this study is to resolve misunderstanding for centroid of a triangle and to clarify several logical problems in finding the centroid of a Polygon. The conclusions are the followings. For a triangle, the misunderstanding that the centroid of a figure is the intersection of two lines that divide the area of the figure into two equal part is more easily accepted caused by the misinterpretation of a median. Concerning the equilibrium of a triangle, the median of it has the meaning that it makes the torques of both regions it divides to be equal, not the areas. The errors in students' strategies aiming for finding the centroid of a polygon fundamentally lie in the lack of their understanding of the mathematical investigation of physical phenomena. To investigate physical phenomena mathematically, we should abstract some mathematical principals from the phenomena which can provide the appropriate explanations for then. This abstraction is crucial because the development of mathematical theories for physical phenomena begins with those principals. However, the students weren't conscious of this process. Generally, we use the law of lever, the reciprocal proportionality of mass and distance, to explain the equilibrium of an object. But some self-evident principles in symmetry may also be logically sufficient to fix the centroid of a polygon. One of the studies by Archimedes, the famous ancient Greek mathematician, gives a solution to this rather awkward situation. He had developed the general theory of a centroid from a few axioms which concerns symmetry. But it should be noticed that these axioms are achieved from the abstraction of physical phenomena as well.

• East Asian mathematical journal
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• v.29 no.4
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• pp.425-447
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• 2013

Since the center of mass of mathematics curriculum in middle school is dealt with only on triangle and it is defined as just an intersection point of median lines without any physical experiments, students sometimes have misconception of the centroid as well as it is difficult to promote divergent thinking that enables students to think the centroids of various figures. To overcome these problems and to instruct effectively the centroid unit in middle school mathematics classroom, this study suggests a teaching and learning method for the unit which uses physical experiments, drawing, and calculation methods sequentially based on the investigation of students' understanding on the centroid of triangle and the analysis of the mathematics textbooks.

• Communications of Mathematical Education
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• v.25 no.4
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• pp.681-691
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• 2011

Mathematics curriculum according to 2009 Revised National Curriculum suggests that school mathematics must cultivate interest and curiosity about mathematics in addition to creative thinking ability of students, and ability and attitude of observing and analyzing many things happening around. Centroid of a triangle in 2007 Revised National Curriculum is defined as 'an intersection point of three median lines of a triangle' and it has been instructed focusing on proof study that uses characteristic of parallel lines and similarity of a triangle. This could not teach by focusing on the centroid itself and there is a problem of planting a miss concept to students. And therefore this writing suggests centroid must be taught according to its essence that centroid is 'a dot that forms equilibrium', and a justification method about this could be different.

• 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.

• Communications of the Korean Mathematical Society
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• v.32 no.1
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• pp.135-145
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• 2017

For a polygon P, we consider the centroid $G_0$ of the vertices of P, the centroid $G_1$ of the edges of P and the centroid $G_2$ of the interior of P. When P is a triangle, (1) we always have $G_0=G_2$ and (2) P satisfies $G_1=G_2$ if and only if it is equilateral. For a quadrangle P, one of $G_0=G_2$ and $G_0=G_1$ implies that P is a parallelogram. In this paper, we investigate the relationships between centroids of quadrangles. As a result, we establish some characterizations for rhombi and show that among convex quadrangles whose two diagonals are perpendicular to each other, rhombi and kites are the only ones satisfying $G_1=G_2$. Furthermore, we completely classify such quadrangles.

• Imaging Science in Dentistry
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• v.51 no.2
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• pp.167-174
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• 2021

Purpose: Sex determination can be done by morphological analysis of different parts of the body. The mastoid region, with its anatomical location at the skull base, is ideal for sex identification. Statistical shape analysis provides a simultaneous comparison of geometric information on different shapes in terms of size and shape features. This study aimed to investigate the geometric morphometry of the inter-mastoid triangle as a tool for sex determination in the Iranian population. Materials and Methods: The coordinates of 5 landmarks on the mastoid process on the 80 cone-beam computed tomographic images(from individuals aged 17-70 years, 52.5% female) were registered and digitalized. The Cartesian x-y coordinates were acquired for all landmarks, and the shape information was extracted from the principal component scores of generalized Procrustes fit. The t-test was used to compare centroid size. Cross-validated discriminant analysis was used for sex determination. The significance level for all tests was set at 0.05. Results: There was a significant difference in the mastoid size and shape between males and females(P<0.05). The first 2 components of the Procrustes shape coordinates explained 91.3% of the shape variation between the sexes. The accuracy of the discriminant model for sex determination was 88.8%. Conclusion: The application of morphometric geometric techniques will significantly impact forensic studies by providing a comprehensive analysis of differences in biological forms. The results demonstrated that statistical shape analysis can be used as a powerful tool for sex determination based on a morphometric analysis of the inter-mastoid triangle.

• Research in Mathematical Education
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• v.25 no.1
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• pp.1-20
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• 2022

Collaborative learning has been highlighted as an effective method of teachers' professional development in various studies. To disclose teachers' discourse threads in the process of collaborative learning for developing their knowledge, this paper adopted two methods including "content analysis" and "time-sequential analysis" of learning analytics. Such analyses were implemented for mining teachers' updated knowledge and the discourse threads in the discussion during collaborative learning. The materials for analysis involved two aspects: one was from the video-taped lesson observation reports written by teachers before and after discussing, and the other was from their discourses during the discussion process. The results proved that teachers' knowledge for teaching the centroid of a triangle was updated in the collaborative learning period, and also revealed the discourse threads of teachers' collaboration contained "requesting information or opinions", "building on ideas", and "providing evidence or reasoning", with the emphasis on "challenging ideas or re-focusing talk"

• Journal of Korea Game Society
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• v.8 no.3
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• pp.69-76
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• 2008

This paper presents a novel rendering algorithm based on sperical coordinate representation of the object. The vertices of the object are transformed into the sperical coordinate system, and we construct additional maps: the centroid and index of the triangle, the memory access table. While OpenGL rendering pipeline touches all vertices of an object, the proposed method takes account of the only visible vertices by examining the visible triangles of the object. Simulation results demonstrated that the proposed method achieve an efficient rendering performace.

• Journal of KIISE:Computer Systems and Theory
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• v.32 no.10
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• pp.520-529
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• 2005

A feature-based 3D modeling algorithm is presented in this paper. Since conventional methods use depth-based techniques, they need much time for the image matching to extract depth information. Even feature-based methods have less computation load than that of depth-based ones, the calculation of modeling error about whole pixels within a triangle is needed in feature-based algorithms. It also increase the computation time. Therefore, the proposed algorithm consists of three phases, which are an initial 3D model generation, model evaluation, and model refinement phases, in order to acquire an efficient 3D model. Intensity gradients and incremental Delaunay triangulation are used in the Initial model generation. In this phase, a morphological edge operator is adopted for a fast edge filtering, and the incremental Delaunay triangulation is modified to decrease the computation time by avoiding the calculation errors of whole pixels and selecting a vertex at the near of the centroid within the previous triangle. After the model generation, sparse vertices are matched, then the faces are evaluated with the size, approximation error, and disparity fluctuation of the face in evaluation stage. Thereafter, the faces which have a large error are selectively refined into smaller faces. Experimental results showed that the proposed algorithm could acquire an adaptive model with less modeling errors for both smooth and abrupt areas and could remarkably reduce the model acquisition time.