• 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.

• 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 $G_1=G_2$.

• 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.

• Journal of the Korean Institute of Intelligent Systems
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• v.19 no.5
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• pp.622-628
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• 2009

A proper walking pattern is to be assigned for a walk of multi-legged robots. For the purpose of identifying a good walking pattern for multi-legged robots, this paper consider a simple model of quadruped robotic walking and analyze its walking balance based on the centroid of foot polygons formed in every step. A performance index to estimate the walking balance is also proposed. Simulation studies show that the centroid trajectory of foot polygons and the walking balance in a common quadruped walking are different according to the walking pattern employed. Based on the walking balance index and a bio-mimetic aspect, a useful walking pattern for quadruped robots is finally addressed.

• The Journal of the Korea Contents Association
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• v.19 no.2
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• pp.186-193
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• 2019

This article compares the weighted geometric median with the centroid, from the question why they use the centroid when they would find the single facility location(the weighted geometric median) which minimize the sum of weighted Euclidean distances in some text books and papers. Firstly, we show that the demand point whose volume of demand exceeds the half of total demand is the weighted geometric median differently from the centroid, and we examine the weighed geometric median when every demand point is located on a line. Meanwhile, we could simply see that the geometric median and the centroid are coincident in the special case when every demand point is located at a vertex of a regular polygon, and every volume of demand is equal. Furthermore, the geometric medians of convex tetragons could be simply attained unlike triangles.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.35 no.1B
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• pp.106-116
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• 2010

Sensor deployment makes an effect on not only covering of the interesting area but also reliable data acquisition and efficient resource management of sensor, so that sensors must be deployed at their better place. In traditional static wireless sensor networks, however, it is impossible to deploy the sensors manually when they are distributed in unexploited, hostile, or disaster areas. Therefore, if each sensor has locomotion capability, it can re-deploy itself using the location information of neighbor sensors. In our previous study, we showed that moving sensors to the centroids of their Voronoi polygon is efficient for extending the coverage area. In this paper, we present an improved potential-field-based sensor self-deployment scheme by combining the centroid of Voronoi polygon with the traditional potential-field scheme. Simulation results show that our scheme can achieve higher coverage in shorter time and less movement than the traditional potential-field scheme.

• ETRI Journal
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• v.44 no.2
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• pp.286-299
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• 2022

Human activity recognition in real time is a challenging task. Recently, a plethora of studies has been proposed using deep learning architectures. The implementation of these architectures requires the high computing power of the machine and a massive database. However, handcrafted features-based machine learning models need less computing power and very accurate where features are effectively extracted. In this study, we propose a handcrafted model based on three-dimensional sequential skeleton data. The human body skeleton movement over a frame is computed through joint positions in a frame. The joints of these skeletal frames are projected into two-dimensional space, forming a "movement polygon." These polygons are further transformed into a one-dimensional space by computing amplitudes at different angles from the centroid of polygons. The feature vector is formed by the sampling of these amplitudes at different angles. The performance of the algorithm is evaluated using a support vector machine on four public datasets: MSR Action3D, Berkeley MHAD, TST Fall Detection, and NTU-RGB+D, and the highest accuracies achieved on these datasets are 94.13%, 93.34%, 95.7%, and 86.8%, respectively. These accuracies are compared with similar state-of-the-art and show superior performance.

• The KIPS Transactions:PartC
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• v.17C no.3
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• pp.251-258
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• 2010

In an indoor localization method taking the lateration-based approach, the location of a target is estimated with the location of anchor points (APs) and the approximated distances between the target and APs using received signal strength (RSS) measurements. The accuracy of distance estimation affects the localization accuracy of a lateration-based method. Since a radio propagation environment varies randomly in time and space, the highest RSSs do not necessarily give the best estimation of the distances between a target and APs. Thus, all APs hearing a target have been used for localization. However, the accuracy of a lateration-based method degrades if more APs beyond a certain threshold are used because the area of polygon with the APs increases. In this paper, we focus on reducing the size of the polygon to further increase the localization accuracy. We use the centroid of the polygon as a reference point to estimate the relative location of a target in the polygon. Once the relative location is estimated, only the APs which are closest to the target are used for localization to reduce the area of the polygon with the APs. We validate the proposed method by implementing an indoor localization system and evaluating the accuracy of the proposed method in the various experimental environments.

• Journal of Institute of Control, Robotics and Systems
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• v.15 no.6
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• pp.626-633
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• 2009

A motion planning algorithm is presented in this paper for a commercialized quadruped walking of robot pet. Stable walking is the basic requirement for a commercial-purpose legged robot. In order to secure the walking stability, modified body sway to the centroid of support polygon is addressed. By representation of walking motion with respect to the world coordinate system rather than body coordinate, it is possible to design the several gaits in unified fashion. The initial gait posture is introduced to maximize the stride and to achieve fast walking. The proposed walking motion planning is verified through computer simulation and experiments.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
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• v.31 no.3
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• pp.193-198
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• 2013

This study proposed a map registration method of a building construction plan drawing with shape matching of cadastral parcel polygon. In general, the drawing contains information about a building boundary and a cadastral parcel boundary. The shape of this cadastral parcel boundary should be same as that of the corresponding parcel polygon object in the KLIS continuous cadastral map. Thus, shape matching between two parcel boundary polygons from the drawing and cadastral map could present transformation parameters. Translation and scaling amounts could be obtained by difference of centroid coordinates and area ratio of the polygons, respectively. Rotation amount could be obtained by the rotation that presents the minimum Turning function dissimilarity of the polygons. The proposed method was applied for building construction plan drawings in eAIS for an urban area in Suwon. To assess positional accuracy of map registration, building polygons in registered drawings and aerial photos were compared. According to the accuracy of the cadastral map which is the reference dataset of the proposed method, the RMSE of corresponding buildings' corners was 0.95m and 2.37m in new and old urban areas, respectively.